1. 0 1 2 3 4 5 6 7 8 0 100 200 300 400 500 600 700 800 Price Generation MER 0 20 40 60 80 100 120 0 100 200 300 400 500 600 Price Generation GEN MER 500 1 300 7 GEN 350 4 100 12 100 100 MER GENERATION PRICE 0 1 100 1 200 1 300 1 400 1 500 1 500 7 550 7 600 7 700 7 800 7 GEN GENERATION PRICE 0 4 50 4 100 4 150 4 200 4 250 4 300 4 350 4 350 12 450 12 450 100 500 100 550 100 There is a total of 1100 MWh of demand in the country. The uniform price of the electricity is $7. 0 20 40 60 80 100 120 0 200 400 600 800 1000 1200 1400 1600 Price Generation Combined offer stack DEMAND=1100MWh Combined offer stack Generation Price 0 1 100 1 200 1 300 1 400 1 500 1 550 1 550 4 600 4 650 4 700 4 750 4 800 4 850 4 850 7 900 7 950 7 1000 7 1050 7 1100 7 1150 7 1150 12 1200 12 1250 12 1250 100 1300 100 1350 100 b. 500MW Minimum: 4X1+12X2+100X3+X4+7X5 0≤X1≤350 Capacity Constraint of generator X1 0≤X2≤100 Capacity Constraint of generator X2 0≤X3≤100 Capacity Constraint of generator X3 0≤X4≤500 Capacity Constraint of generator X4 0≤X5≤300 Capacity Constraint of generator X5 The minimizing cost calculation is performed using solver in Microsoft excel. Generator GEN MER LINE X1 X2 X3 X4 X5 fGM Costs(Price/MWh) 4 12 100 1 7 0 Capacity(MWh) 0 0 0 0 300 -500 Node balance: demand Node North 1 1 1 0 0 1 900 Node South 0 0 0 1 1 -1 200 Lower bounds 0 0 0 0 0 -500 Upper bounds 350 100 100 500 300 500 Total Cost 2100 North Island South Island Target Cell (Min) Cell Name Original Value Final Value $C$17 objective value gen2 0 2100 Adjustable Cells Cell Name Original Value Final Value $B$6 variable values gen1 0 0 $C$6 variable values gen2 0 0 $D$6 variable values gen3 0 0 $E$6 variable values mer1 0 0 $F$6 variable values mer2 0 300 $G$6 variable values fGM 0 -500 Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$6 variable values gen1 0 4 4 1E+30 4 $C$6 variable values gen2 0 12 12 1E+30 12 $D$6 variable values gen3 0 100 100 1E+30 100 $E$6 variable values mer1 0 1 1 1E+30 1 $F$6 variable values mer2 300 7 7 1E+30 7 $G$6 variable values fGM -500 0 0 1E+30 0 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $E$9 North island mer1 0 0 0 0 1E+30 $E$10 South island mer1 1 0 1 0 1E+30 $E$13 mer1 0 0 0 0 1E+30 Based on the result from Solver calculation, the minimum total cost for this system is $2100. Only the MER(X5) generator is operated. From the above table, shadow price remains unchanged. c. GEN K=105, 55, 25MW MER Minimum: 4X1+12X2+100X3+X4+7X5  If K=105MW (Maximum capacity of HAY-HLY link is 105MW): Node1-HLY X1+X2+X3+ fHLY_HAY- fHLY_OTA Node2-HAY fBEN_HAY- fHLY_HAY- fHAY_OTA Node3-OTA fHLY_OTA+ fHAY-OTA Node4-BEN X4+X5- fBEN_HAY Kirchhoff fHAY_OTA+ fHLY_HAY- fHLY_OTA Capacity constraint on (BEN-HAY) link -500≤ fBEN-HAY≤500 Capacity constraint on (HAY-HLY) link -105≤f HAY-HLY≤105 Capacity constraint on generator X1 0≤ X1≤350 Capacity constraint on generator X2 0≤X2≤100 Capacity constraint on generator X3 0≤X3≤100 Capacity constraint on generator X4 0≤X4≤500 Capacity constraint on generator X5 0≤X5≤300 HLY OTA HAY BEN Demand=900MWh Minimizing cost calculation is performed using Solver in Microsoft Excel: Generator GEN MER LINE X1 X2 X3 X4 X5 fHLYOTA fHAYOTA fHLYHAY fBENHAY Costs(Price/MWh) 4 12 100 1 7 0 0 0 0 Capacity(MWh) 350 50 0 400 300 500 400 100 500 Node1-HLY 1 1 1 0 0 -1 0 -1 0 Node2-HAY 0 0 0 1 1 -1 -1 1 0 Node3-OTA 0 0 0 0 0 1 1 0 0 Node4-BEN 0 0 0 1 1 0 0 0 -1 Kirchhoff 0 0 0 0 0 -1 1 1 0 Lower bound 0 0 0 0 0 -105 -500 Upper bound 350 100 100 500 300 105 500 Total cost 4500 Target Cell (Min) Cell Name Original Value Final Value $C$17 objective value Gen2 4500 4500 Adjustable Cells Cell Name Original Value Final Value $B$6 variable values Gen1 350 350 $C$6 variable values Gen2 50 50 $D$6 variable values Gen3 0 0 $E$6 variable values Mer1 400 400 $F$6 variable values Mer2 300 300 $G$6 variable values fHLY-OTA 500 500 $H$6 variable values fHAY-OTA 400 400 $I$6 variable values fHLY-HAY 100 100 $J$6 variable values fBEN-HAY 500 500 Constraints Cell Name Cell Value Formula Status Slack $K$14 Kirchoff -2.84217E-14 $K$14=$L$14 Not Binding 0 $K$9 HLY -2.84217E-14 $K$9=$L$9 Not Binding 0 $K$10 HAY 0 $K$10=$L$10 Not Binding 0 $K$11 OTA 900 $K$11=$L$11 Not Binding 0 $K$12 BEN 200 $K$12=$L$12 Not Binding 0 $J$6 variable values fBEN-HAY 500 $J$6<=$J$16 Binding 0 $J$6 variable values fBEN-HAY 500 $J$6>=$J$15 Not Binding 1000 $E$6 variable values Mer1 400 $E$6>=$E$15 Not Binding 400 $J$6 variable values fBEN-HAY 500 $J$6>=$J$15 Not Binding 1000 $B$6 variable values Gen1 350 $B$6>=$B$15 Not Binding 350 $B$6 variable values Gen1 350 $B$6<=$B$16 Binding 0 $C$6 variable values Gen2 50 $C$6>=$C$15 Not Binding 50 $C$6 variable values Gen2 50 $C$6<=$C$16 Not Binding 50 $D$6 variable values Gen3 0 $D$6<=$D$16 Not Binding 100 $D$6 variable values Gen3 0 $D$6>=$D$15 Binding 0 $E$6 variable values Mer1 400 $E$6<=$E$16 Not Binding 100 $F$6 variable values Mer2 300 $F$6>=$F$16 Binding 0 $F$6 variable values Mer2 300 $F$6<=$F$16 Binding 0 $J$6 variable values fBEN-HAY 500 $J$6<=$J$16 Binding 0 $I$6 variable values fHLY-HAY 100 $I$6>=$I$15 Not Binding 205 $H$6 variable values fHAY-OTA 400 $H$6>=$H$15 Not Binding 400 $I$6 variable values fHLY-HAY 100 $I$6<=$I$16 Not Binding 5 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $K$14 Kirchoff -2.84217E14 0 0 205 5 $K$9 HLY -2.84217E14 12 0 50 50 $K$10 HAY 0 12 0 50 2.5 $K$11 OTA 900 12 900 50 5 $K$12 BEN 200 1 200 100 400  If K=55 MW (Maximum capacity of HAY-HLY link is 55MW) Node1-HLY X1+X2+X3+ fHLY_HAY- fHLY_OTA Node2-HAY fBEN_HAY- fHLY_HAY- fHAY_OTA Node3-OTA fHLY_OTA+ fHAY-OTA Node4-BEN X4+X5- fBEN_HAY Kirchhoff fHAY_OTA+ fHLY_HAY- fHLY_OTA Capacity constraint on (BEN-HAY) link -500≤ fBEN-HAY≤500 Capacity constraint on (HAY-HLY) link -55≤f HAY-HLY≤55 Capacity constraint on generator X1 0≤ X1≤350 Capacity constraint on generator X2 0≤X2≤100 Capacity constraint on generator X3 0≤X3≤100 Capacity constraint on generator X4 0≤X4≤500 Capacity constraint on generator X5 0≤X5≤300 Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$6 variable values Gen1 350 -8 4 8 1E+30 $C$6 variable values Gen2 50 0 12 88.00000027 8 $D$6 variable values Gen3 0 88.00000005 100 1E+30 88.00000005 $E$6 variable values Mer1 400 0 1 6 1E+30 $F$6 variable values Mer2 300 6 7 1E+30 6 $G$6 variable values fHLY-OTA 500 0 0 11 1E+30 $H$6 variable values fHAY-OTA 400 0 0 1E+30 11 $I$6 variable values fHLY-HAY 100 0 0 5.5 1E+30 $J$6 variable values fBEN-HAY 500 -11 0 11 1E+30 Minimizing cost calculation is performed using Solver in Microsoft Excel: Generator GEN MER LINE X1 X2 X3 X4 X5 fHLYOTA fHAYOTA fHLYHAY fBENHAY Costs(Price/MWh) 4 12 100 1 7 0 0 0 0 Capacity(MWh) 350 50 0 500 200 500 400 100 500 Node1-HLY 1 1 1 0 0 -1 0 -1 0 Node2-HAY 0 0 0 1 1 -1 -1 1 0 Node3-OTA 0 0 0 0 0 1 1 0 0 Node4-BEN 0 0 0 1 1 0 0 0 -1 Kirchhoff 0 0 0 0 0 -1 1 1 0 Lower bound 0 0 0 0 0 -55 -500 Upper bound 350 100 100 500 300 55 500 Total cost 3900 Target Cell (Min) Cell Name Original Value Final Value $C$17 objective value Gen2(X2) 4500 3900 Adjustable Cells Cell Name Original Value Final Value $B$6 variable values Gen1(X1) 350 350 $C$6 variable values Gen2(X2) 50 50 $D$6 variable values Gen3(X3) 0 0 $E$6 variable values Mer1(X4) 400 500 $F$6 variable values Mer2(X5) 300 200 $G$6 variable values fHLY-OTA 500 500 $H$6 variable values fHAY-OTA 400 400 $I$6 variable values fHLY-HAY 100 100 $J$6 variable values fBEN-HAY 500 500 Constraints Cell Name Cell Value Formula Status Slack $F$16 upper bound Mer2(X5) 300 $F$16<=$F$16 Binding 0 $I$16 upper bound fHLY-HAY 55 $I$16<=$I$16 Binding 0 $K$9 HLY 4.26326E-14 $K$9=$L$9 Not Binding 0 $K$10 HAY -1.13687E-13 $K$10=$L$10 Not Binding 0 $K$11 OTA 900 $K$11=$L$11 Not Binding 0 $K$12 BEN 200 $K$12=$L$12 Not Binding 0 $K$14 Kirchoff 4.26326E-14 $K$14=$L$14 Not Binding 0 $B$6 variable values Gen1(X1) 350 $B$6<=$B$16 Binding 0 $B$6 variable values Gen1(X1) 350 $B$6>=$B$15 Not Binding 350 $C$6 variable values Gen2(X2) 50 $C$6<=$C$16 Not Binding 50 $C$6 variable values Gen2(X2) 50 $C$6>=$C$15 Not Binding 50 $D$6 variable values Gen3(X3) 0 $D$6<=$D$16 Not Binding 100 $D$6 variable values Gen3(X3) 0 $D$6>=$D$15 Binding 0 $E$6 variable values Mer1(X4) 500 $E$6<=$E$16 Binding 0 $E$6 variable values Mer1(X4) 500 $E$6>=$E$15 Not Binding 500 $F$6 variable values Mer2(X5) 200 $F$6>=$F$15 Not Binding 200 $H$6 variable values fHAY-OTA 400 $H$6>=$I$15 Not Binding 455 $J$6 variable values fBEN-HAY 500 $J$6<=$J$16 Binding 0 $J$6 variable values fBEN-HAY 500 $J$6>=$J$15 Not Binding 1000 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $F$16 upper bound Mer2(X5) 300 0 300 1E+30 0 $I$16 upper bound fHLY-HAY 55 0 55 1E+30 0 $K$9 HLY 4.26326E-14 12 0 50 50 $K$10 HAY -1.13687E13 12 0 50 50 $K$11 OTA 900 12 900 50 50 $K$12 BEN 200 7 200 1E+30 200 $K$14 Kirchoff 4.26326E-14 0 0 1.95477E+13 455  If K=25 MW (Maximum capacity of HAY-HLY link is 25MW) Node1-HLY X1+X2+X3+ fHLY_HAY- fHLY_OTA Node2-HAY fBEN_HAY- fHLY_HAY- fHAY_OTA Node3-OTA fHLY_OTA+ fHAY-OTA Node4-BEN X4+X5- fBEN_HAY Kirchhoff fHAY_OTA+ fHLY_HAY- fHLY_OTA Capacity constraint on (BEN-HAY) link -500≤ fBEN-HAY≤500 Capacity constraint on (HAY-HLY) link -25≤f HAY-HLY≤25 Capacity constraint on generator X1 0≤ X1≤350 Capacity constraint on generator X2 0≤X2≤100 Capacity constraint on generator X3 0≤X3≤100 Capacity constraint on generator X4 0≤X4≤500 Capacity constraint on generator X5 0≤X5≤300 Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$6 variable values Gen1(X1) 350 -8 4 8 1E+30 $C$6 variable values Gen2(X2) 50 0 12 88.00000027 5 $D$6 variable values Gen3(X3) 0 88.00000005 100 1E+30 88.00000005 $E$6 variable values Mer1(X4) 500 -6 1 6 1E+30 $F$6 variable values Mer2(X5) 200 0 7 5 6 $G$6 variable values fHLY-OTA 500 0 0 5 1E+30 $H$6 variable values fHAY-OTA 400 0 0 1E+30 5 $I$6 variable values fHLY-HAY 100 0 0 2.5 1E+30 $J$6 variable values fBEN-HAY 500 -5 0 5 1E+30 Minimizing cost calculation is performed using Solver in Microsoft Excel: Generator GEN MER LINE X1 X2 X3 X4 X5 fHLYOTA fHAYOTA fHLYHAY fBENHAY Costs(Price/MWh) 4 12 100 1 7 0 0 0 0 Capacity(MWh) 350 87.5 0 500 162.5 462.5 437.5 25 462.5 Node1-HLY 1 1 1 0 0 -1 0 -1 0 Node2-HAY 0 0 0 1 1 -1 -1 1 0 Node3-OTA 0 0 0 0 0 1 1 0 0 Node4-BEN 0 0 0 1 1 0 0 0 -1 Kirchhoff 0 0 0 0 0 -1 1 1 0 Lower bound 0 0 0 0 0 -25 -500 Upper bound 350 100 100 500 300 25 500 Total cost 4087.5 Target Cell (Min) Cell Name Original Value Final Value $C$17 objective value Gen2(X2) 3900 4087.5 Adjustable Cells Cell Name Original Value Final Value $B$6 variable values Gen1(X1) 350 350 $C$6 variable values Gen2(X2) 50 87.5 $D$6 variable values Gen3(X3) 0 0 $E$6 variable values Mer1(X4) 500 500 $F$6 variable values Mer2(X5) 200 162.5 $G$6 variable values fHLY-OTA 500 462.5 $H$6 variable values fHAY-OTA 400 437.5 $I$6 variable values fHLY-HAY 100 25 $J$6 variable values fBEN-HAY 500 462.5 Constraints Cell Name Cell Value Formula Status Slack $K$9 HLY 6.36646E-11 $K$9=$L$9 Not Binding 0 $K$10 HAY 9.61791E-11 $K$10=$L$10 Not Binding 0 $K$11 OTA 900 $K$11=$L$11 Not Binding 0 $K$12 BEN 200 $K$12=$L$12 Not Binding 0 $K$14 Kirchoff 1.04592E-11 $K$14=$L$14 Not Binding 0 $B$6 variable values Gen1(X1) 350 $B$6<=$B$16 Binding 0 $B$6 variable values Gen1(X1) 350 $B$6>=$B$15 Not Binding 350 $C$6 variable values Gen2(X2) 87.5 $C$6<=$C$16 Not Binding 12.5 $C$6 variable values Gen2(X2) 87.5 $C$6>=$C$15 Not Binding 87.5 $D$6 variable values Gen3(X3) 0 $D$6<=$D$16 Not Binding 100 $D$6 variable values Gen3(X3) 0 $D$6>=$D$15 Binding 0 $E$6 variable values Mer1(X4) 500 $E$6<=$E$16 Binding 0 $E$6 variable values Mer1(X4) 500 $E$6>=$E$15 Not Binding 500 $F$6 variable values Mer2(X5) 162.5 $F$6<=$F$16 Not Binding 137.5 $F$6 variable values Mer2(X5) 162.5 $F$6>=$F$15 Not Binding 162.5 $I$6 variable values fHLY-HAY 25 $I$6<=$I$16 Binding 0 $I$6 variable values fHLY-HAY 25 $I$6>=$I$15 Not Binding 50 $J$6 variable values fBEN-HAY 462.5 $J$6<=$J$16 Not Binding 37.5 $J$6 variable values fBEN-HAY 462.5 $J$6>=$J$15 Not Binding 962.5 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $K$9 HLY 6.36646E11 12 0 12.5 87.5 $K$10 HAY 9.61791E11 7 0 37.5 162.5 $K$11 OTA 900 9.5 900 25 175 $K$12 BEN 200 7 200 137.5 162.5 $K$14 Kirchoff 1.04592E11 -2.5 0 75 25 According to the result of solver equation, it can be seen that when the line connecting HAY and HLY has a capacity of K=105 MW, the total minimum cost for this system is $4500. At North Island, the price of electricity at node HAY, HLY, OTA is $12. Meanwhile, in south island, the price of electricity at node BEN is $1. The allowable increase at North Island is no more than 50MW. According to the result of solver equation, it can be seen that when the line connecting HAY and HLY has a capacity of K=55 MW, the total minimum cost for this system is $3900. At North Island, the price of electricity at node HAY, HLY, OTA is $12. Meanwhile, in south island, the price of electricity at node BEN is $7. The allowable increase at North Island is no more than 50MW. According to the result of solver equation, it can be seen that when the line connecting HAY and HLY has a capacity of K=25 MW, the total minimum cost for this system is $4087.5. At North Island, the price of electricity at node HAY, HLY, OTA is $12. Meanwhile, in south island, the price of electricity at node BEN is $1. The price of electricity of node HAY Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$6 variable values Gen1(X1) 350 -8 4 8 1E+30 $C$6 variable values Gen2(X2) 87.5 0 12 88.00000023 5 $D$6 variable values Gen3(X3) 0 88.00000001 99.99999997 1E+30 88.00000001 $E$6 variable values Mer1(X4) 500 -6 1 6 1E+30 $F$6 variable values Mer2(X5) 162.5 0 7 5 6 $G$6 variable values fHLY-OTA 462.5 0 0 5 1E+30 $H$6 variable values fHAY-OTA 437.5 0 0 1E+30 5 $I$6 variable values fHLY-HAY 25 -2.5 0 2.5 1E+30 $J$6 variable values fBEN-HAY 462.5 0 0 5 1E+30 decreased from $12 to $7 per MWh. Compared to 105MW, the node price at BEN has increased to $7. The spring water effect occurs when there is a constrained circuit in the transmission loop. In this effect, prices are sensitive to minor changes to input data.