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lecture5 - The Power Flow Solution Most common and...

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Power Systems I The Power Flow Solution Most common and important tool in power system analysis also known as the “Load Flow” solution used for planning and controlling a system assumptions: balanced condition and single phase analysis Problem: determine the voltage magnitude and phase angle at each bus determine the active and reactive power flow in each line each bus has four state variables: voltage magnitude voltage phase angle real power injection reactive power injection
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Power Systems I The Power Flow Solution Each bus has two of the four state variables defined or given Types of buses: Slack bus (swing bus) voltage magnitude and angle are specified, reference bus solution: active and reactive power injections Regulated bus (generator bus, P-V bus) models generation-station buses real power and voltage magnitude are specified solution: reactive power injection and voltage angle Load bus (P-Q bus) models load-center buses active and reactive powers are specified (negative values for loads) solution: voltage magnitude and angle
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Power Systems I Newton-Raphson PF Solution Quadratic convergence mathematically superior to Guass-Seidel method More efficient for large networks number of iterations required for solution is independent of system size The Newton-Raphson equations are cast in natural power system form solving for voltage magnitude and angle, given real and reactive power injections
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Power Systems I Newton-Raphson Method A method of successive approximation using Taylor’s expansion Consider the function: f ( x ) = c , where x is unknown Let x [0] be an initial estimate, then Δ x [0] is a small deviation from the correct solution Expand the left-hand side into a Taylor’s series about x [0] yeilds ( ) c x x f = Δ + ] 0 [ ] 0 [ ( ) ( ) c x dx f d x dx df x f = + Δ Υ੟ Φ੯ Τ੏ ΢ਯ Σਿ Ρਟ + Δ Υ੟ Φ੯ Τ੏ ΢ਯ Σਿ Ρਟ + L 2 ] 0 [ 2 2 2 1 ] 0 [ ] 0 [
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Power Systems I Newton-Raphson Method Assuming the error, Δ x [0] , is small, the higher-order terms are neglected, resulting in where rearranging the equations ( ) ] 0 [ ] 0 [ ] 0 [ ] 0 [ x dx df c c x dx df x f Δ Υ੟ Φ੯
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