DynCtrl12 Heffron Phillips 2012



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Small-Signal Stability and Power System StabilizerDynamics 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 EEH – Power Systems Laboratory 2 x(= 0) x0 x t =  Rate of change of each state is a linear combination of all states:   x1   a11 a12   x1   x  = a 2   21 a22   x2      = a11 x1 + a12 x2 x1  = a21 x1 + a22 x2 x2  Transformation to diagonal form in order to derive solution easily:  z1 = λ1 z1 = z1 (0) ⋅ eλ1t z1 EEH – Power Systems Laboratory 3 .Review: Closed-Loop Stability State space formulation of dynamical system  Autonomous dynamical linear system with initial condition:  = Ax. ......... λn ) φi ⋅ λi = A ⋅ φi ⇒ ( A − λi I ) ⋅ φi = 0 det( A − λi I ) = 0 λi .eigenvalues φi .Review: Closed-Loop Stability State space formulation of dynamical system  Our aim is to transform the equation to the “easy“ form:   z1  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 −   1 Φ z = Φ⋅ A ⋅ ⋅ z Λ  z = Λ⋅z Φ =[φ1 ..φn ] Λ =diag (λ1 ........ φ2 ... λ2 ......right eigenvectors EEH – Power Systems Laboratory 4 . ).2 > 0 : The corresponding mode is unstable (growing oscillation).Review: Closed-Loop Stability Eigenvalues. Then holds: λ1 < 0 : The corresponding mode is stable (decaying exponential).2 < 0 : The corresponding mode is stable (decaying oscillation). Then: Re λ1. Re λ1. λ1 = 0 : The corresponding mode has integrating characteristics. The following dynamic properties can be established:  Oscillation frequency: f =  Damping ratio: ω 2π ζ = −σ 5 σ 2 + ω2 EEH – Power Systems Laboratory . oscillation frequency and damping ratio  Let λ1 be a real eigenvalue of matrix A . stability.2 σ ± jω be a complex conjugate pair of eigenvalues of A .2 = 0 : The corresponding mode is critically stable (undamped osc. =  Let λ1. Re λ1. λ1 > 0 : The corresponding mode is unstable (growing exponential). 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 ∆δ = ∆ω s EEH – Power Systems Laboratory 6 . 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 ∆δ = ∆ω s EEH – Power Systems Laboratory 7 .Heffron-Phillips Model Purpose:  Simplified representation of synchronous machine. Singel Machine Infinite Bus (SMIB) Generator terminals Power line Generator ∆eF AVR set t ut Infinite bus (Voltage magnitude and phase constant) u EEH – Power Systems Laboratory 8 . 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 ∆δ = ∆ω s EEH – Power Systems Laboratory 9 . Heffron-Phillips Model Electrical torque change EEH – Power Systems Laboratory 10 . Heffron-Phillips Model … including the composition of the electric torque: Approximation of torque with power: After linearization and some substitutions: EEH – Power Systems Laboratory 11 . 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 12 . 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 13 . Heffron-Phillips Model Full model: Influence of torque angle on internal voltage EEH – Power Systems Laboratory 14 . Heffron-Phillips Model Simulink implementation EEH – Power Systems Laboratory 15 . Dynamic Analysis of the Heffron-Phillips Model Splitting between synchronizing and damping torque ∆ω K Damp ∆Te Exercise 3! K Sync ∆δ ∆Te K Sync ⋅ ∆δ + K Damp ⋅ ∆ω = EEH – Power Systems Laboratory 16 . 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 λ1.385 Kdamp 0 EEH – Power Systems Laboratory Synchronizing and damping torque coefficients λ1.016 ± 6.2 Ksync 0.2 s Real 0 Imaginary Damping Ratio - f [Hz] 1.757 17 . 5333 0 18 ± 6.2 λ3 Real – 0.411 0 Synchronizing and damping torque coefficients due to field circuit λ1.204 Imaginary Damping Ratio 0.020 Kdamp 1.7651 f [Hz] 1.109 – 0.0170 1.0008 – 0.0 s Ksync – 0.2 λ3 EEH – Power Systems Laboratory .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. 0 f [Hz] 1.Dynamic Analysis of the Heffron-Phillips Model SMIB including excitation system Eigenvalues Real 0.8837 – 33.7864 0 0 Synchronizing and damping torque coefficients due to exciter s λ1.0816 1.6038 0 0 EEH – Power Systems Laboratory 19 .0 1.8103 – 7.8342 –18.2 λ3 λ4 ± 10.2731 – 19.4567 Imaginary Damping Ratio – 0.7167 0 0 λ1.2 λ3 λ4 Ksync 0.0126 Kdamp -10. 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! EEH – Power Systems Laboratory 20 . torque EEH – Power Systems Laboratory 21 .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: Tuning parameter for damping torque increase Washout filter: Suppress effect of low-frequency speed changes Phase compensation: Provide phase-lead characteristic to compensate for lag between exciter input and el. Power System Stabilizer Block diagram EEH – Power Systems Laboratory 22 . Power System Stabilizer Effect on the system dynamics EEH – Power Systems Laboratory 23 . 2 λ3.6071 ±12.Power System Stabilizer Effect on the system dynamics Eigenvalues Real – 1.072 Kdamp 22.0052 – 19.2 λ3.761 290.7970 – 39.8394 - f [Hz] 1.16 0.69 – 13.4 λ5 λ6 ± 6.306 –1.0406 - λ1.21 – 1.30 Kdamp – 8.163 0 0 24 EEH – Power Systems Laboratory .838 – 30.1504 0.7388 Imaginary Damping Ratio 0.8213 0 0 Synchronizing and damping torque coefficients due to exciter Synchronizing and damping torque coefficients due to PSS s λ1.0516 2.4 λ5 λ6 Ksync – 0.00 0 0 s λ1.145 10.2 λ3.27 1.4 λ5 λ6 Ksync 0.0969 – 0. PSS design and testing  Date and time: Tuesday. timing is tight!  Attendance is compulsory for the “Testat“. Please notify us in case you cannot attend  substitute task.Coming up … Exercise 3: Power System Stabilizer  Contents: Stability analysis of Heffron-Phillips Model. 29 May 2012  Handouts will be sent around one week in advance. Please prepare the exercise at home. EEH – Power Systems Laboratory 25 .
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