Expand the summations in (12.4.14) for N _ 2, and verify

Open PowerWorld Simulator case Problem 12_3. This case models the system from Example 12.1 except with the rate feedback gain constant, Kf , has been set to zero and the simulation end time was increased to 30 seconds. Without rate feedback the system voltage response will become unstable if the amplifier gain, Ka, becomes too large. In the simulation this instability will be indicated by undamped oscillations in the terminal voltage (because of the limits on Vr the response does not grow to in- finity but rather bounces between the limits). Using transient stability simulations, iteratively determine the approximate value of Ka at which the system becomes unstable. The value of Ka can be on the Generator Information Dialog, Stability, Exciters page.

One of the disadvantages of the IEEET1 exciter is following a fault the terminal voltage does not necessarily return to its pre-fault value. Using PowerWorld Simulator case Problem 12_3 determine the pre-fault bus 4 terminal voltage and field voltage. Then use the simulation to determine the final, post-fault values for these fields for Ka _ 100, 200, 50, and 10. Referring to Figure 12.3, what is the relationship between the reference voltage, and the steady-state terminal voltage and the field voltage?

An area of an interconnected 60-Hz power system has three turbine-generator units rated 200, 300, and 500 MVA. The regulation constants of the units are 0.03, 0.04, and 0.06 per unit, respectively, based on their ratings. Each unit is initially operating at one-half its own rating when the load suddenly decreases by 150 MW. Determine (a) the unit area frequency response characteristic b on a 100- MVA base, (b) the steady-state increase in area frequency, and (c) the MW decrease in mechanical power output of each turbine. Assume that the reference power setting of each turbine-governor remains constant. Neglect losses and the dependence of load on frequency.

Each unit in Problem 12.5 is initially operating at one-half its own rating when the load suddenly increases by 100 MW. Determine (a) the steady-state decrease in area frequency, and (b) the MW increase in mechanical power output of each turbine. Assume that the reference power setting of each turbine-generator remains constant. Neglect losses and the dependence of load on frequency.

Each unit in Problem 12.5 is initially operating at one-half its own rating when the frequency increases by 0.005 per unit. Determine the MW decrease of each unit. The reference power setting of each turbine-governor is fixed. Neglect losses and the dependence of load on frequency.

Repeat Problem 12.7 if the frequency decreases by 0.005 per unit. Determine the MW increase of each unit.

An interconnected 60-Hz power system consisting of one area has two turbinegenerator units, rated 500 and 750 MVA, with regulation constants of 0.04 and 0.05 per unit, respectively, based on their respective ratings. When each unit carries a 300-MVA steady-state load, let the area load suddenly increase by 250 MVA. (a) Compute the area frequency response characteristic b on a 1000-MVA base. (b) Calculate Df in per-unit on a 60-Hz base and in Hz.

Open PowerWorld Simulator case Problem 12_10. The case models the system from Example 12.4 except 1) the load increases is a 50% rise at bus 6 for a total increase of 250 MW (from 500 MW to 750 MW), 2) the value of R for generator 1 is changed from 0.05 to 0.04 per unit. Repeat Example 12.4 using these modified values.

Open PowerWorld Simulator case Problem 12_11, which includes a transient stability representation of the system from Example 6.13. Each generator is modeled using a two-axis machine model, an IEEE Type 1 exciter and a TGOV1 governor with R _ 0:05 per unit (a summary of the generator models is available by selecting either Stability Case Info, Transient Stability Generator Summary which includes the generator MVA base, or Stability Case Info, Transient Stability Case Summary). The contingency is the loss of the generator at bus 50, which initially has 42.1 MW of generation. Analytically determine the steady-state frequency error in Hz following this contingency. Use PowerWorld Simulator to confirm this result; also determine the magnitude and time of the largest bus frequency deviation.

Repeat Problem 12.12 except first double the H value for each of the machines. This can be most easily accomplished by selecting Stability Case Info, Transient Stability Case Summary to view the summary form. Right click on the line corresponding to the Machine Model class, and then select Show Dialog to view an editable form of the model parameters. Compare the magnitude and time of the largest bus frequency deviations between Problem 12.12 and 12.11

For a large, 60 Hz, interconnected electrical system assume that following the loss of two 1400 MW generators (for a total generation loss of 2800 MW) the change in frequency is 0:12 Hz. If all the on- line generators that are available to participate in frequency regulation have an R of 0.05 per unit (on their own MVA base), estimate the total MVA rating of these units.

A 60-Hz power system consists of two interconnected areas. Area 1 has 1200 MW of generation and an area frequency response characteristic b1 _ 600 MW/Hz. Area 2 has 1800 MW of generation and b2 _ 800 MW/Hz. Each area is initially operating at one-half its total generation, at Dptie1 _ Dptie2 _ 0 and at 60 Hz, when the load in area 1 suddenly increases by 400 MW. Determine the steady-state frequency error and the steady-state tie-line error Dptie of each area. Assume that the reference power settings of all turbine-governors are fixed. That is, LFC is not employed in any area. Neglect losses and the dependence of load on frequency.

Repeat Problem 12.14 if LFC is employed in area 2 alone. The area 2 frequency bias coecient is set at Bf 2 _ b2 _ 800 MW/Hz. Assume that LFC in area 1 is inoperative due to a computer failure.

Repeat Problem 12.14 if LFC is employed in both areas. The frequency bias coecients are Bf 1 _ b1 _ 600 MW/Hz and Bf 2 _ b2 _ 800 MW/Hz.

Rework Problems 12.15 through 12.16 when the load in area 2 suddenly decreases by 300 MW. The load in area 1 does not change.

On a 1000-MVA common base, a two-area system interconnected by a tie line has the following parameters: Area 1 2 Area Frequency Response Characteristic b1 _ 0:05 per unit b2 _ 0:0625 per unit Frequency-Dependent Load Coecient D1 _ 0:6 per unit D2 _ 0:9 per unit Base Power 1000 MVA 1000 MVA Governor Time Constant tg1 _ 0:25 s tg2 _ 0:3 s Turbine Time Constant tt1 _ 0:5 s tt2 _ 0:6 s The two areas are operating in parallel at the nominal frequency of 60 Hz. The areas are initially operating in steady state with each area supplying 1000 MW when a sudden load change of 187.5 MW occurs in area 1. Compute the new steady-state frequency and change in tie-line power flow (a) without LFC, and (b) with LFC.

The fuel-cost curves for two generators are given as follows: C1P1_ _ 600 _ 15 ? P1 _ 0:05 ? P1_ 2 C2P2_ _ 700 _ 20 ? P2 _ 0:04 ? P2_ 2 Assuming the system is lossless, calculate the optimal dispatch values of P1 and P2 for a total load of 1000 MW, the incremental operating cost, and the total operating cost.

Rework problem 12.19 except assume that the limit outputs are subject to the following inequality constraints: 200 a P1 a 800 MW 100 a P2 a 500 MW

Rework problem 12.19 except assume the 1000 MW value also includes losses, and that the penalty factor for the first unit is 1.0, and for the second unit 0.95.

The fuel-cost curves for a two generators power system are given as follows: C1P1_ _ 600 _ 15 ? P1 _ 0:05 ? P1_ 2 C2P2_ _ 700 _ 20 ? P2 _ 0:04 ? P2_ 2 While the system losses can be approximated as PL _ 2 104 P1_ 2 _ 3 104 P2_ 2 4 104 P1P2 MW 686 C If the system is operating with a marginal cost (l) of $60/hr, determine the output of each unit, the total transmission losses, the total load demand, and the total operating cost.

Expand the summations in (12.4.14) for N _ 2, and verify the formula for qPL=qPi given by (12.4.15). Assume Bij _ Bji

Given two generating units with their respective variable operating costs as: C1 _ 0:01P2 G1 _ 2PG1 _ 100 $=hr for 25 aPG1 a150 MW C2 _ 0:004P2 G2 _ 2:6PG2 _ 80 $=hr for 30a PG2 a 200 MW Determine the economically optimum division of generation for 55 aPL a350 MW. In particular, for PL _ 282 MW, compute PG1 and PG2. Neglect transmission losses

Resolve Example 12.8, except with the generation at bus 2 set to a fixed value (i.e., modeled as o_ of AGC). Plot the variation in the total hourly cost as the generation at bus 2 is varied between 1000 and 200 MW in 5-MW steps, resolving the economic dispatch at each step. What is the relationship between bus 2 generation at the minimum point on this plot and the value from economic dispatch in Example 12.8? Assume a Load Scalar of 1.0.

Using PowerWorld case Example 12_10, with the Load Scalar equal to 1.0, determine the generation dispatch that minimizes system losses (Hint: Manually vary the generation at buses 2 and 4 until their loss sensitivity values are zero). Compare the operating cost between this solution and the Example 12.10 economic dispatch result. Which is better?

Repeat Problem 12.26, except with the Load Scalar equal to 1.4.

Using LP OPF with PowerWorld Simulator case Example 12_11, plot the variation in the bus 5 marginal price as the Load Scalar is increased from 1.0 in steps of 0.02. What is the maximum possible load scalar without overloading any transmission line? Why is it impossible to operate without violations above this value?

Load PowerWorld Simulator case Problem 12_30. This case models a slightly modi- fied version of the 37 bus case from Example 6.13 with generator cost information, but also with two of the three lines between buses BLT69 and UIUC69 open. When the case is loaded the Total Cost field shows the economic dispatch solution, which results in an overload on the remaining line between buses BLT69 and UIUC69. Before solving the case, select LP OPF, OFP Buses to view the bus LMPs, noting that they are all identical. Then Select LP OPF, Primal LP to solve the case using the OPF, and again view the bus LMPs. Verify the LMP at the UIUC69 bus by manually changing the load at the bus by one MW, and then noting the change in the Total Cost field. Repeat for the DEMAR69 bus. Note, because of solution convergence tolerances the manually calculated results may not exactly match the OFP calculated bus LMPs.