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systems analysis design

Questions and Answers of Systems Analysis Design

Solve for \(x_{1}\) and \(x_{2}\) in the system of equations given by\[\begin{array}{r}x_{2}-3 x_{1}+1.9=0 \\x_{2}+x_{1}^{2}-3.0=0\end{array}\]using the Gauss method with an initial guess of
Solve \(x^{2}-4 x+1=0\) using the Jacobi iterative method with \(x(0)=1\). Continue until (6.2.2) is satisfied with \(\varepsilon=0.01\). Check using the quadratic formula.
Try to solve Problem 6.2 using the Jacobi and Gauss-Seidel iterative methods with the value of \(\mathrm{A}_{33}\) changed from 14 to 0.14 and with \(x_{1}(0)=x_{2}(0)=\) \(x_{3}(0)=0\). Show that
Using the Jacobi method (also known as the Gauss method), solve for \(x_{1}\) and \(x_{2}\) in the following system of equations.\[\begin{array}{r}x_{2}-3 x_{1}+1.9=0
Use the Gauss-Seidel method to solve the following equations that contain terms that are often found in power flow equations.\[\begin{gathered}x_{1}=(1 /(-20 j)) *\left[(-1+0.5 j)
Find a root of the following equation by using the Gauss-Seidel method: (use an initial estimate of \(x=2\) ) \(f(x)=x^{3}-6 x^{2}+9 x-4=0\).
Use the Jacobi method to find a solution to \(x^{2} \cos x-x+0.5=0\). Use \(x(0)=1\) and \(\varepsilon=0.01\). Experimentally determine the range of initial values that results in convergence.
Determine the poles of the Jacobi and Gauss-Seidel digital filters for the general two-dimensional problem (N=2)(N=2) :Then determine a necessary and sufficient condition for convergence of these
Use Newton-Raphson to find a solution to the polynomial equation \(f(x)=y\) where \(y=0\) and \(f(x)=x^{3}+8 x^{2}+2 x-40\). Start with \(x(0)=1\) and continue until (6.2.2) is satisfied with
Repeat 6.18 using \(x(0)=-2\).Problem 6.18Use Newton-Raphson to find a solution to the polynomial equation \(f(x)=y\) where \(y=0\) and \(f(x)=x^{3}+8 x^{2}+2 x-40\). Start with \(x(0)=1\) and
Use Newton-Raphson to find one solution to the polynomial equation \(f(x)=y\), where \(y=7\) and \(f(x)=x^{4}+3 x^{3}-15 x^{2}-19 x+30\). Start with \(x(0)=0\) and continue until (6.2.2) is satisfied
Repeat Problem 6.20 with an initial guess of \(x(0)=4\).Problem 6.20Use Newton-Raphson to find one solution to the polynomial equation \(f(x)=y\), where \(y=7\) and \(f(x)=x^{4}+3 x^{3}-15 x^{2}-19
For Problem 6.20, plot the function \(f(x)\) between \(x=0\) and 4. Then provide a graphical interpretation why points close to \(x=2.2\) would be poorer initial guesses.Problem 6.20Use
Use Newton-Raphson to find a solution towhere \(x_{1}\) and \(x_{2}\) are in radians.(a) Start with \(x_{1}(0)=1.0\) and \(x_{2}(0)=\) 0.5 and continue until (6.2.2) is satisfied with
Solve the following equations by the Newton-Raphson method:\[\begin{array}{r}2 x_{1}+x_{2}^{2}-8=0 \\x_{1}^{2}-x_{2}^{2}+x_{1} x_{2}-3=0\end{array}\]Start with an initial guess of \(x_{1}=1\) and
The following nonlinear equations contain terms that are often found in the power flow equations:\[\begin{aligned}& f_{1}(x)=10 x_{1} \sin x_{2}+2=0 \\& f_{2}(x)=10\left(x_{1}ight)^{2}-10 x_{1} \cos
Repeat Problem 6.25 except using x1(0)=0.25x1(0)=0.25 and x2(0)=0x2(0)=0 radians as an initial guess.Problem 6.25The following nonlinear equations contain terms that are often found in the power flow
For the Newton-Raphson method, the region of attraction (or basin of attraction) for a particular solution is the set of all initial guesses that converge to that solution. Usually initial guesses
Consider the simplified electric power system shown in Figure 6.22 for which the power flow solution can be obtained without resorting to iterative techniques. (a) Compute the elements of the bus
In Example 6.9, double the impedance on the line from bus 2 to bus 5 . Determine the new values for the second row of \(\boldsymbol{Y}_{\text {bus }}\). Verify your result using PowerWorld Simulator
Determine the bus admittance matrix ( \(\boldsymbol{Y}_{\text {bus }}\) ) for the three-phase power system shown in Figure 6.23 with input data given in Table 6.11 and partial results in Table 6.12.
For the system from Problem 6.30, assume that a 75-Mvar shunt capacitance (three phase assuming one per unit bus voltage) is added at bus 4 . Calculate the new value of \(\mathrm{Y}_{44}\).
For a two-bus power system, a \(0.7+j 0.4 \) per unit load at bus 2 is supplied by a generator at bus 1 through a transmission line with series impedance of \(0.05+j 0.1 \) per unit. With bus 1 as
Repeat Problem 6.32 with the slack bus voltage changed to \(1.0 \angle 30^{\circ}\) per unit.Problem 6.32For a two-bus power system, a \(0.7+j 0.4 \) per unit load at bus 2 is supplied by a generator
For the three-bus system whose \(\boldsymbol{Y}_{\text {bus }}\) is given, calculate the second iteration value of \(\mathrm{V}_{3}\) using the Gauss-Seidel method. Assume bus 1 as the slack (with
Repeat Problem 6.34 except assume the bus 1 (slack bus) voltage of \(V_{1}=\) \(1.05 \angle 0^{\circ}\).Problem 6.34For the three-bus system whose \(\boldsymbol{Y}_{\text {bus }}\) is given,
The bus admittance matrix for the power system shown in Figure 6.24 is given byWith the complex powers on load buses 2, 3, and 4 as shown in Figure 6.24, determine the value for \(\mathrm{V}_{2}\)
The bus admittance matrix of a three-bus power system is given bywith \(\mathrm{V}_{1}=1.0 \angle 0^{\circ}\) per unit; \(\mathrm{V}_{2}=1.0\) per unit; \(\mathrm{P}_{2}=60 \mathrm{MW} ;
A generator bus (with a 1.0 per unit voltage) supplies a \(180 \mathrm{MW}, 60 \mathrm{Mvar}\) load through a lossless transmission line with per unit (100 MVA base) impedance of \(j 0.1 \) and no
Repeat Problem 6.38 except use an initial voltage guess of \(1.0 / 30^{\circ}\).Problem 6.38A generator bus (with a 1.0 per unit voltage) supplies a \(180 \mathrm{MW}, 60 \mathrm{Mvar}\) load through
Repeat Problem 6.38 except use an initial voltage guess of \(0.25 \angle 0^{\circ}\).Problem 6.38A generator bus (with a 1.0 per unit voltage) supplies a \(180 \mathrm{MW}, 60 \mathrm{Mvar}\) load
Determine the initial Jacobian matrix for the power system described in Problem 6.34.Problem 6.34For the three-bus system whose \(\boldsymbol{Y}_{\text {bus }}\) is given, calculate the second
Use the Newton-Raphson power flow to solve the power system described in Problem 6.34. For convergence criteria, use a maximum power flow mismatch of 0.1 MVA.Problem 6.34For the three-bus system
For a three-bus power system, assume bus 1 is the slack with a per unit voltage of 1.0∠0∘1.0∠0∘, bus 2 is a PQ bus with a per unit load of \(2.0+j 0.5 \), and bus 3 is a PV bus with 1.0 per
Repeat Problem 6.43 except with the bus 2 real power load changed to 1.0 per unit.Problem 6.43or a three-bus power system, assume bus 1 is the slack with a per unit voltage of 1.0∠0∘1.0∠0∘,
Load PowerWorld Simulator case Example 6_11; this case is set to perform a single iteration of the Newton-Raphson power flow each time Single Solution is selected. Verify that initially the Jacobian
Load PowerWorld Simulator case Problem 6_46. Using a 100 MVA base, each of the three transmission lines have an impedance of \(0.05+j 0.1 \) p.u. There is a single \(180 \mathrm{MW}\) load at bus 3,
As was mentioned in Section 6.4, if a generator's reactive power output reaches its limit, then it is modeled as though it were a PQ bus. Repeat Problem 6.46, except assume the generator at bus 2 is
Load PowerWorld Simulator case Problem 6_46. Plot the reactive power output of the generator at bus 2 as a function of its voltage setpoint value in 0.005 p.u. voltage steps over the range between
Open PowerWorld Simulator case Problem 6_49. This case is identical to Example 6.9, except that the transformer between buses 1 and 5 is now a tap-changing transformer with a tap range between 0.9
Use PowerWorld Simulator to determine the Mvar rating of the shunt capacitor bank in the Example 6_14 case that increases \(\mathrm{V}_{2}\) to 1.0 per unit. Also determine the effect of this
Use PowerWorld Simulator to modify the Example 6_9 case by inserting a second line between bus 2 and bus 5. Give the new line a circuit identifier of " 2 " to distinguish it from the existing line.
Open PowerWorld Simulator case Problem 6_52. Open the \(69 \mathrm{kV}\) line between buses REDBUD69 and PEACH69 (shown towards the bottom of the oneline). With the line open, determine the amount of
Open PowerWorld Simulator case Problem 6_53. Plot the variation in the total system real power losses as the generation at bus PEAR 138 is varied in \(20 \mathrm{MW}\) blocks between \(0
Repeat Problem 6.53, except first remove the \(69 \mathrm{kV}\) line between LOCUST69 and PEAR69.Problem 6.53Open PowerWorld Simulator case Problem 6_53. Plot the variation in the total system real
Using the compact storage technique described in Section 6.8, determine the vectors DIAG, OFFDIAG, COL, and ROW for the following matrix: S= 17 -9.1 25 0 -8.1 0 -1.1 -6.1 -9.1 -2.1 -7.1 0 0 -8.1 9 0
For the triangular factorization of the corresponding \(\boldsymbol{Y}_{\text {bus }}\), number the nodes of the graph shown in Figure 6.9 in an optimal order.Figure 6.9 385 MW -37 Mvar slack One 65%
Compare the angles and line flows between the Example 6_17 case and results shown in Tables 6.6, 6.7, and 6.8.Table 6.6Table 6.7Table 6.8Example 6_17Determine the dc power flow solution for the five
Redo Example 6.17 with the assumption that the per-unit reactance on the line between buses 2 and 5 is changed from 0.05 to 0.03.Example 6.17Determine the dc power flow solution for the five bus
Open PowerWorld Simulator case Problem 6_59, which models a seven-bus system using the de power flow approximation. Bus 7 is the system slack. The real power generation/load at each bus is as shown,
Using the PowerWorld Simulator case from Problem 6.59, if the rating on the line between buses 1 and 2 is \(150 \mathrm{MW}\), the current flow is \(101 \mathrm{MW}\) (from bus 1 to bus 3 ), and the
PowerWorld Simulator cases Problem 6_61_PQ and 6_61_PV model a sevenbus power system in which the generation at bus 4 is modeled as a Type 1 or 2 wind turbine in the first case and as a Type 3 or 4
The fuel-cost curves for two generators are given as follows:\[\begin{aligned}& \mathrm{C}_{1}\left(\mathrm{P}_{1}ight)=600+18 \cdot \mathrm{P}_{1}+0.04 \cdot\left(\mathrm{P}_{1}ight)^{2} \\&
Rework Problem 6.62, except assume that the limit outputs are subject to the following inequality constraints:\[\begin{aligned}& 200 \leq \mathrm{P}_{1} \leq 800 \mathrm{MW} \\& 100 \leq
Rework Problem 6.62, except assume the \(1000 \mathrm{MW}\) value also includes losses, and the penalty factor for the first unit is 1.0 and for the second unit 0.95.Problem 6.62The fuel-cost curves
The fuel-cost curves for a two-generator power system are given as follows:\[\begin{aligned}& \mathrm{C}_{1}\left(\mathrm{P}_{1}ight)=600+15 \cdot \mathrm{P}_{1}+0.05
Expand the summations in (6.12.14) for \(N=2\), and verify the formula for \(\partial \mathrm{P}_{\mathrm{L}} / \partial \mathrm{P}_{i}\) given by (6.12.15). Assume \(\mathrm{B}_{i j}=\mathrm{B}_{j
Given two generating units with their respective variable operating costs as\[\begin{array}{ll}\mathrm{C}_{1}=0.01 \mathrm{P}_{\mathrm{G} 1}^{2}+2 \mathrm{P}_{\mathrm{G} 1}+100 \$ / \mathrm{hr} &
Resolve Example 6.20, except with the generation at bus 2 set to a fixed value (i.e., modeled as off of Automatic Generation Control). Plot the variation in the total hourly cost as the generation at
Using PowerWorld case Example 6_22 with the Load Scalar equal to 1.0, determine the generation dispatch that minimizes system losses. Manually vary the generation at buses 2 and 4 until their loss
Repeat Problem 6.69, except with the Load Scalar equal to 1.4.Problem 6.69Using PowerWorld case Example 6_22 with the Load Scalar equal to 1.0, determine the generation dispatch that minimizes system
Using LP OPF with PowerWorld Simulator case Example 6_23, 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
Load PowerWorld Simulator case Problem 6_72. This case models a slightly modified, lossless version of the 37-bus case from Example 6.13 with generator cost information, but also with the transformer
The asymmetrical short-circuit current in series \(\mathrm{R}-\mathrm{L}\) circuit for a simulated solid or "bolted fault" can be considered as a combination of symmetrical (ac) component that is a
Even though the fault current is not symmetrical and not strictly periodic, the rms asymmetrical fault current is computed as the rms ac fault current times an "asymmetry factor," which is a function
The amplitude of the sinusoidal symmetrical ac component of the three-phase short-circuit current of an unloaded synchronous machine decreases from a high initial value to a lower steady-state value,
The duration of subtransient fault current is dictated by time ____________ constant and that of transient fault current is dictated by time ____________ constant.
The reactance that plays a role under steady-state operation of a synchronous machine is called ____________.
The dc-offset component of the three-phase short-circuit current of an unloaded synchronous machine is different in the three phases and its exponential decay is dictated by ____________.
Generally, in power-system short-circuit studies, for calculating subtransient fault currents, transformers are represented by their ____________ transmission lines by their equivalent ____________
In power-system fault studies, all nonrotating impedance loads are usually neglected.(a) True(b) False
Can superposition be applied in power-system short-circuit studies for calculating fault currents?(a) Yes(b) No
Before proceeding with per-unit fault current calculations, based on the single-line diagram of the power system, a positive-sequence equivalent circuit is set up on a chosen base system.(a) True(b)
The inverse of the bus-admittance matrix is called a ____________ matrix.
For a power system, modeled by its positive-sequence network, both busadmittance matrix and bus-impedance matrix are symmetric.(a) True(b) False
The bus-impedance equivalent circuit can be represented in the form of a "rake" with the diagonal elements, which are ____________ and the non-diagonal (off-diagonal) elements, which are ____________.
A circuit breaker is designed to extinguish the arc by ____________.
Power-circuit breakers are intended for service in the ac circuit above ____________ V.
In circuit breakers, besides air or vacuum, what gaseous medium, in which the arc is elongated, is used?
Oil can be used as a medium to extinguish the arc in circuit breakers.(a) True(b) False
Besides a blast of air/gas, the arc in a circuit breaker can be elongated by ____________.
For distribution systems, standard reclosers are equipped for two or more reclosures, whereas multiple-shot reclosing in EHV systems is not a standard practice.(a) True(b) False
Breakers of the \(115 \mathrm{kV}\) class and higher have a voltage range factor \(\mathrm{K}=\) ____________, such that their symmetrical interrupting current capability remains constant.
A typical fusible link metal in fuses is ____________, and a typical filler material is
The melting and clearing time of a current-limiting fuse is usually specified by a ____________ curve.
In the circuit of Figure 7.1, \(\mathrm{V}=277\) volts, \(\mathrm{L}=2 \mathrm{mH}, \mathrm{R}=0.4 \Omega\), and \(\omega=2 \pi 60 \mathrm{rad} / \mathrm{s}\). Determine (a) the rms symmetrical
Repeat Example 7.1 with \(\mathrm{V}=4 \mathrm{kV}, \mathrm{X}=2 \Omega\), and \(\mathrm{R}=1 \Omega\)Example 7.1A bolted short circuit occurs in the series R–L circuit of Figure 7.1 with V = 20
In the circuit of Figure 7.1, let \(\mathrm{R}=0.125 \Omega ., \mathrm{L}=10 \mathrm{mH}\), and the source voltage is \(e(\mathrm{t})=151 \sin (377 \mathrm{t}+\alpha) \mathrm{V}\). Determine the
Consider the expression for \(i(t)\) given by\[i(t)=\sqrt{2} \mathrm{I}_{\mathrm{rms}}\left[\sin \left(\omega t-\theta_{z}ight)+\sin \theta_{z} \cdot e^{-(\omega R / X) t}ight]\]where
If the source impedance at a \(13.2-\mathrm{kV}\) distribution substation bus is \((0.5+\) j1.5) \(\Omega\) per phase, compute the rms and maximum peak instantaneous value of the fault current for a
A 1000-MVA, \(20-\mathrm{kV}, 60-\mathrm{Hz}\), three-phase generator is connected through a \(1000-\mathrm{MVA}, 20-\mathrm{kV}, \Delta / 345-\mathrm{kV}\), Y transformer to a \(345-\mathrm{kV}\)
For Problem 7.6, determine(a) the instantaneous symmetrical fault current in kA in phase \(a\) of the generator as a function of time, assuming maximum dc offset occurs in this generator phase,
A 300-MVA, 13.8-kV, three-phase, \(60-\mathrm{Hz}\), Y-connected synchronous generator is adjusted to produce rated voltage on open circuit. A balanced three-phase fault is applied to the terminals
Two identical synchronous machines, each rated \(60 \mathrm{MVA}\) and \(15 \mathrm{kV}\) with a subtransient reactance of 0.1 p.u., are connected through a line of reactance 0.1 p.u. on the base of
Recalculate the subtransient current through the breaker in Problem 7.6 if the generator is initially delivering rated MVA at 0.80 p.f. lagging and at rated terminal voltage.Problem 7.6A 1000-MVA,
Solve Example 7.3 parts (a) and (c) without using the superposition principle. First calculate the internal machine voltages \(E_{g}^{\prime \prime}\) and \(E_{m}^{\prime \prime}\) using the prefault
Equipment ratings for the four-bus power system shown in Figure 7.14 are as follows: A three-phase short circuit occurs at bus 1, where the prefault voltage is \(525 \mathrm{kV}\). Prefault load
For the power system given in Problem 7.12, a three-phase short circuit occurs at bus 2, where the prefault voltage is \(525 \mathrm{kV}\). Prefault load current is neglected. Determine the(a)
Equipment ratings for the five-bus power system shown in Figure 7.15 are as follows:A three-phase short circuit occurs at bus 5, where the prefault voltage is 15kV15kV. Prefault load current is
For the power system given in Problem 7.14, a three-phase short circuit occurs at bus 4 , where the prefault voltage is \(138 \mathrm{kV}\). Prefault load current is neglected. Determine(a) the

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