ECE 4 Power System Operations & Planning 7 Transient Stability Spring 5 Instructor: Kai Sun
References Saaat s Chapter.5 ~. EPRI Tutorial s Chapter 7 Kunur s Chapter 3
Transient Stability The ability of the power system to maintain synchronism when subjecte to a severe transient isturbance such as a fault on transmission facilities, loss of generation, or loss of a large loa. The system response to such isturbances involves large excursions of generator rotor angles, power flows, bus voltages, an other system variables. Stability is influence by the nonlinear characteristics of the system If the resulting angular separation between the machines in the system remains within certain bouns, the system maintains synchronism. If loss of synchronism ue to transient instability occurs, it will usually be evient within 3 secons of the initial isturbances 3
Single Machine Infinite Bus (SMIB) System P e = =P max sin P max = E E B /X T Swing Equation: H w t = P - P = P -P m e m max sin P a is calle Accelerating Power P m = P e The rotor will accelerate if P m increases, or P e ecreases 4
Power Angle Relationship P e = =P max sin Question: What if both circuits are out of service? 5
Response to a step change in P m H = P a = Pm- Pe = Pm-P w t max sin Consier a suen increase in P m : P m P m. New equilibrium point b satisfying P e ( )=P m a b: Due to the rotor s inertia, cannot jump from to, so P a =P m -P e ( )> an r increases from. When b is reache, P a = but r >, so continues to increase. b c: > an P a <, so r ecreases until c. At c, r = an reaches the peak value max. P a r a > =, b = r, max, c < =, = max, c: At c, the rotor starts to ecelerate (since P a <) with r < an ecreases. With all resistances (amping) neglecte, an r oscillate aroun the new equilibrium point b with a constant amplitue. 6
Equal Area Criterion (EAC) If max exists where /t=: w D wr = ( Pm - Pe) H ò H w wr ( Pm Pe) w D = ò - æ w ö D = ( m- e) ç w ò çè ø max r J P P = max At max, r = an the integral= Moment of inertia in p.u. = ò +ò max ( - ) ( P - P ) P P m e m e (Note: all losses are neglecte) = area A - area A = 7
Equal Area Criterion (EAC): The stability is maintaine only if a ecelerating area A the accelerating area A can be locate above P m (from b to, i.e. the Unstable Equilibrium Point or UEP). SEP UEP If A < A, will continue increasing at UEP (since r > ), so it will lose stability. For the case with a step change in P m, the new P m oes matter for transient stability. 8
Transient stability limit for a step change of P m Following a step change P m P m, solve the transient stability limit of P m : Assume A = A in orer to solve the limit of P m P P P P m ò ò max ( - )- sin = sin - ( - ) max max m max ( - ) P = P (cos -cos ) max m max max - P + P (cos - cos ) =-P (cos -cos )-P m max max max m max At the UEP, Pm = Pmax sinmax ( - )sin + cos = cos max max max SEP UEP Solve max to calculate the transient stability limit for a step change of P m : P = P sin m max max = p-max Questions: Can we increase P m beyon that transient limit? If yes, how much further can we increase P m? 9
Solve max by the Newton Raphson metho ( - )sin + cos = cos max max max The nonlinear function form: Select an initial estimate: f ( ) = cos = c max p/< < p ( k ) max Calculate iterative solutions by the N R algorithm: + = +D ( k ) ( k) ( k) max max max where c- f( ) c- f( ) D = = ( k) ( k) ( k ) max max max ( k) ( k) f ( max - )cosmax max ( k ) max Give a solution when a specific accuracy is reache, i.e. + - e ( k ) ( k) max max
Response to a three phase fault P e = =P max sin P e,uring fault << P e, post-fault P e,post-fault <P e,pre-fault for a permanent fault (cleare by tripping the fault circuit) or P e, post-fault =P e,pre-fault for a temporary fault
Stable Unstable
Critical Clearing Angle (CCA) Saaat s Sec..6 (Example.5) Consier a simple case A three phase fault at the sening en P e, uring fault = if all resistances are neglecte Critical Clearing Angle c ò ò c max P = ( P sin -P) m max m A A Integrating both sies: ( ) c P - = P (cos -cos )-P ( - ) m c max c max m max c Pm cos c = ( max - c) + cos P max max Pm = ( p- - )-cos P max c = - 3
Critical Clearing Time (CCT) Solve the CCT from the CCA: Since P e, uring fault = for this case, uring the fault: H w t = P m t = w Pm t Pmt t H ò = w H w 4H = Pt m + t c = 4 H ( c -) w P m 4
For a more general case: P e (uring fault)> P 3max (pre-fault) A 3 A (post-fault) P 3max P max A (fault-on) P max s ( u ) ( ) c max P - P sn i = P sin - P ( - ) m c -ò ò max 3max m max c c cos c = P ( - ) + P cos -P cos P - P m max c 3max max max 3max max A = V ke ( c ), the kinetic energy at c A + A 3 = V( c )=V ke ( c )+V pe ( c ), total energy at c A + A 3 = V pe ( u )=V cr, i.e. the largest potential energy If an only if V( c ) V cr (i.e. A A ), the generator is stable 5
Factors influencing transient stability How heavily the generator is loae. The generator output uring the fault. This epens on the fault location an type The fault clearing time The post fault transmission system reactance The generator reactance. A lower reactance increases peak power an reuces initial rotor angle. The generator inertia. The higher the inertia, the slower the rate of change in angle. This reuces the kinetic energy gaine uring fault; i.e. the accelerating area A is reuce. The generator internal voltage magnitue (E ). This epens on the fiel excitation The infinite bus voltage magnitue E B 6
EAC for a Two Machine System Two interconnecte machines respectively with H an H The system can be reuce to an equivalent SMIB system w w t H H = ( P m- Pe ) = Pa w w t H H = ( P m- Pe) = Pa w P P ( ) t t t H H a a = - = - w HH H P - HP H P - HP a a m m e e = = - H + H t H+ H H+ H H + H H P - HP H w t = P -P m, e, P e ò,max, w H ( P - P ) = m, e, P m,,max 7
Methos for Transient Stability Analysis Analyzing a system s transient stability following a given contingency Time omain simulation: At present, the most practical available metho of transient stability analysis is time omain simulation in which the nonlinear ifferential equations are solve by using step by step numerical integration techniques. Direct methos: Those methos etermine stability without explicitly solving the system ifferential equations. The methos are base on Lyapunov s secon metho, efine a Transient Energy Function (TEF) as a possible Lyapunov function, an compare the TEF to a critical energy, enote by V cr, to juge stability EAC is a irect metho for a SMIB or two machine system 8
Numerical Integration Methos The ifferential equations to be solve in power system stability analysis are nonlinear ODEs (orinary ifferential equations) with known initial values x=x an t=t x t f ( x, t) where x is the state vector of n epenent variables an t is the inepenent variable (time). Our objective is to solve x as a function of t Explicit Methos In these methos, the value of x at any value of t is compute from the knowlege of the values of x from only the previous time steps, e.g. Euler metho an R-K methos Implicit Methos These methos use interpolation functions (involving future time steps) for the expression uner the integral, e.g. the Trapezoial Rule 9
Euler Metho The Euler metho is equivalent to using the first two terms of the Taylor series expansion for x aroun the point (x, t ), referre to as a first-orer metho (error is on the orer of t ) Approximate the curve at x=x an t=t by its tangent x(t ) x x t f ( xt, ) x t x f ( x, t ) x t x x t t x x x x x x t t x t t At step i+ x x x t i i t x The stanar Euler metho results in inaccuracies because it uses the erivative only at the beginning of the interval as though it applie throughout the interval i
Moifie Euler (ME) Metho Moifie Euler metho consists of two steps: (a) Preictor step: p x x t x t x Slope at the beginning of t x (t ) x c x p x x t f ( xt, ) t The erivative at the en of the t is estimate using x p x t x p p f( x, t ) Estimate slope at the en of t t t (b) Corrector step: x x x x t p x t t p x x t c c i x i x x t x x i i t It is a secon-orer metho (error is on the orer of t 3 ) Step size t must be small enough to obtain a reasonably accurate solution, but at the same time, large enough to avoi the numerical instability with the computer, e.g. increasing roun-off errors.
Runge Kutta (R K) Methos General formula of the n orer R-K metho: (error is on the orer of t 3 ) k f( x, t ) t k f( x k, t t) t x x ak ak At Step i+: x(t ) x + k x x f ( xt, ) t t t t k f ( x, t ) t i i x i x ak a k i k f( x k, t t) t i i The ME metho is a special case with a =a =/, = = General formula of the 4 th orer R-K metho: (error is on the orer of t 5 ) x i x i ( k k k3 k4 ) 6 k f( xi, ti) t k t k f( xi, ti ) t k t k3 f( xi, ti ) t k f( x k, t t) t 4 i 3 i
Numerical Stability of Explicit Integration Methos Numerical stability is relate to the stiffness of the set of ifferential equations representing the system The stiffness is measure by the ratio of the largest to smallest time constant, or more precisely by max / min of the linearize system. Stiffness in a transient stability simulation increases with moeling more etails (more small time constants are concerne). Explicit integration methos have weak stability numerically; with stiff systems, the solution blows up unless a small step size is use. Even after the fast moes ie out, small time steps continue to be require to maintain numerical stability 3
Implicit Methos Implicit methos use interpolation functions for the expression uner the integral. Interpolation implies the function must pass through the yet unknown points at t The simplest implicit integration metho is the Trapezoial Rule metho. It uses linear interpolation. The stiffness of the system being analyze affects accuracy but not numerical stability. With larger time steps, high frequency moes an fast transients are filtere out, an the solutions for the slower moes is still accurate. For example, for the Trapezoial rule, only ynamic moes faster than f(x n,t n ) an f(x n+,t n+ ) are neglecte. f x(t )=x(t )+ A + B B f(x,t ) f(x,t ) Δt x=x+ f x,t Δt x = x + f x,t + +f x,t f x, t n+ n n n n+ n+ A t Compare to ME metho: t x x f x,t + f x,t n p n n n n n+ t t 4
Overall System Equations The overall system equations are expresse in the general form comprising a set of first-orer ifferential equations (ynamic evices) an a set of algebraic equations (evices an network) where x V I Y N state vector of the system bus voltage vector x=f(x,v) I(x,V) = YN V DE AE current injection vector noe amittance matrix. It is constant except for changes introuce by network-switching operations; symmetrical except for issymmetry introuce by phase-shifting transformers 5
Solution of the Equations Schemes for the solution of equations DE an AE are characterize by the following factors The manner of interface between the DE an AE. Either a partitione approach or a simultaneous approach may be use The integration metho, i.e. an implicit metho or explicit metho, use to solve the DE. The metho use to solve the AE (power flow analysis), e.g. the Newton-Raphson metho. Most commercialize power system simulation programs provie the Moifie Euler, n orer R-K, 4 th orer R-K an Trapezoial Rule methos 6
A Simplifie Moel for Multi Machine Systems Consier these classic simplifying assumptions: Each synchronous machine is represente by a voltage source E with constant magnitue E behin X (neglecting armature resistances, the effect of saliency an the changes in flux linkages) The mechanical rotor angle of each machine coincies with the angle of E The governor s actions are neglecte an the input powers P mi are assume to remain constant uring the entire perio of simulation Using the pre-fault bus voltages, all loas are converte to equivalent amittances to groun. Those amittances are assume to remain constant (constant impeance loa moels) Damping or asynchronous powers are ignore. Machines belonging to the same station swing together an are sai to be coherent. A group of coherent machines is represente by one equivalent machine 7
Solve the initial power flow an etermine the initial bus voltage phasors V i. Terminal currents I i of m generators prior to isturbance are calculate by their terminal voltages V i an power outputs S i, an then use to calculate E i All loas are converte to equivalent amittances: To inclue voltages behin X i, a m internal generator buses to the n-bus power system network to form a n+m bus network (groun as the reference for voltages): E X E X X m E m Y bus nxn Y reuce bus m m 8
Noe voltage equation with groun as reference I bus is the vector of the injecte bus currents V bus is the vector of bus voltages measure from the reference noe Y bus is the bus amittance matrix : Y ii (iagonal element) is the sum of amittances connecte to bus i Y ij (off-iagonal element) equals the negative of the amittance between buses i an j Compare to the Y bus for power flow analysis, aitional m internal generator noes are ae an Y ii (i n) is moifie to inclue the loa amittance at noe i 9
To simplify the analysis, all noes other than the generator internal noes are eliminate as follows I where I I m where ij is the angle of Y ij nees to be upate whenever the network is change. 3