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lecture8 - Power Systems I Inequality Constraints in...

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Unformatted text preview: Power Systems I Inequality Constraints in Optimization l Practical problems contain inequality as well as equality constraints l Minimize the cost function u subject to the equality constraints u and the inequality constraints l The Lagrange multiplier is extended to include the inequality constraints by introducing the m-dimensional vector of undetermined quantities ( ) n x x x f L , , 2 1 ( ) k i x x x g n i , , 2 , 1 , , 2 1 L L = = ( ) m j x x x u n j , , 2 , 1 , , 2 1 L L = Power Systems I Kuhn-Tucker Method m j u m j u L k i g L n l x L u g f L j j j j j i i l j j j i i i , , 1 & , , 1 , , 1 , , 1 L L L L = > = = = = = = = = + + = l The unconstrained cost function becomes l The resulting necessary conditions for contrained local minima of L are the following Power Systems I Example l Use the Kuhn-Tucker method to determine the minimum distance from the origin of the x-y plane to a circle described by constrained by u The minimum distance is obtained by minimizing the distance squared ( )( ) ()( )( ) () () 2 2 2 2 2 2 , 12 2 , 25 6 8 , 25 6 8 y x y x f y x y x u y x y x g or y x + = + = + = = + Power Systems I Example () ()( )( ) () ( )( ) [ ] [ ] ( ) ( ) ( )( ) 12 2 25 6 8 6 2 2 2 8 2 2 12 2 25 6 8 12 2 , 25 6 8 , , 2 2 2 2 2 2 2 2 2 2 = + = = + = = + + = = + + = + + + + + = + + = + = = + = + = y x L y x L y y y L x x x L y x y x y x u g f L y x y x u y x y x g y x y x f The cost function The resulting necessary conditions for constrained local minima of L Power Systems I u eliminating from the first two equations u substituting for y in the third equation yields x y y x 4 3 2 12 2 16 = = ( ) () ( ) 3 4 min 3 , 9 , 12 1 , 3 , 4 : 12 & 4 75 25 16 25 25 6 4 3 8 2 2 2 = = = = = = = + = + y x and extrema x x x x x x Example Power Systems I Economic Dispatch with Generator Limits l The power output of any generator should not exceed its rating nor be below the value for stable boiler operation u Generators have a minimum and maximum real power output limits l The problem is to find the real power generation for each plant such that cost are minimized, subject to: u Meeting load demand - equality constraints u Constrained by the generator limits - inequality constraints l The Kuhn-Tucker conditions (min) (max) (max) (min) i i i i i i i i i i i i i P P dP dC P P dP dC P P P dP dC = = < < = Power Systems I Example l Neglecting system losses, find the optimal dispatch and the total cost in $/hr for the three generators and the given load demand and generation limits MW 975 225 100 350 150 450 200 009 . 8 . 5 200 006 . 5 . 5 400 ] MWhr / [$ 004 . 3 . 5 500 3 2 1 2 3 3 3 2 2 2 2 2 1 1 1 = + + = + + = + + = Demand P P P P P P C P P C P P C Power Systems I...
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lecture8 - Power Systems I Inequality Constraints in...

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