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of actual losses to the losses that would be recovered if the marginal loss rates were applied to all generators.
2 $18,232 = $20 x 500 + $21 x 392. $122 = $18,232 - $18,110.
3 The figures for the scaled marginal cost approach are derived as follows. Without losses, the 892-MW system load is met by 744I MW and 147G MW from the western and eastern generators, respectively. The marginal loss rates for these generators are 0.02197 (=0.05 x 392/892) and -0.02803 (=-0.05 x 500/892), respectively, where 0.05 is the marginal line loss for this case in which line flows average 250 MW. Total marginal losses are 16.33 MW (=0.02197 x 744I) and -4.13 MW (=-0.02803 x 147G), respectively. Scaled losses are 8.36 MW (=6.25 x 16.33/(16.33-4.13)) and -2.11 MW (=6.25 x -4.13/(16.33-4.13)), respectively. Break-even prices are $20.22 = $20 x (753J)/ 744I and $20.70 = $21 x (145J)/147G, respectively.
4 Two-thirds of any load increment at bus 1 flows to bus 3 over line 1-3. One-third of any load increment at bus 2 flows to bus 3 over line 1-3. Decreasing bus 1 output by 1 MW therefore decreases flow on line 1-3 by two-thirds of a megawatt, while increasing bus 2 output by 2 MW increases flow on line 1-3 by that same two-thirds of a megawatt. In simultaneously decreasing bus 1 output by 1 MW and increasing bus 2 output by 2 MW, flow on line 1-3 is left unchanged, while supply increases by the 1 MW needed to serve a 1-MW load increment at bus 3.
5 In the context of the efficient dispatch of Figure 4, a 1-MW increase in the capacity of line 1-3 would allow generator A2 to reduce output by 3 MW while generator B1 increased output by 3 MW. The cost change attributable to a 1-MW relaxation of the line 1-3 constraint thus is -3 x $25 + 3 x $15 = -$30.
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