Unit Trip April 20, 2011, 13:40:53 CDT. McDonald Angle Relative to U.T. Austin Second

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1 _35_EE394J_Spring11 Order_Illustrator.doc Ringdown Analysis of Voltage Phase Angle Using 30 Point-per- Synchrophasor Data and the Excel Solver Values in the yellow boxes of the spreadsheet are entered by the user. Blue boxes are computed by the spreadsheet. Starting values for A, B, T 1, Tau 1, C, T 2, Tau 2, and T damp in the eight columns beginning with Column C will be adjusted by the Excel Solver. The problem is to curve fit the ringdown with t T ) = + ( ) ( 1 1) / Tau θ t t t A B A e 1 ) + ( start stop 1 where + C e t T2 ) / Tau2 ( ( ) sin ω d 2 2. t T / ( 1) Tau1 1 A + B A) ( 1 e ) is the exponential term that transitions the steady-state angle from initial value A degrees to final asymptote value B degrees with time constant τ 1. t T2 ) / Tau2 C e ( ( ) sin ω 2 2 is the classic damped sinusoid. d Page 1 of 5

2 The Solution Steps _35_EE394J_Spring11 Order_Illustrator.doc Paste the synchrophasor data into the spreadsheet as follows: sample number goes in Col O, and phase angle data goes in Col P. Make sure no old data remain in these columns. The time column is then filled-in by the spreadsheet assuming 30 points per second. Make sure that formula columns Q through Y are at least as long as the data pasted-in Col s O and P. It is best to leave the length = 3600 so that two full minutes of synchrophasor data can be entered if needed. The measured data will appear in the lower graph of the spreadsheet. Paste Paste Sample Angle Time Adjust the X-axis scale of the top graph until the ringdown fills about one-half of the screen. A 10-second window usually works nicely. Change the title of the top graph to match the study case. Enter zero in the yellow box for variable C so that the damped sinewave is not yet part of the analysis problem. Enter analysis Start Sec. and Stop Sec. in the two left-most yellow boxes. The start time should begin about ½ second prior to the onset of the ringdown. Stop Sec. is when the idealized ringdown is about to show signs of governor action. Start Sec. and Stop Sec. will not be adjusted by the Excel Solver. Enter estimates for initial steady-state angle A and final steady-state angle asymptote B. Enter estimates for exponential starting time T 1 and time constant Tau 1, adjusting them until there is an approximate visual match in the steady-state angle transition (without the sinusoid) Page 2 of 5

3 _35_EE394J_Spring11 Order_Illustrator.doc Next, enter estimates for C, T 2, Tau 2, and T damp. C is the peak magnitude of the damped sinusoid (can be positive or negative). T 2 is the time where the damped sinusoidal begins. Tau 2 is the time constant of the damped sinusoid s envelope. T damp is the time period of the damped resonant frequency estimate it using the time between peaks. Adjust the estimates until there is an approximate visual match Next, invoke the Excel Solver to minimize the least squared error (in cell M3) by adjusting the eight variables: A, B, T 1, Tau 1, C, T 2, Tau 2, and T damp (i.e., cells C3 through J3) Page 3 of 5

4 _35_EE394J_Spring11 Order_Illustrator.doc It is sometimes necessary to adjust Stop Sec. to obtain a better match for the pre-governor-action ringdown response. The main objective of this analysis is to obtain the damped resonant frequency, and the F damp normalized damping coefficient Zeta. Both are computed by the spreadsheet and displayed in the blue boxes. Page 4 of 5

5 _35_EE394J_Spring11 Order_Illustrator.doc Additional Example. McDonald Ringdown, May 8, 2011, 20:53 GMT Unit Trip May 8, 2011, 20:53 GMT. McDonald Angle Relative to U.T. Austin Manual Starting Point for Excel Solver Unit Trip May 8, 2011, 20:53 GMT. McDonald Angle Relative to U.T. Austin Output of Excel Solver Page 5 of 5