DynCtrl12 Heffron Phillips 2012 [PDF]

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Small-Signal Stability and Power System Stabilizer Dynamics and Control of Electric Power Systems



Contents    



Review: Closed-Loop Stability Third-Order Model of the Synchronous Machine Heffron-Phillips Model Dynamic Analysis of the Heffron-Phillips Model    



Split between damping and synchronizing torque SMIB with classical generator model SMIB including field circuit dynamics SMIB including excitation system



 Power System Stabilizer  Block diagram  Effect on system dynamics



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Review: Closed-Loop Stability State space formulation of dynamical system  Autonomous dynamical linear system with initial condition:



= x Ax, x(= t 0)= x0  Rate of change of each state is a linear combination of all states:  x1   a11 a12   x1   x  =  a  x  a  2   21 22   2  = x1 a11 x1 + a12 x2 = x2 a21 x1 + a22 x2



 Transformation to diagonal form in order to derive solution easily: z1 = λ1 z1 = z1 z1 (0) ⋅ eλ1t EEH – Power Systems Laboratory



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Review: Closed-Loop Stability State space formulation of dynamical system  Our aim is to transform the equation to the “easy“ form:







 z1  λ1 0   z1   z  =  0 λ  ⋅  z  ⇔ z = Λ ⋅ z  2  2  2 Linear coordinate transformation:



x = Φ⋅z x = Φ ⋅ z  This is equivalent to:



Φ ⋅ z= A ⋅ Φ ⋅ z



z = Λ ⋅ z



−1 ⋅ A ⋅ Φ⋅z z = Φ  Λ



Φ =[φ1 , φ2 .....φn ] Λ =diag (λ1 , λ2 ....., λn )



φi ⋅ λi = A ⋅ φi ⇒ ( A − λi I ) ⋅ φi = 0 det( A − λi I ) = 0



λi ........eigenvalues φi .........right eigenvectors



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Review: Closed-Loop Stability Eigenvalues, stability, oscillation frequency and damping ratio  Let λ1 be a real eigenvalue of matrix A . Then holds: λ1 < 0 : The corresponding mode is stable (decaying exponential). λ1 > 0 : The corresponding mode is unstable (growing exponential). λ1 = 0 : The corresponding mode has integrating characteristics.



 Let λ1,2= σ ± jω be a complex conjugate pair of eigenvalues of A . Then: Re λ1,2 < 0 : The corresponding mode is stable (decaying oscillation). Re λ1,2 > 0 : The corresponding mode is unstable (growing oscillation). Re λ1,2 = 0 : The corresponding mode is critically stable (undamped osc.). The following dynamic properties can be established:  Oscillation frequency: f =  Damping ratio:



ζ =



ω 2π



−σ



σ 2 + ω2



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Third-Order Model of the Synchronous Machine  Voltage deviation in d- and q-axis:



with  Linearized swing equation:



= ∆ω



1 (∆Tm − ∆Te ) 2 Hs + K D



2π f 0 ∆= ∆ω δ s



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Heffron-Phillips Model Purpose:  Simplified representation of synchronous machine, suitable for stability studies: “Small Signal Stability”  linearized model Basis: Electrical torque change  Third-order Model of synchronous machine Starting point for derivation:  Single-Machine Infinite-Bus (SMIB) System  Linearized generator swing equation:



1 (∆Tm − ∆Te ) = ∆ω 2 Hs + K D



2π f 0 ∆= ∆ω δ s EEH – Power Systems Laboratory



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Singel Machine Infinite Bus (SMIB) Generator terminals



Power line Generator



∆eF



AVR



ut



Infinite bus (Voltage magnitude and phase constant)



set t



u



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Heffron-Phillips Model Purpose:  Simplified representation of synchronous machine, suitable for stability studies: “Small Signal Stability”  linearized model Basis: Electrical torque change  Third-order Model of synchronous machine Starting point for derivation:  Single-Machine Infinite-Bus (SMIB) System  Linearized generator swing equation:



1 (∆Tm − ∆Te ) = ∆ω 2 Hs + K D



2π f 0 ∆= ∆ω δ s EEH – Power Systems Laboratory



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Heffron-Phillips Model



Electrical torque change



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Heffron-Phillips Model … including the composition of the electric torque:



Approximation of torque with power:



After linearization and some substitutions:



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Heffron-Phillips Model … including the effect of the field voltage equation:



Influence of torque angle on internal voltage



Field voltage equation:



After linearization and some substitutions:



with: EEH – Power Systems Laboratory



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Heffron-Phillips Model … including the model of the terminal voltage magnitude: ∆eF + K 4 ∆δ



Influence of torque angle on internal voltage



−∆eF



−∆eF



Terminal voltage: Linearization and substitution:



with EEH – Power Systems Laboratory



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Heffron-Phillips Model Full model:



Influence of torque angle on internal voltage



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Heffron-Phillips Model Simulink implementation



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Dynamic Analysis of the Heffron-Phillips Model Splitting between synchronizing and damping torque ∆ω



K Damp



∆Te



K Sync



Exercise 3!



∆δ



∆= Te K Sync ⋅ ∆δ + K Damp ⋅ ∆ω



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Dynamic Analysis of the Heffron-Phillips Model SMIB with classical generator model (mechanical damping torque KD = 0)



Eigenvalues on imaginary axis  system is critically stable



Eigenvalues



λ1,2



Real 0



Imaginary



± 6.385



Damping Ratio



f [Hz]



-



1.016



Synchronizing and damping torque coefficients



s



λ1,2



Ksync



Kdamp



0.757



0 EEH – Power Systems Laboratory



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Dynamic Analysis of the Heffron-Phillips Model SMIB including field circuit dynamics



Eigenvalues moved to the left because field circuit adds damping torque



Eigenvalues



λ1,2 λ3



Real – 0.109



Imaginary



± 6.411



– 0.204



0



Synchronizing and damping torque coefficients due to field circuit



Damping Ratio



f [Hz]



0.0170



1.020



1.0 s



λ1,2 λ3



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Ksync



Kdamp



– 0.0008



1.5333



– 0.7651



0 18



Dynamic Analysis of the Heffron-Phillips Model SMIB including excitation system Eigenvalues



λ1,2 λ3 λ4



Real



Imaginary



Damping Ratio



f [Hz]



– 0.0816



1.7167



± 10.7864



0.8837 – 33.8342



0



1.0



0



–18.4567



0



1.0



0



Synchronizing and damping torque coefficients due to exciter



s



λ1,2 λ3 λ4



Ksync



Kdamp



0.2731



-10.6038



– 19.8103



0



– 7.0126



0



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Dynamic Analysis of the Heffron-Phillips Model SMIB including excitation system



 Generator tripping  might eventually result in Blackout! Eigenvalues moved to the right by the excitation system  System is unstable!



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Power System Stabilizer  Purpose: provide additional damping torque component in order to prevent the system from becoming unstable  Approach: insert feedback between angular frequency and voltage setpoint  Block diagram: Gain:



Washout filter:



Phase compensation:



Tuning parameter



Suppress effect



Provide phase-lead characteristic



for damping torque



of low-frequency



to compensate for lag between



increase



speed changes



exciter input and el. torque



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Power System Stabilizer Block diagram



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Power System Stabilizer Effect on the system dynamics



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Power System Stabilizer Effect on the system dynamics Eigenvalues



λ1,2 λ3,4 λ5 λ6



Real – 1.0052 – 19.7970



Imaginary



Damping Ratio



f [Hz]



0.1504



1.0516



0.8394



2.0406



± 6.6071 ±12.8213



– 39.0969



0



-



-



– 0.7388



0



-



-



Synchronizing and damping torque



Synchronizing and damping torque



coefficients due to exciter



coefficients due to PSS



s



λ1,2 λ3,4 λ5 λ6



Ksync



Kdamp



0.21



– 8.69



– 1.27



– 13.00



1.16



0



0.30



0



s



λ1,2 λ3,4 λ5 λ6



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Ksync



Kdamp



– 0.145



22.761



10.838



290.163



– 30.306



0



–1.072



0 24



Coming up … Exercise 3: Power System Stabilizer  Contents: Stability analysis of Heffron-Phillips Model, PSS design and testing  Date and time: Tuesday, 29 May 2012  Handouts will be sent around one week in advance. Please prepare the exercise at home, timing is tight!  Attendance is compulsory for the “Testat“. Please notify us in case you cannot attend  substitute task.



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