, and ignoring all load currents, determine

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1 ECE43 Test 3 Dec 8, 5 Q. (33 pts.) The Zbus for the above 3-bus network with bus as reference, in per unit, is given to be 3.87 j.798 j.8 j Z.798 j.87 j.8 j bus.8 j.8 j j Assuming that the prefault values of all buses are per unit, that is V V V o, 3 and ignoring all load currents, determine a) (9 pts) the short-circuit current, in per unit, that would flow out of bus 3 through a fault of zero fault impedance connected between bus 3 and the reference bus b) (5 pts) estimate the voltages of buses,, and 3 during the short-circuit fault on bus c) (9 pts) If the impedance of the branch z directly connecting buses and is j.5 per unit and you are asked to remove z from the Zbus given. Determine the augmented matrix in this process, that is the intermediate matrix before you perform a Kron s reduction to get the update Zbus. Do NOT perform the Kron s reduction. Assuming prefault bus voltages are all unity V pf V pf V 3pf (a) Short-circuit current at bus 3 modelled by superimposing a equal and opposite prefault voltage V 3pf ΔV 3 V 3pf Z th3 Zbus 33 j I 3sc Z th3 j (b) With a direct short-circuit on bus 3, the new bus voltages can be estimated using

2 V V V 3 V pf ΔV V pf ΔV V 3pf ΔV 3 Zbus Zbus Zbus 3 Zbus Zbus Zbus 3 Zbus 3 Zbus 3 Zbus 33 I pf I pf I 3pf ΔI 3 For a direct short at bus 3, ΔV 3 = V 3pf and ΔI 3 = I 3sc. The prefault currents produce the prefault voltages, leaving ΔV ΔV V 3pf Zbus Zbus Zbus 3 Zbus Zbus Zbus 3 Zbus 3 Zbus 3 Zbus 33 I 3sc ΔV ΔV V 3pf V 3pf Zbus 3 Zbus 33 Zbus 3 Zbus 33 Using Zbus matrix values that were computed earlier, the bus voltages during the fault is given by V V V 3 Zbus 3 V pf Zbus 33 Zbus 3 V pf Zbus 33.. (c) To remove the link z between buses and, add a link of equal but opposite impedance value z a z.5j vadd4 Zbus Zbus z bb Zbus Zbus Zbus z a.93j Zbusaug stack augment( Zbus vadd4) augment vadd4 T z bb Answer asked for double-checking Zbusaug.87j.798j.44j.798j.87j.44j j.44j.44j.93j eliminate( Z nrow ncol pv) for for Zreduce i nrow j ncol Zreduce ij Z ij Z ipv Z pvj Z pvpv

3 Zbus eliminate( Zbusaug 3 3 4) Zbus.96j.64j.64j.96j j

4 Q. (34 pts.) -bus system The -bus system with line impedance z=j. per unit is to be operated with the following desired operating condition: The bus type and their scheduled/desired operating condition are given in the table below: Bus No. i Bus type Bus voltages in pu Generation into bus P gi + jq gi in pu Loading on bus P di + jq di in pu Magnitude Angle P gi Q gi P di Q di V i i Slack bus.??.. Gen bus.?.6?..4 a) (5 pts) Starting with an initial guess of. o o V and V, perform one complete iteration of updating V and Q. g b) (9 pts) Also compute the Sg of the slack generator at the end of this first iteration. Desired operation conditions: V. S d. j. V sch P g.6 S d. j.4 z j. y z 5j By inspection, the Y-bus is Y y Y y Y Y Y y Y 5j 5j 5j 5j

5 Equations to compute net power injected Gen bus, given P g and S d S d P d jq d P P g P d Calculate Q ImV Y V Q g Q Q d S P jq Slack bus I Y V Y V S V I S g S S d Equations for updating voltage of Gen bus S V Y V Y V 8 arg V π Update voltage of bus to scheduled value using V V sch e j arg V (a) Perform one Gaussian iteration Initial and scheduled values V sch. V V sch e j S d..j V sch. V V sch e j P g.6 S d.4j First store values for voltage convergence check later V old V P P g Re S d.4 Q ImV Y V Y V. S P jq.4.j Q g Q Im S d.3 Iterating for voltage of Gen bus S V Y V Y.8j argv V π Adjust voltage magnitude back to scheduled value

6 V V sch e j arg V j Compute Q Q ImV Y V Y V.838 Q g Q Im S d.36 Check change in V's for convergence V V old.798 Calculate slack bus injection and generation I Y V Y V j S V I.467.8j S g S S d.667.8j

7 Q.3 (33 pts.) The fuel cost in dollars per hour and allowable operating range of two generators in the same power station are given as FP ( ).5P P5 $/hour P MW FP ( ).4P 6P 98 $/hour P MW where the output powers of the two generators, P and P, are in MW. If the total demand to be met by the power stations, that is P t = P + P, is 8 MW, a) (8 pts) Determine the incremental fuel cost of each generator and the total operating fuel cost in $ per hour. b) (5 pts) Determine the incremental fuel cost, $ per MWh, for the next MW of total demand beyond 8 MW? Given conditions, with P's in MW.5P F P F P P 5 $/h P min P max.4p 6P 98 P min P max $/h Minimum and maximum total demand from station P tmin 3 P tmax 8 a ) Obtain the incremental fuel cost equations by differentiating F and F with respect to P and P, respectively P λ P λ P.8 P 6 $/MWh $/MWh P t At a total load P t 8 Using an initial estimates P 9 P P t P 9 Given P.8P λ = P P λ = 6 = P t λ λ P Soln Find P P λ P Soln P Soln λ Soln $/MWh

8 Above solution is feasible because P and P do not exceed their limits. b ) λ P F P F t F P.3 4 $/h r For the next MW of additional demand P t 8 Given P.8P λ = P P λ = 6 = P t 77. Soln Find P P λ P Soln 77. P Soln λ Soln $/MWh F P Total fuel cost to generate more MW F t F P.39 4 $ Thus, the differential cost to generate the extra MW F t F t $ Note that the difference in total fuel cost is equal to the average value of the incremental costs λ λ $/MWh Using alternate solution method for parts a) and b) a b a.8 b 6 a t b b a a.4444 b t a t a a P t 8 For a total demand λ a t P t b t $/MWh λ b λ b Check limits P P a a Total fuel cost F P.3 4 F t F P For a total demand P t 8 λ a t P t b t 89. $/MWh λ b λ b Check limits P 77. P a a

9 Total fuel cost F P.39 4 F t F P Differential cost to generate the extra MW F t F t λ λ Note that the difference in total fuel cost is equal to the average value of the incremental costs