Linearized Analysis of the Synchronous Machine for PSS Chapter 6 ...

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Excitation system torque: This is the electrical torque that results from the K 5 and K 6 loops, as shown in Fig. 7 below. Fig. 7 This torque may be expressed based on the block diagram as: K 2 K 3 Δ Texc = Ge ( s) ( ΔVref − ΔVt ) 1 + sτ ' d 0 K 3 Ignoring ΔV ref (it represents manual changes in the voltage setting), and using (from eq. (**)): we obtain K ΔVt = K5Δδ + K6ΔE' q K G ( s ) ( − K Δδ − K ΔE' ) 2 3 ΔTexc = e 5 6 1+ sτ' d 0 K3 We want to express each torque as a function of ∆δ or ∆ω. But the last expression has a ∆E’ q . We can address this by noticing the q 12
elation (from the block diagram) that ΔT exc =K 2 ΔE’ q ΔE’ q =ΔT exc /K 2 , and so we can write that K 2K = 1 + sτ ' 3 ΔTexc d 0 Solving for ΔT exc yields K 3 G e ⎛ ( s) ⎜− ⎝ K 5 Δδ − K K 6 2 ΔT exc ⎞ ⎟ ⎠ − K 2 K 3K 5 Δ Texc = Ge ( s) + sτ ' K + K K G ( s) 1 d 0 3 3 6 e ( Δδ ) On substituting equation (****) for the static excitation transfer function G e (s), we obtain, after some algebra, the following: ΔT exc = − K (1 + sT 2 3 5 A F [( )( ) ]( ) ( ) ( Δ δ ) 1 + sT 1 + sT + K K 1 + sτ ' K + K K K 1 + sT F A F K A K K d 0 3 ) 3 6 A F The above relation appears quite challenging to analyze, but we can simplify the task greatly by observing that the denominator is third order. Thus, it will be possible to write the above relation as: − K 2 K 3K 5K A (1 + sTF ) ΔT = ( )( )( ) ( Δδ ) exc s + p1 s + p2 s + p3 where p i are the poles. We may have 3 real poles or 1 real with 2 complex. In general, we express each pole as p i =σ i +jω i . Thus, the transfer function becomes: ΔT exc = − 3 5 A F ( )( )( ) ( Δ δ ) s + σ + jω s + σ + jω s + σ + jω 1 K 1 2K K K ( 1 We want to evaluate the transfer function at s=jω osc , where ω osc is the frequency of oscillation of concern (we assume this frequency to be an interarea oscillation). Therefore, 2 + 2 sT ) 3 3 13
Page 1 and 2: Linearized Analysis of the Synchron
Page 3 and 4: The one-axis model is a 3-state mod
Page 5 and 6: In the above equations (*) and (**)
Page 7 and 8: Fig. 2 We will use this block diagr
Page 9 and 10: Δω axis Positive angle ΔT d ΔT
Page 11: Δω axis Positive angle ΔT I +ΔT
Page 15 and 16: Repeating our transfer function:
Page 17 and 18: The transfer function K S G lead (s
Page 19 and 20: However, this is not very easy to d

Linearized Analysis of the Synchronous Machine for PSS Chapter 6 ...

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