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Unformatted text preview: Lecture 13 Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS 2 Announcements Be reading Chapter 6, also Chapter 2.4 (Network Equations). HW 5 is 2.38, 6.9, 6.18, 6.30, 6.34, 6.38; do by October 6 but does not need to be turned in. First exam is October 11 during class. Closed book, closed notes, one note sheet and calculators allowed. Exam covers thru the end of lecture 13 (today) An example previous exam (2008) is posted. Note this is exam was given earlier in the semester in 2008 so it did not include power flow, but the 2011 exam will (at least partially) 3 MultiVariable Example 1 2 2 2 1 1 2 2 2 2 1 2 1 2 1 1 1 2 2 2 1 2 x Solve for = such that ( ) 0 where x f ( ) 2 8 f ( ) 4 First symbolically determine the Jacobian f ( ) f ( ) ( ) = f ( ) f ( ) x x x x x x x x x x = = + = = + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ x f x x x x x J x x x 4 Multivariable Example, cont’d 1 2 1 2 1 2 1 1 1 2 1 2 1 2 1 2 2 (0) 1 (1) 4 2 ( ) = 2 2 Then 4 2 ( ) 2 2 ( ) 1 Arbitrarily guess 1 1 4 2 5 2.1 1 3 1 3 1.3 x x x x x x x x x f x x x x x f + ∆ =  ∆ + =  = =  J x x x x x 5 Multivariable Example, cont’d 1 (2) (2) 2.1 8.40 2.60 2.51 1.8284 1.3 5.50 0.50 1.45 1.2122 Each iteration we check ( ) to see if it is below our specified tolerance 0.1556 ( ) 0.0900 If = 0.2 then we wou ε ε = =  = x f x f x ld be done. Otherwise we'd continue iterating. 6 Possible EHV Overlays for Wind AEP 2007 Proposed Overlay 7 NR Application to Power Flow * * * * i 1 1 We first need to rewrite complex power equations as equations with real coefficients S These can be derived by defining Recal i n n i i i ik k i ik k k k ik ik ik j i i i i ik i k V I V Y V V Y V Y G jB V V e V θ θ θ θ θ = = = = = ÷ + = ∠ ∑ ∑ @ @ @ j l e cos sin j θ θ θ = + 8 Real Power Balance Equations * * i 1 1 1 i 1 i 1 S ( ) (cos sin )( ) Resolving into the real and imaginary parts P ( cos sin ) Q ( sin cos ik n n j i i i ik k i k ik ik k k n i k ik ik ik ik k n i k ik ik ik ik Gi Di k n i k ik ik ik i k P jQ V Y V V V e G jB V V j G jB V V G B P P V V G B θ θ θ θ θ θ θ = = = = = = + = = = + = + = = ∑ ∑ ∑ ∑ ∑ ) k Gi Di Q Q = 9 NewtonRaphson Power Flow i 1 In the NewtonRaphson power flow we use Newton's method to determine the voltage magnitude and angle at each bus in the power system....
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 Fall '08
 Overbye,T

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