Zone 1. Zone 3. Zone 2. PROBLEM 1 (40 points) Fixed Load Auction (No transmission limits considered):

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1 EE5721 Practice Final Exam PROBLEM 1 (40 points) A power system consists of three zones. Each zone will be represented by a single generator and a single load on a single bus as shown below: GA GC LA LC Zone 1 GB Zone 3 LB Zone 2 The generators are labeled GA, GB and GC and the loads are labeled LA, LB and LC. The schedule of generation and load in this system is determined by an auction. We shall consider two variations of input data for the auctions. In the first, load will be considered fixed in that it will be satisfied (that is, the load is supplied) no matter what the generator s sell prices are (assuming that the transmission system and the generators have adequate capacity to meet the load). The second auction will consider the load to be variable in that the load shall submit a willingness to pay price to purchase blocks of energy. If the price of energy is below the load s willingness to pay price it is satisfied (again assuming that there is sufficient transmission and generation capacity). If the price of energy from the generators is above the load s willingness to pay price - then the load is not satisfied (that is, the load block is not supplied). Fixed Load Auction (No transmission limits considered): Generators submit blocks of energy for sale with a price for each block. The power exchange arranges all the generator sell blocks from lowest price to highest price. The load is summed for the entire system. It then starts the process of finding a clearing price as follows: Step down the generation bids block by block, until there is sufficient generation to meet the load, then stop. The last block of generation may be completely used of it may only be partially used. The generation price for this last block of energy is the clearing price for the entire auction.

2 EE5721 Practice Final Exam Problem 1 Part A) Loads are fixed, and the generation blocks and prices are given in the table below. Note that each generation block is labeled GA1, GA2,..., GC5. We shall assume that the transmission system capacity is adequate for any transfer of power from any zone to any other zone. Find the clearing price for this auction. Indicate which blocks of generation are actually sold. Calculate the flows on the transmission paths from zone 1 to zone 2 and from zone 2 to zone 3. Generator: GA Generator: GB Generator: GC Block Block Price Block Block Price Block Block Price Number MW $/MWh Number MW $/MWh Number MW $/MWh GA GB GC GA GB GC GA GC GA GC GC Load: LA Load: LB Load: LC 300 MW 600 MW 450 MW Data for Problem 1 Part A Variable Load Auction (No transmission limits considered): Generators submit blocks of energy for sale with a price for each block. Loads submit bids to purchase blocks of energy with a price or willingness to pay for each block. The power exchange arranges all the generator sell blocks from lowest price to highest price and arranges the load purchase blocks from highest price down to lowest price. It then starts the process of finding a clearing price as follows: 1) Match the lowest sell price for a generator block to the highest purchase price for a load block. If the selling price of the generation block is below the purchasing price of the block of load then the load block is satisfied. The clearing price is the price of the generation block. 2) Match the next highest sell price for a generator block to the next lowest purchase price for a load block. If the sell price of generation is below the purchase price for load, the load is satisfied and the clearing price for the entire auction now becomes the generation sell price of this most recently sold block of generation. 3) Continue, block by block, until the sell price for a generator block is greater than the purchase price for a load block - then stop. The generation block price for the last match where generator sell price is below load purchase price is the clearing price for the entire auction.

3 EE5721 Practice Final Exam PROBLEM 1 Part B) The Generator Blocks and their prices (same as in part A) and the Load Blocks and their prices are given below: Generator: GA Generator: GB Generator: GC Block Block Price Block Block Price Block Block Price Number MW $/MWh Number MW $/MWh Number MW $/MWh GA GB GC GA GB GC GA GC GA GC GC Load: LA Load: LB Load: LC Block Block Price Block Block Price Block Block Price Number MW $/MWh Number MW $/MWh Number MW $/MWh LA LB LC LA LB LC LB LC LB Data for Problem 1 Part B Note that now both the generation and load blocks are each given an identifier. Find the clearing price for this auction assuming that there are no limits on the power that can be transmitted over the interconnections between zones. Identify which blocks of each generator are sold and which blocks of each load are satisfied. Identify which blocks of load go unsatisfied and explain why. Finally, calculate the flows on the transmission paths from zone 1 to zone 2 and from zone 2 to zone 3. Auctions with Transmission Limits When transmission limits are imposed, the same matching procedure as on the previous pages is used, however, each time a block of generation is matched to a block of load the transmission flows are calculated for the addition of the sale of that block. If a transmission limit is hit, then the matching procedure will only sell as much of that block as will make the transmission path go to its limit. At that point the system is split into two separate auctions - one auction matches unfilled load in the zone(s) on one side of the congested path to unused generation in the same zone(s), the second auction matches unfilled load to unused generation on the remainder of the network located on the other side of the congested path.

4 EE5721 Practice Final Exam Problem 1 Part C Assume that the transmission path from zone 1 to zone 2 is limited to only 100 MW flow in either direction while the transmission path from zone 2 to zone 3 is still unlimited. What is the clearing price in each zone and what are the transmission flows, if the loads are fixed (same block and price data as Problem 1 Part A). Problem 1 Part D Assume again that the transmission path zone 1 to zone 2 is limited to only 100 MW in either direction while the transmission path from zone 2 to zone 3 is still unlimited. What is the clearing price in each zone and what are the transmission flows, if the loads are variable (same block and price data as Problem 1 Part B).

5 Solution for Parts A and B, no transmission limits Part B Solution, with variable load Part A solution, all loads fixed Gen Asking Prices Load Bids Fixed Load Variable Load Cum MW Cum MW Clearing Price Difference Clear Price GB LB GA LA GA LC GB LA GA LB GC LC GA LB GC LB GC * LC GC GC Part C Solution with transmission limits fixed loads Gen Asking Prices Load Blocks Clear Price Transmission GB LB GB LB GB LB GA LA GA LA GA LA GA LA GA LA GA LA GB LB GB LB GB LB GA LB GA LB clear price zone A GA not sold GC LC GC LC GC LC GA not sold GA not sold GA not sold GC LC GC LC GC LC GC LC GC LC GC LC GC LB GC LB GC LB GC LB Clear price zone B&C Final Flows GC not sold GC not sold

6 Part D Solution with transmission limits Variable loads Gen Asking Prices Load Bids Transmission Difference Clearing Price GB LB GB LB GB LB GA LA GA LA GA LA GA LC GA LC Zone A GA LA GA LA GA LA Zone A Clearing Price GA not sold GA not sold GA not sold GA not sold Zones B & C GB LC GB LB GB LB GC LB GC LC GC LC GC LC Zones B&C Clear Price GC not sold* LB not satisfied* GC not sold* LB not satisfied* GC not sold LB not satisfied GC not sold LB not satisfied GC not sold LB not satisfied GC not sold LB not satisfied GC not sold LC not satisfied GC not sold LC not satisfied GC not sold LC not satisfied GC not sold GC not sold * Never said what to do if gen price equals load price, above assumes these not sold, if you assume they are sold then the clear price is still but LB3 gets 100 (not 0) and GC2 sells 150 (not 50).

7 PROBLEM 2 Given the following network: Bus 1 Bus 2 Bus 3 m2 m3 m12 θ 1 = 0 m23 The network is to be modeled with a DC power flow with line reactances as follows: (assume 100 MVA base) x 12 = 0.15 pu x 23 = 0.2 pu The meters are all of the same type with a standard deviation of σ = 0.05 pu for each. The measured values are: m12 = 2.46 pumw m2 = 1.58 pumw m23 =.684 pumw m3 = 1.74 pumw 1) Find the phase angles which result in a best fit to the measured values. 2) Find the value of the residual function J. 3) Calculate estimated generator output the generator, the estimated load at each load, and the estimated power flow on each line. 4) Are there any errors in the measurements? If you think so explain which meters are apt to be in error and why. 5) If in part 4 a measurement error was found, remove that measurement and resolve for the angles, flows, and residual J.

8 H = θ θ = 2 = θ The four measurements calculated using the values of theta estimate are: m12 = m2 = m23 = m3 = These result in a residual ot J = The individual errors for each measurement are: m12 error = m2 error = m23 error = m3 error = Dropping the m3 measurement due to its large error: θ θ = 2 = θ Measured quantities are: m12 = m2 = m23 = The resulting residual is J = The errors are: m12 error = m2 error = m23 error = The large drop in J indicates that this is a much better estimate.

9 PROBLEM 3 Four generators G1, G2, G3 and G4 are located at a large oil refinery. Ggnerators G1 and G2 are connected together on one bus and G3 and G4 are connected to another bus. The buses are connected together with a short tie line. The refinery draws a load of 2000 MW as shown below: P ref GOV P ref GOV G1 G3 P ref GOV P ref G2 GOV G4 LOAD The parameters for the four generators are: G1 and G2: 500MVA, R1 = 0.05 G3: 1000MVA, R2 = 0.02 G4: 200MVA, R3 = 0.01 where the values for R1, R2, R3 and R4 are all in per unit referenced to their generator s MVA as base. When you solve the problem, change all generator MW s, the load MW and the R values to a common system base of 100 MVA. At time t = 0 all four generators are supplying power to the load and the frequency is at 60 Hz. Assume the generators supply the following: P1 and P2 = 450 MW, P3 = 950 MW and P4 = 150 MW. Just after t=0 the load drops suddenly from 2000 to 1500 MW. Assuming no Automatic Generation Control action (no LFC) - only governor action. a) What is the steady state frequency reached after the load drop? b) How much power flows on the tie line and in what direction? b) What is the new MW output of each generator after the frequency reaches a stable value? Now assume that G1 and G2 and their bus are controlled by an Automatic Generation Control (AGC) system. What is the Area Control Error (ACE) when the frequency stabilizes (assume = 1000MW/Hz and that the tie schedule is 1100 MW. B f

10 Convert to common base: Rnew 1 = ( 0.05) = 0.01 Similarly: R2=0.01, R3 = 0.002, and R4 = β = = β f = load f = pu = 0.333Hz new frequency is pu = Hz P -1 1 = f = -100*(.00555)= pu=-55.5MW R 1 Similarly: delp2=-55.5 MW, delp3=-277mw, delp4 = -111 MW

11 New Generation is: Pnew 1 = = and P2new = 394.5, P3new=673, and P4new = 39 the tie flow goes to 712 MW from right to left. By the convention of interchange being positive going out, this means the schedule is now negative 712 MW and the original schedule was negative 1100 MW ACE = ( (-1100))-1000( 0.333) = 55