Electric Torques. Damping and Synchronizing Torques. Mechanical Loop. Synchronous Machine Model

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Lynne Wells
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1 Electric Torques Dapig a ychroizig Torques ohae. El-harkawi Departet of Electrical Egieerig Uiversity of Washigto eattle, W Eail: elsharkawi@ee.washigto.eu echaical torque fro turbie is coverte ito electrical torque The electric torque has two copoets: ychroizig torque (i phase with power agle) Dapig torque (i phase with spee) The lack of either torque reer the syste ustable Power syste cotrollers (stabilizers) are use to ehace these torques ohae El-harkawi, Uiversity of Washigto ohae El-harkawi, Uiversity of Washigto ychroous achie oel echaical Loop echaical loop T D 77 T D 77 Electrical loop e f o e f 5 v T T e 6 T s ohae El-harkawi, Uiversity of Washigto ohae El-harkawi, Uiversity of Washigto

2 echaical Loop echaical Loop T T e T e T s T T T D Dapig Torque s D ychroizig Torque 77 T T s T e T T T D D 77 D 77 T haracteristic Equatio ohae El-harkawi, Uiversity of Washigto 5 ohae El-harkawi, Uiversity of Washigto 6 echaical Loop echaical Loop T T T D D where 77 D 77 ohae El-harkawi, Uiversity of Washigto Natural frequcey Dapig oefficiet T T where D j 77 ohae El-harkawi, Uiversity of Washigto 8 are the roots of the characteristic equatio

3 echaical Loop echaical Loop T D 77 T D 77 j Where j is the apig is the ape frequcy j j j ohae El-harkawi, Uiversity of Washigto 9 ohae El-harkawi, Uiversity of Washigto 0 j Roots 0 0 j ohae El-harkawi, Uiversity of Washigto j T echaical Loop T D ssue T 77 ohae El-harkawi, Uiversity of Washigto is a step iput T The tie oai solutio of the equatio is t t si t e ta Tie

4 Tie Respose of echaical Loop Lack of Dapig Torque 0 Tie Tie Lack of Dapig Torque 0 Lack of ychroizig Torque Whe is very sall, or is very sall li li si t t e Tie ohae El-harkawi, Uiversity of Washigto 6

5 Lack of ychroizig Torque ychroous achie oel echaical loop T D 77 Electrical loop e f o e f 5 v 6 Tie ohae El-harkawi, Uiversity of Washigto 7 ohae El-harkawi, Uiversity of Washigto 8 Electrical Loop without VR T e T e Electrical loop e f o 5 v e f Electrical loop e f o 5 v e f 6 6 ohae El-harkawi, Uiversity of Washigto 9 ohae El-harkawi, Uiversity of Washigto 0 5

6 Te Te j j T e o o j G j o G j j o o j 80 ta o G j T e o o j 80 ta o o G 0 80 o orer frequecy 80 o -5 o 0 90 o ohae El-harkawi, Uiversity of Washigto ohae El-harkawi, Uiversity of Washigto T e o G 0 80 o orer frequecy 80 o -5 o 0 90 o 0 Dapig torque is positive ychroizig torque is egative T e T s T 0 T e T s T Total apig torque 0 Total sychroizig torque 0 T apig torque fro echaical loop T e apig torque fro electrical loop T s sychroizig torque fro echaical loop T se sychroizig torque fro electrical loop ohae El-harkawi, Uiversity of Washigto ohae El-harkawi, Uiversity of Washigto 6

7 t corer frequecy 0 T e T s T Total apig torque 0 Total sychroizig torque 0 Exaple T e T s T The syste is stable ohae El-harkawi, Uiversity of Washigto 5 ohae El-harkawi, Uiversity of Washigto 6 T e e f o e f 6 With VR, we ca igore ohae El-harkawi, Uiversity of Washigto 7 5 v T e e f T T e o e f 6 e ohae El-harkawi, Uiversity of Washigto 8 5 v 5 o o o 6 o 7

8 T e 5 T e 5 e f o e f v e f o e f v T T Te e j e j j 6 ; 80 ta j j Replace by j(oscillatio frequecy) Te j j 6 80 ta Whe 5 >0 ychroizig Torque is egative up to = 0 T e T s T ohae El-harkawi, Uiversity of Washigto 9 ohae El-harkawi, Uiversity of Washigto 0 T e e f o 6 e f v 5 Negative Ipact of VR 0 T Te j j ta Whe 5 <0 Dapig Torque is egative up to = T T s T e 0 > VR coul ehace oe of the torques but coul reuce the other ohae El-harkawi, Uiversity of Washigto ohae El-harkawi, Uiversity of Washigto 8

9 j Dapig a ychroizig Torques i ulti-achie yste ij T i i D i ii i 77 i Electric Torques o i th achie T ei ij j ij e fj j e fj ij ii e fi ii ii oi ii e fi i i v i 5ii 5ij 6ij j e fj The agle betwee achie buses eteries the agitue of the sychroizig a/or apig torques / ij ij e fj j 6ii ohae El-harkawi, Uiversity of Washigto ohae El-harkawi, Uiversity of Washigto 9

The state space model needs 5 parameters, so it is not as convenient to use in this control study.

The state space model needs 5 parameters, so it is not as convenient to use in this control study. Trasfer fuctio for of the odel G θ K ω 2 θ / v θ / v ( s) = = 2 2 vi s + 2ζωs + ω The followig slides detail a derivatio of this aalog eter odel both as state space odel ad trasfer fuctio (TF) as show