Elec t r ic a l Mac hines 302-2006Chapt er 7 (Lec t ures 8-10)

SYNCHRONOUS MACHINE DYNAMICS

Dr. Moham m ad A.S. MasoumElectrical & Computer Engineering Department

Curtin University of TechnologyPerth, West Australia

Email: m.masoum@curtin.edu.au

SYNCHRONOUS MACHINE DYNAMICS7.1. Classification of Stability Problems7.2. Angle Stability7.3. Synchronous Machine Dynamics

7.3.1. The Swing Equation [3]7.3.2. Normalized Swing Equation [1-2]7.3.3. The Power-Angle Equation [1-2]7.3.4. Synchronous Machine Power Coefficients [1-2]7.3.5. Response to a Step Change in Pm [1-2]

7.4. Linear Analysis of Swing Equation [3]7.5 Equal-Area Criterion of Stability (Nonlinear Analysis of

Swing Equation) [1-3]7.6. Numerical Integration Methods to Solve Power

System Stability problems [1-2][1]. P. Kundur, Power System Stability and Control, McGraw-Hill, 1994.[2]. W.D. Stevenson, Elements of Power System Analysis, 1995.[3]. A.E. Fitzgerald &, Electric Machinery, Fifth Edition, McGraw-Hill, 1990.

Power System Stability- property of the power system that enables it to remain in a state of operating equilibrium under normal operating conditions and to regain an acceptable state of equilibrium after being subjected to a disturbance. Since power systems rely on synchronous machines (SM) for electrical power generation a necessary condition for stability is that: all synchronous machines remain in synchronism PS stability can also be defined as: property enabling synchronous machines to respond to a disturbance and return to their normal operation Instability- loss of synchronism

7.1. Class i f ic at ion of St ab i l i t y Problem s

Hunting- periodic variations in applied torque (e.g., to steam generators) periodic speed variations periodic variations in system voltage (V) & frequency (F) periodic variations in V & F of motors (connected to the system) loss of synchronism Class i f i c at ion of St abi l i t y problem s:PSS problems may be classified as [1]: Angle Stability Voltage Stability Frequency (Mid- and Long-Term) StabilityEach category can be divided to [1-2]: Small-Signal (Dynamic) Stability Transient Stability

Transient St abi l i t y : Determines if system remains in synchronism following a major disturbance (e.g., transmission fault, sudden load change, loss of generation, line switching) Need to model: Generators + their excitation systems + turbine governing control systems Must solve nonlinear differential equations (by direct or iterative procedures) Two types:

-- First-Swing Stability: for 1st second after a system fault (simple generator model & no control model)

-- Multi-Swing Stability: system analysis over longperiod of time (more sophisticated machine model)

Sm al l -Signal (Dynam ic ) St ab i l i t y : Determines if system remains in synchronism following a small disturbance (e.g., small load and/or generation variations) Excitation + turbine governing systems are replaced by synchronous machine model analysis of flux linkage variation Nonlinear differential equations are replaced by a set of linear equationsUsual Assum pt ions (t o fac i l i t a t e c om put at ion): Only synchronous F (no dc-off set, no harmonics) Use of symmetrical components Constant generator voltage Use phasor + load-flow + positive sequence network

Pow er Syst em St abi l i t y

Angle St abi l i t y Vol t age St abi l i t yFrequenc y St abi l i t y

Ability to maintain synchronism Torque balance of Synchronous machines

Transient St ab i l i t y

Sm al l -Signa lSt ab i l i t y

Large-disturbances First-swing aperiodic drift Study period: up to 10 s

Non-osc i l la t ory Osc i l la t ory

Mid-t erm St ab i l i t y Long-t erm St ab i l i t y Severe upsets, large voltage & frequency Fast & slow dynamics Study period: to several minutes

Uniform system frequency Slow dynamics Study period: to tens of minutes

Ability to maintain steady acceptable voltage

Reactive power balance

Large-Dist urbanc e Vol t age St abi l i t y

Sm al l -Dist urbanc e Vol t age St abi l i t y

Large disturbance switching events

Dynamics of ULTC and Loads

Coordination of protections & controls

Steady-state P/Q-V relations Stability margins, Q reserve

Ability to remain in operating equilibrium Equilibrium btw opposing forces

Insufficientsynchronoustorque

Insufficient damping torque Unstable control action

Rotor angle stability is the ability of interconnected synchronous machines of a power system to remain in synchronism & involves the study of the inherent electromechanical oscillation need to know, how the output of synchronous machines vary as their rotors oscillate.Sync hronous Mac hine Charac t er is t ic s : When rotor (containing field windings, excited by dc current) isdriven by a prime mover (turbine), the rotating magnetic field (of rotor) induces ac voltages in the stator armature windings. Frequency (F) of resulting V & I is synchronized with rotor mechanical speed. Resulting stator currents also produce a rotating mmf that, under steady-state operation, rotates at the same speed as rotor with an angular separation G (depending on electrical output torque Te (or power P).

7.2. Angle St abi l i t y

Pow er-Angle Relat ionship: The highly nonlinear (approximately sinusoidal) relation btw

interchange power and rotor angular positions of synchronous machines, has a bearing on PSS:

)XXXX(,sinX

EEP MLGTT

MG G

(Generator) (Motor)

The St ab i l i t y Phenom ena: Stability is a condition of equilibrium btw opposing forces. Interconnected synchronous machines maintain synchronism

with one another through restoring forces, which acts whenever there are forces tending to accelerate or decelerate one or more machine wrt other machines.

Under steady-state, there is equilibrium btw input mechanical torque & output electrical torque (or power) of each machine speed remains constant.

If one generator temporary runs faster than another the resulting angular difference transfers part of the load from theslow machine to the fast machine reduces speed difference & angular position.

However, the power-angle relation is highly nonlinear beyond a certain limit, an increase in angle is accompanied by a decrease in P further increase in angle instability

SM Torque (or Power) , following a perturbation:

-Synchronous Torque ( ): in phase with rotor angle perturbation-Damping Torque ( ): in phase with speed deviation

PSS depends on the existence of both torque components for each machine:

--lack of sufficient Tsyn instability through a periodic driftin rotor angle

--lack of sufficient Tdamp oscillatory instability Angle stability phenomena is characterized in:

(a) small-signal (or small-disturbance) stability(b) transient stability

Z'G' ' Dse TTTG'sT

Z'DT

Sm al l -Signal (Dynam ic ) Angle St abi l i t y : The ability of the power system to maintain synchronism

under small disturbances (e.g., small variations in loads & generation)

Disturbances are considered sufficiently small, for linearization of system equations

Two types of instability- (i) steady increase in angle (lack of Tsyn), -(ii) rotor oscillation of increasing amplitude (lack of Tdamp).

System response to small-signal disturbances depend on a number factors (initial operating state, transmission system strength, type of generator excitation control)

For example, for a generator connected radially to a large power system,instability may be due lack of Ts (with constant field voltage) or Td (with a voltage regulator).

Generator with constant field voltage

Generator with automatic voltage regulator

(excitation control)

Nat ure of sm al l -d is t urbanc e response for a s ing le m ac hine-in f in i t e bus syst em

Z'G' ' Dse TTT

Transient Angle St abi l i t y (THIS CHAPTER): The ability of the power system to maintain

synchronism when subjected to a severe transient disturbance (e.g., different types of transmission line short circuits, bus or transformer faults, loss of generation, loss of a large load) .

The resulting system response involves large excursion rotor angles, power flows, bus voltages and other system variables & is influenced by the nonlinear power-angle relationship

If resulting angular separation btw machines remains within certain bounds, system maintains synchronism

Loss of Synchronism (due transient stability) will usually be evident within 2 to 3 sec of initial disturbance

Transient Angle St abi l i t y (Cont inue): The fault is usually assumed to be cleared by opening

of appropriate breakers. The term dynamic stability has been widely used as a

class of rotor angle stability, denoting different aspects by different authors therefore, we will not use it here

Stability study-period is usually limited to 3-5 seconds

An example of a synchronous machine behavior to transient disturbance in shown below:

Stable: angle increases to a max, then deceases & oscillates with decreasing magnitude until reaches a steady-state

First-swing instability: rotor angle steadily increases until lost of synchronism, caused by insufficient Ts

Stable on first swing, but becomes unstable as a result of growing

oscillation as end state is approached

7.3. Sync hronous Mac hine Dynam ic sOf central importance for PSS analysis are rotational inertia equation (describing effects of unbalance btw Telectromagnetic &Tmechanical of individual SM).

Steady-state condition: Tm=Te Ta=0 no acceleration (or deceleration) constant speed = Zsynchronous = Zo

generator motor

combined (generator + prime mover) moment

of inertia (kg.m2)angular velocity of rotor (mech. rad/s)

Based on elementary principle in dynamics:Taccelerating=Ta =(moment of inertia)(angular acceleration)

7.3.1. The Sw ing Equat ion [3 ]

7.3.2. Norm al ized Sw ing Equat ion [1-2]

can be normalized in terms of

rearranging

since:

per unit inertia constant (H), defined as kinetic energy in watt-seconds at rated speed divided by based VA:

angular velocity of rotor (mech. rad/s)

rated angular velocityof rotor (mech. rad/s)

angular velocity of rotor (ele. rad/s)

rated angular velocityof rotor (ele. rad/s)

angular velocity of rotor (mech. rad/s)rated angular velocityof rotor (mech. rad/s)

number of field poles

If G=rotor angular position (ele. rad) wrt a synchronously rotating reference & Go=its value at t=0:

Substituting in gives:

It is often desirable to include a component for damping torque, not included in Te by adding a term proportional to speed deviation:

Sw ing Equat ion: represents swings of rotor angle Gduring disturbances. Graph of solution: Sw ing Curve

angular velocity of rotor (ele. rad/s)

damping factor (putorque/pu speed)

rated angular velocityof rotor (ele. rad/s)

rotor angular position (ele. rad)

unit inertia constant (watt-sec/VA at rated speed)

Fur t her Considerat ion of The Sw ing Equat ion: The swing equation may be written in terms of pu power:

and s = o (rated angular velocity of rotor)

should have consistent units (mech. or ele. rad/s)

in ele. rad in ele. deg.

For large systems (with many machines), minimize number of swing equations by assuming disturbances affect machines so that their rotors swings together combine machines into one single equivalent machine. Consider a power plant with two generators connected on the same bus:

since rotors swing togetherG1= G2= G

[2]

Machines which swing together are called coherent. When G and Zs =Zo are expressed in electrical degrees or radians, the swing equations of coherent machines can be combined together (even when they have different rated speed) For any pair of non-coherent machines :

An application of these equations is a two-system machine system: a generator (machine 1) and a Synchronous motor (machine 2) connected by a pure reactance line:

where:

&

7.3.3. The Pow er Angle Equat ion [1-2]In for generator, Pm will be consideredconstant (e.g., electrical conditions are expected to change before turbine reacts) Pe determines whether rotor accelerates, decelerates, or remains at syn. speed (Pe=Pm) For transient stability each syn. machine is represented by:

transient internal voltage

transient reactance

each machine must be considered relative to the system of which it is a part phasor angles are

measured wrt the common system reference

Consider a generator (bus#1) supplying power through a transmission line to a receiving-end system (bus#2):

synchronous generator

receiving-endsystem

infinite-bus or synchronous

motor

Transm iss ionnet w ork

includes transient reactance of generators

transient internal generator voltage

Using with :

Similar equations apply to bus#2.

Let and then:

where

For a zero-resistance network ( and are zero):

where

The Power-Angle Equation(its graph is called: the power-angle curve)

The operating point= intersection of P-G curve & Pm curve (e.g., G=28.44 & G=151.6) However, G=151.6 is not valid since: at an acceptable operating point, generator should not lose synchronism due to temporary changes in electrical output power:

Let us examine this:

7.3.4. Sync hronizing Pow er Coef f ic ient s

also

Note that is the slop of the P-G curve at Goand is called:

Synchronizing Power Coefficient =

Therefore:

swing equation for incremental rotor-angle variations

A linear, second-order differential equation (its solution depends on algebraic sign of Sp)

Solution of depends on sign of Sp:--- If >0: = simple harmonic motion (represented by oscillation of undamped swing pendulum)

is solution to which is

STABLE (e.g., at G= 28.44) sinusoidal oscillations, undamped angular frequency:

--- If

7.3.5. Response t o a St ep Change in Pm [1-2]power-angle variations

rotor angle time response

(1) step change in mechanical input power

(2) rotor angle cannot change instantly from Go to G1 (rotor inertia).

Pm>Pe rotor accelerates a o b

(3) Pacceleratingis zero (Pm=Pe) but Zr > ZsynG continues to

increase

(4) for G>G1 we have Pe>Pm

rotor decelerates

(6) rotor angle oscillates indefinitely about the new equilibrium angle G1 with a constant amplitude (R=0)

(5) at some peak value Gmax speed recovers to Zsyn (but Pe>Pm1) rotor continues to decelerate (speed dropping below

Zsyn cob

(7) if A2>A1 (that is G>GL ) lost of stability (since Pm1>Pe and the net torque is accelerating rather than decelerating

7.4. L inear Analys is of Sw ing Equat ion [3 ]

[3]

[3]

7.5. Equal -Area Cr i t er ion of St ab i l i t y Swing eq: nonlinear solutions can not be explicitly found (even for single-machine infinite-bus) require computer techniques. A direct solution approach (without solving eqs) for stability of two-machine system: equal-area criterion

fault

operating point before fault

Consider the system shown:--- before fault: A (closed) & B (open) Example 14.3

(generator operating with Go=Pm=Pe)--- during fault: three-phase fault at P--- after fault: A (open) to clear fault

Fault at to (Go): Pe=0 (a b) & Pm0 due to Pm - Pe>0Ekinetic rotor speed G until fault clears at tc (Gc) (b c) During fault (to< t

At the instant of fault clearing (tc ):--increase in rotor speed:

--separationangle:

At tc (fault is cleared): Pe Ppoint-d >Pm Paccelerating

from a to f: Pm>Pe Zm increases again until reaches synchronism at f. Point f is such that A3=A4 In absence of damping: rotor continues to oscillate in the sequence f-a-e, e-a-f, etc with Zsyn at e and f

7.5.1. Der ivat ion of Equal -Area Cr i t er ion (EAC): Define relative angular velocity of rotor Zr= dG/dt= Z - Zs differentiate it & substitute it into swing equation to get:

multiply by Zr=dG/dt

multiply by dt& integrate

Zr1=0 when G=G1Zr2=0 when G=G2 Zr1=Zr2=0

Apply last equation to points a & e (G1=Go & G2=Gx):

Since rotor speed is synchronous at Gx and Gy A3=A4

Equation applies to any two points G1and G2 at which rotor speed is synchronous

directly proportional to increase in kinetic energy of rotor while it is accelerating

directly proportional to decrease in kinetic energy of rotor while it is decelerating

Equal-Area Criterion states: whatever kinetic energy is added to rotor following a fault must be removed after the fault to restore rotor to synchronous speed

7.5.2. Cr i t ic a l Clear ing T im e: A1 depends on time taken to clear the fault If long delay in clearing fault A1 A2 G x For long delay in clearing fault such that G swing beyond Gmax (e.g., Gx>Gmax) INSTABILITY (due to Paccelerating >0) There is a Critical Clearing Angle (Gcr) for clearing the fault in order to satisfy Equal-Area Criterion for stability The required time is called: Critical Clearing Time (tcr)

critical clearingangle

Com put ing Gc r and t c r :

Set A1 =A2 and transpose terms:

critical clearingangle

(since Pe=0)

Substituting and gives:

Substituting G cr into gives:

critical clearing angle

critical clearing

time

7.5.3.Fur t her Appl ic at ion Equal -Area Cr i t er ion: So far, we have assumed Pe=0 during the fault. When power is transmitted during a fault (e.g., when the fault effects only one line of a parallel transmission system):

By evaluating areas A1 and A2, we would find:

A literal form solution for tcr is not possible in this case. When: r1=0 & r2=1

(before fault)

(after fault)

(during fault)

7.5.4. Response t o a Shor t -Ci rc u i t Faul t : Three-phase fault at F(cleared by opening circuit breakers)

equivalentcircuit

response to a fault cleared at tc1 sec Stable

response to a fault cleared at tc2 sec Unstable

[3]

[3]

7.5.5. Fac t ors In f luenc ing Transient St abi l i t y :Transient stability of generator depends on: How heavily the generator is loaded. Generator output during fault (which depends on fault

location & type). The fault clearing time. The post-fault transmission system reactance. Generator reactance (lower X higher Pmax lower Ginitial ) Generator inertia (higher inertia lower dG/dt reduces

kinetic energy gain during fault (e.g., A1 is reduced) Generator internal voltage magnitude (E) which depends on field excitation.

The infinite bus voltage magnitude.

7.6. Num er ic a l In t egrat ion Met hods t o Solve Pow er Syst em St ab i l i t y problem s

For stability analysis, nonlinear ordinary differential equations (with known initial values) are to be solved:

There are many numerical integration techniques applicable to the solution of this equation including [1]:-- Euler Method-- Modified Euler Method-- Runge-Kutta (R-K) Methods

For further information see reference [1]

e.g., is of form:

where X = state vectort = independent variable

7.7. Mul t i -Mac hine Class ic a l St abi l i t y St udies When a multi-machine system operates under electro-mechanical transient conditions, inter-machine oscillations occur btw machines through transmission system. Each machine acts as a single oscillating source (foscillation=1-2Hz) superimposed upon nominal frequency swing equation will reflect the combined presence of many such oscillations

System frequency: not unduly perturbed from its nominal value The 60-Hz network parameters are still applicable. Additional assumptions for Classical stability Models):--Pm= constant (for each machine) --negligible damping power --each machine presented by constant transient reactancein series with a constant transient internal voltage

--mechanical rotor angle coincides with G (ele. rotor angle)--loads are presented as shunt impedances; determined byconditions immediately prior to transient conditions

For transient stability study; system conditions before fault& network configuration during and after its occurrence must be known Steps for multi-machine stability studies:--St ep1: steady-state pre-fault conditions (load-flow)

bus-admittance matrix includes transient reactance of generators&shunt load admittances

--St ep2: pre-fault network representation is determinedand modified to account for fault & post-faultconditions. Only generator internal buses haveinjections other buses can be eliminated

voltagebusVloadthe)jQP(

whereV

QjPY

L

LL

2L

LLL

IXjVEknownareV&Q,P

dt

t

c c

During&after the fault, power flow into network from each generator is calculated by the corresponding P-G equations For example, for the network shown:

Similar equations for Pe2 & Pe3 with Yij values from the 3u3 bus-admittance matrices (fault or post-fault conditions) The P-G equations form part of the swing equations:

Solution depends on location& duration of fault, & YBUS which results when the faulted line is removed The basic procedures used in digital computers are revealed by following examples

[2]

[2]

Each of the P-G equations obtained in the previous two examples are of the form:

where the bracketed right-hand side term represents the accelerating power on the rotor. Accordingly, we may write them in the form:

where