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 .