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systems analysis design

Questions and Answers of Systems Analysis Design

For the three-phase power system of Fig. 1.10, the following specifications have been given. Determine the voltage drop of the line in percent:Line : \(\mathbf{Z}=(10+j 40) \Omega /\) phase Load :
In the power system of Fig. 1.11, calculate the impedance of the load in per unit (p.u.) for the following specifications. In this problem, \(20 \mathrm{kV}\) (in the generator side) and 3 MVA are
In the single-phase power bus of Fig. 1.12, the characteristics of the loads are as follows. Determine the total power factor of the bus:\[\begin{gathered}\text { Load } 1: P_{1}=25 k W, Q_{1}=25 k V
In the single-phase power bus of Fig. 1.13, determine the capacitance of the shunt capacitor that needs to be connected to the bus to adjust its power factor at one for the following
In the single-phase power system of Fig. 1.14, three loads have been connected to the power bus in parallel. Determine the capacitance of the shunt capacitor that needs to be connected to the bus to
In the power system of Fig. 1.15, determine the reactive power of the shunt capacitor to keep the voltage of its bus at 1 p.u. In this problem, assume that \(\cos \left(\sin ^{-1}(0.1)ight) \equiv
In the three-phase power system of Fig. 1.16, two balanced three-phase loads with the star and delta connections have been connected to a three-phase power supply. Calculate the line voltage of the
In the power system of Fig. 1.17, \(\delta=15^{\circ}\). If the value of \(\delta\) increases and \(E_{1}\) and \(E_{2}\) are kept constant, which one of the following choices is correct? In this
Three loads with the following specifications, resulted from the load flow simulation, have been connected to the power bus shown in Fig. 1.18. If all the loads are modeled by an admittance,
At the end of a three-phase power system, \(400 \mathrm{~V}, 50 \mathrm{~Hz}\), three capacitor banks (with triangle configuration) have been connected to the system. Determine the capacitance of
The single-line diagram of a balanced three-phase power system is shown in Fig. 1.19. In this problem \(S_{B}=100 M V A\) and \(V_{B}=22 \mathrm{kV}\) in the first bus. Calculate the impedance seen
In the power system of Fig. 1.20, calculate the current of the load in per unit (p.u.) for the following specifications. In this problem, \(100 \mathrm{~V}\) (in the generator side) and \(1
For a set of linear algebraic equations in matrix format, \(\mathbf{A x}-\mathbf{y}\), for a unique solution to exist, \(\operatorname{det}(\mathbf{A})\) should be ___________.
For an \(N \times N\) square matrix \(\mathbf{A}\), in \((N-1)\) steps, the technique of Gauss elimination can transform into an ___________ matrix.
For the iterative solution to linear algebraic equations \(\mathbf{A x}-\mathbf{y}\), the \(\mathbf{D}\) matrix in the Jacobi method is the Gauss-Siedel is the __________ portion of \(\mathbf{A}\).
Is convergence guaranteed always with Jacobi and Gauss-Siedel methods, as applied to iterative solutions of linear algebraic equations?(a) Yes(b) No
For the iterative solutions to nonlinear algebraic equations with the Newton-Raphson method, the Jacobian matrix \(\mathbf{J}(i)\) consists of the partial derivatives. Write down the elements of
For the Newton-Raphson method to work, one should make sure that \(\mathbf{J}^{-1}\) exists.(a) True(b) False
The Newton-Raphson method in four steps makes use of Gauss elimination and back substitution.(a) True(b) False
The number of iterations required for convergence is dependent/independent of the dimension \(N\) for Newton-Raphson method. Choose one.
The swing bus or slack bus is a reference bus for which \(\mathrm{V}_{1} / \delta_{1}\), typically \(1.0 \angle 0^{\circ}\) per unit, is input data. The power flow program computes __________. Fill
Most buses in a typical power flow program are load buses, for which \(\mathrm{P}_{k}\) and \(\mathrm{Q}_{k}\) are input data. The power flow program computes __________.
For a voltage-controlled bus \(k\), __________ are input data, while the power flow program computes __________.
When the bus \(k\) is a load bus with no generation and inductive load, in terms of generation and load, \(\mathrm{P}_{k}=\) __________, and \(\mathrm{Q}_{k}=\) __________.
Starting from a single-line diagram of a power system, the input data for a power flow problem consists of __________, __________, and __________.
Nodal equations \(\boldsymbol{I}=\boldsymbol{Y}_{\text {bus }} \boldsymbol{V}\) are a set of linear equations analogous to \(\boldsymbol{y}=\boldsymbol{A x}\).(a) True(b) False
Because of the nature of the power flow bus data, nodal equations do not directly fit the linear-equation format, and power flow equations are actually nonlinear. However, the Gauss-Siedel method can
The Newton-Raphson method is most well suited for solving the nonlinear power flow equations.(a) True(b) False
By default, PowerWorld Simulator uses __________ method for the power flow solution.
Prime-mover control of a generator is responsible for a significant change in __________, whereas excitation control significantly changes __________.
From the power flow standpoint, the addition of a shunt-capacitor bank to a load bus corresponds to the addition of a positive/negative reactive load. Choose the right word.
Tap-changing and voltage-magnitude-regulating transformers are used to control bus voltages and reactive power flows on lines to which they are connected.(a) True(b) False
A matrix, which has only a few nonzero elements, is said to be __________.
Sparse-matrix techniques are used in Newton-Raphson power flow programs in order to reduce computer __________ and __________ requirements.
Reordering buses can be an effective sparsity technique in power flow solutions.(a) True(b) False
While the fast decoupled power flow usually takes more iterations to converge, it is usually significantly faster than the Newton-Raphson method.(a) True(b) False
The "dc" power flow solution, giving approximate answers, is based on completely neglecting the \(\mathrm{Q}-\mathrm{V}\) equation and solving the linear real-power balance equations.(a) True(b) False
Using Gauss elimination, solve the following linear algebraic equations:\[\begin{aligned}-25 x_{1}+10 x_{2}+10 x_{3}+10 x_{4} & =0 \\5 x_{1}-10 x_{2}+10 x_{3} & =2 \\10 x_{1}+5 x_{2}-10 x_{3}+10
Using Gauss elimination and back substitution, solve 8 2 1 4 6 2 3 4 14 X X 3 4 2
Rework Problem 6.2 with the value of 8 changed to 4.Problem 6.2Using Gauss elimination and back substitution, solve 8 2 1 4 6 2 3 4 14 X X 3 4 2
What is the difficulty in applying Gauss elimination to the following linear algebraic equations?\[\begin{aligned}-5 x_{1}+5 x_{2} & =5 \\10 x_{1}-10 x_{2} & =-5\end{aligned}\]
Show that, after triangularizing \(\mathbf{A x}=\mathbf{y}\), the back substitution method of solving \(\mathbf{A}^{(N-1)} \mathbf{x}=\mathbf{y}^{(N-1)}\) requires \(N\) divisions, \(N(N-1) / 2\)
Solve Problem 6.2 using the Jacobi iterative method. Start with \(x_{1}(0)=\) \(x_{2}(0)=x_{3}(0)=0\), and continue until (6.2.2) is satisfied with \(\varepsilon=0.01\).Eq (6.2.2)Problem 6.2Using
Repeat Problem 6.6 using the Gauss-Seidel iterative method. Which method converges more rapidly?Problem 6.6Using the Jacobi iterative method. Start with \(x_{1}(0)=\) \(x_{2}(0)=x_{3}(0)=0\), and
Express the following set of equations in the form of (6.2.6), and then solve using the Jacobi iterative method with \(\varepsilon=0.05\) and with \(x_{1}(0)=1\), and \(x_{2}(0)=1, x_{3}(0)=0\).Eq
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

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