Nodal pricing exerciseFor the above system:(a) Determine the economic dispatch assuming there are no transmissionconstraints(b) Determine the system marginal price(c) Determine the constrained economic dispatch based on the given lineparameters and capacities(d) Identify overloaded lines(e) Correct the overloads by re-dispatching generation in the most economicfashion(f) Compute the nodal prices for the constrained system(g) Compute the congestion cost(h) Finally, for case (c) write the equations for the objective function and theconstraints that you would need to solve using optimisation techniques to findthe dispatch and nodal prices(i) Solve the problem in (h) using an optimisation routine (in MATLAB, EXCELetc) and compare your results with what you found in (c) and (f)Nodal pricing exerciseSolution(a) Determine the economic dispatch assuming there are no transmission constraintsSincethemarginalcostsareconstant,thegenerators aredispatchedstartingwiththe least marginal cost until the load is met.Generator Node Cost($/MWh)Capacity(MW)Dispatch(MW)Demand(MW)GA 1 8 140 140 50GB 1 10 285 270GC 2 20 90 0 60GD 3 15 85 0 300Total 410MW 410MW(b) Determine the system marginal priceThesystemmarginalpriceis $10/MWh (Fromthemostexpensivepartiallyloaded generator)(c) Determine the constrained economic dispatch based on the given line parametersand capacitiesThelineflowsaredeterminedfromKirchhoffsvoltage(KVL)andnodalbalance(KCL) equationsFigure 1KCL equationsNode 1: GA + GB F1 F3 50 = 0 (1)Node 1: GC + F2 F1 60 = 0 (2)Node 1: GD + F3 + F2 300 = 0 (3)One loop hence one KLV equationKVL: 0.2*F1 + 0.1*F2 0.2*F3 = 0 (4)Substituting for known variables, we have;Nodal pricing exerciseNode 1: 360 F1 F3 = 0 (5)Node 1: F2 F1 60 = 0 (6)Node 1: F3 + F2 300 = 0 (7)From (5), F1 = 360 F3 (8)From (4) F1 = (0.1*F2 0.2*F3)/0.2 = 0.5F2 + F3 (9)Equating (8) and (9); 360 F3 = 0.5F2 + F3 (10)From (10), 2F3 0.5F2 = 360 (11)Multiply (7) by 2 2*(F3 + F2) = 2*(300) 2F3 2F2 = 600 (12)Add (12) and (11) to give 2.5F2 = 240; givingF2=240/2.5 = 96MWTherefore F3 and F1 can be computed from (7) and (5) respectivelyF3=300-F2=300-96=204MWF1=360-F3= 360-204=156MW(d) Identify overloaded linesFromlinecapacitiesline1-2isoverloadedsinceitisratedat120MWandtheflow is 156MW(e) Correct the overloads by re-dispatching generation in the most economic fashionTheoverloadinline 1-2canbecorrectedbyre-dispatchingeitherGDorGC.Checkeachoneinturnand comparethecoststochoosethemosteconomicsolution.Case1 UseGDtoreduceflowinline1-2by36MWto120MWtheratedlinecapacity.Figure 2For the above scenario we will haveY = F3 + F2Nodal pricing exercise, ), )MW YY F F90 5 182 . 05 . 036362 . 0 1 . 0 2 . 02 . 02 1= - = - = =+ +- = =Cost of re-dispatch = 90*15= 135Case 2 Use GC to reduce flow in line 1-2 by 36MW to 120MW the rated linecapacity:Figure 3, ), )MW XX F60 5 123 . 05 . 036362 . 0 1 . 0 2 . 01 . 0 2 . 01= - = - = =+ + +- =Table 1MarginalCost/MWhUnconstrainedEconomicdispatchCase 1constraineddispatchCase 2constraineddispatchGA 8 140 140 140GB 10 270 180 210GC 20 0 0 60GD 15 0 90 0Total(MW) 410 410 410Cost ($) 3,820 4,270 4,420Cost ofconstraints($) 0 450 600The most economic way to eliminate the constraint online 1-2 would be to usecase 1. However GD capacity is limited to 85MW hence some use would have tobe made of generator GC.Hence with GD output equal to 85 the line 1-2 will still be overloaded by:MW 2 34 365 . 02 . 085 36 = = - Hence we need to dispatch generator GC by:Nodal pricing exercise, ), )MW XX F33 . 33 . 05 . 0222 . 0 1 . 0 2 . 01 . 0 2 . 01= - = =+ + +- =dispatch would beHence final dispatch will be as shown in Table 2 belowTable 2MarginalCost/MWh(A)UnconstrainedEconomicdispatch(B)Case 1constraineddispatchGA 8 140 140GB 10 270 181.67GC 20 0 3.33GD 15 0 85Total(MW) 410 410Cost ($) 3,820 4,278Cost ofconstraints($) = (B)-(A) 0 458(f) Compute the nodal prices for the constrained systemDue to line 1-2 being congested, the nodal prices will be different at each node.Node 1:The nodal price at node 1 is equal to 10$/MWh because generator GB is not fullyloaded and hence the next MW at this node will be supplied by GBNode 2:The generator GC at node 2 is also not fully loaded and hence the next MWdemand at this node will be supplied from this node. Therefore the nodal price atthis node is 20$/MWhNode 3:Since the generator at node 3 is fully loaded, the next MW demand at this nodemust be supplied from node 1 and node 2. Hence the nodal price is found as alinear combination of the nodal prices at nodes 1 and 2.Nodal pricing exerciseFigure 4Figure 5From Figure 4 and Figure 5, compute nodal prices as follows:0 2 . 0 4 . 0112 2 12 1= A = A A= A + AF P PMW P P2212 15 . 04 . 02 . 02 . 0 4 . 0PPPP PA =A= AA = AMW PMW PPP P P31325 . 111 ) 5 . 0 1 (5 . 0 1 11222 1 2= A = = A= + AA = A = ATherefore; the nodal price for node 3 is:MWh MCMC P MC P MC/ $ 67 . 163502032103132 2 1 1 3= = - + - =A + A =(g) Compute the congestion costCongestion is given from final dispatch table 2 as $458(h) Finally, for case (c) write the equations for the objective function and theconstraints that you would need to solve using optimisation techniques to find thedispatch and nodal pricesNodal pricing exerciseFigure 6(a) Objective functioniNgiTi G C Min =1= GD GC GB GA Min 15 20 10 8 : + + +whereNg is the number of generatorsSubject to the following constraints:(b) KCL equations0 300 3 20 60 2 10 50 3 1= + += += +F F GDF F GCF F GB GA(c) KVL equation0 3 2 . 0 2 1 . 0 1 2 . 0 = + F F F(d) Generator limits85 090 0285 0140 0s ss ss ss sGDGCGBGA(e) Line flow limits250 3 250130 2 130120 1 120s s s s s s FFF(i) Solve the problem in (h) using an optimisation routine (in MATLAB, EXCEL etc)and compare your results with what you found in (c) and (f)Nodal pricing exerciseResult from ExcelMicrosoft Excel 11.0 Sensitivity ReportWorksheet: [Solver.xls]Sheet2Report Created: 31/12/2011 02:09:20Adjustable CellsFinal ReducedCell Name Value Gradient$F$3 GA solution 140 -2$F$4 GB solution 181.6666667 0$F$5 GC solution 3.333333333 0$F$6 GD solution 85 -1.666666667$F$9 F1 solution 120 -16.66666667$F$10 F2 solution 63.33333333 0$F$11 F3 solution 151.6666667 0ConstraintsFinal LagrangeCell Name Value Multiplier$C$14 KCL1 min -5.68434E-14 10$C$15 KCL2 min 0 20 Nodal Prices$C$16 KCL3 min 0 16.66666667$C$17 KVL min 0 33.33333284