CHAPTER 5
Power Flow Studies—2
5.1 Introduction
This chapter is a continuation of Chapter 4 and discusses power flow solution by the Newton–Raphson (NR), and the decoupled and fastdecoupled methods. Though the Gauss–Seidel method is computationally much easier, it has limitations when applied to largesized power systems involving more number of unknowns. The methods presented in this chapter are useful for such systems. NR method is very accurate when compared to other methods and guarantees convergence in five to seven iterations irrespective of the size of power system. We compare the NR method with the GS method towards the end of this chapter to appreciate the difference between the two types of power flow solutions. Since the NR method is computationally difficult, the method is simplified with suitable assumptions, leading to the decoupled method and further simplification leading to the fastdecoupled method. Figure 5.1 gives these details.
Fig 5.1 Power Flow Solution Methods
Before applying the NR method for power flow solutions, it would be helpful to briefly look at the general procedure for solving simultaneous algebraic equations as dealt with in the following section.
5.2 Newton–Raphson Method
The NR method can be applied for linear or nonlinear algebraic equations. The method can be easily understood for singlevalued functions.
5.2.1 NR Method for SingleValued Functions
Consider a singlevalued function described by
The solution of Equation (5.1) is the value of ‘x’ at which f (x) = 0.
Start with a guess for x as x^{0}. Now, we assume the first iteration value of x, x^{(1)} as the solution where,
The increment ∆x^{(0)} is not known, but can be estimated by expanding the above equation as a Taylor's series approximation as:
can be obtained by partial differentiating f(x) with respect to x and then by substituting x = x^{0}.
The assumption is that though x^{0} is not the exact solution, it is very close to the real solution. Therefore ∆x^{(0)} is very small and the higher order terms like ∆^{2}x^{(0)}, ∆^{3}x^{(0)} … being still smaller, can be neglected. Based on this, Equation (5.3) reduces to:
From the above, the value of ∆x^{(0)} is:
The first iteration value of x^{(1)} now can be calculated as x^{(1)} = x^{(0)} + ∆x^{(0)} and in general the (r + 1)^{th} iteration value of x is x^{(r + 1)} = x^{r} + ∆x^{r}, where:
and
The iteration process shall be terminated when the following convergence condition is satisfied:
, is the error specified
Example 5.1
Find the root of the equation f (x) = x^{2} – 3x + 2 by using Newton–Raphson method
Solution:
Differentiate f(x) with respect to x as:
Let initial approximation x^{0} = 0
Using Equation (5.6), the first iteration value of x is,
Similarly, consecutive iteration values are
When x = 1, f(1) = 0 condition is satisfied. Hence the root of the equation is x = 1
5.2.2 NR Method for MultiValued Function
Let us apply the NR method for the system of equations where the number of unknowns is more than one. As an example, consider two algebraic equations with two unknown functions x_{1} and x_{2} as:
The iteration process is started with guess values for x_{1} = x_{1}^{0} and x_{2} = x_{2}^{0}.
The next iteration values of x_{1} and x_{2} can be obtained by providing error increments to x_{1} and x_{2} as ∆x_{1}^{0} and ∆x_{2}^{0}.
such that Equation (5.7) can be satisfied simultaneously as:
The error increments can be obtained by expanding Equation (5.8) from Taylor's series approximation as said before and then by neglecting the higher order terms.
Thus,
where denotes the partial derivatives of evaluated at x_{1} = x_{1}^{0} while keeping x_{2} as constant. Similarly other terms in the Equation (5.9) can be evaluated. The matrix form of Equation (5.9) appears as shown below:
The condensed form of above equation can be written as
In Equation (5.10), [J] matrix contains partial derivative terms and it is known as the Jacobian matrix and [∆X] matrix is an increment matrix which is required. In the Equation (5.10) all the matrices except the increment matrix are unknown and can be obtained as:
The increments can be used to update the x_{1} and x_{2} values.
Continue the iteration process till the (r + 1)^{th} iteration:
where,
and terminate, when the following convergence conditions are satisfied simultaneously.
Equation (5.13) can be generalized for nunknown variables of nsimultaneous algebraic equations as:
Example 5.2
Use the Newton–Raphson method to solve
Assume x_{1}^{0} = 2 and x_{2}^{0} = –1.
Update the values of x_{1} and x_{2}, perform one iteration
Solution:
Consider Equation (5.9.3). The Jacobian matrix elements are:
The coefficient matrix elements are:
The Jacobian Matrix [J] is
and its inverse is:
The coefficient matrix is:
Now, the increment matrix can be obtained as follows:
The updated values of x_{1} and x_{2} are:
5.3 Power Flow Solution by Newton–Raphson Method
The general procedure for solving simultaneous algebraic equations by Newton–Raphson method is described in Section 5.2. Now, we shall apply the same to power flow problems. NR method can be applied to the power flow problem in two ways, depending upon how bus voltages are expressed. Bus voltages may be expressed in the polar form or in the rectangular form.
5.3.1 NR Method when Bus Voltages are Expressed in the Polar Form
Recall the static power flow equations that were derived in Chapter 4.
It can be observed in the above equations that the injected P_{i} and Q_{i} at each bus in an nbus power system are functions of n bus voltage magnitudes V and another n number of phase angles (δ), totaling 2n bus quantities.
Since both P_{i} and Q_{i} are the functions of 2n quantities, if any one or more quantities changes, the value of both P_{i} and Q_{i} changes. The change in V and δ can be written with the help of Taylor's series as:
and
for i = 1, 2,…, n.
We start the NR method for the Case1 study where PV buses are not present. Bus1 is slack bus and the rest i = 2,…, n are PQ buses.
The following discussion modifies Equations (5.17) and (5.18).
 In an nbus power system, Bus1 is generally designated as slack bus. For the slack bus, V_{1} and δ_{1} are specified. As the specified quantities do not change, the increments (error) ∆V_{1} and ∆δ_{1} are zeroes.
 ∆P and ∆Q denoted are power mismatches, and they represent the difference between specified powers and calculated powers. These are nonzero values owing to an error in V and δ values. In the case of PQ buses, specified powers in addition to calculated powers are available and power mismatches can be determined. However, for the slack bus these cannot be determined as the specified powers are not available.
In view of the above, Equations (5.17) and (5.18) can be modified to:
for i = 2, 3,…,n
The above equations can be written as:
In Equations (5.21) and (5.22), ∆V_{k} is replaced by for the sake of convenience.
For an nbus power system, Equations (5.23) and (5.24) can be written in the matrix form as:
The condensed form of Equation (5.25) is:
In Equation (5.26), the Jacobian matrix is shown partitioned with submatrices H, N, J and L. Any general i^{th} row and k^{th} column element of these submatrices are:
Derivation of Jacobian elements
In this section the Jacobian elements are derived while the bus voltages are expressed in polar form. Recall the equation for complex power injected into the i^{th} bus:
In the above equation, let
Substituting above quantities in Equation (4.13):
Expressing the above equation in rectangular form:
Separating the real and imaginary terms in the above equation gives:
By separating the i^{th} term in Equation (5.27) and (5.28), they can be written as:
Equations (5.29) and (5.30) can be used to derive the Jacobian elements.
Diagonal elements of Hmatrix
Adding Equations (5.31) and (5.30),
From the above,
Offdiagonal elements of Hmatrix
Diagonal terms of Nmatrix
Multiply both sides of Equation (5.34) by V_{i}
Subtracting Equation (5.29) from (5.34)
From the above
Offdiagonal term of Nmatrix
Diagonal term of Jmatrix
Subtracting Equation (5.38) from (5.29),
Offdiagonal terms of Jmatrix
Diagonal term of Lmatrix
Multiplying Equation (5.41) by V_{i}
Subtracting Equation (5.30) from (5.42) yields
From the above,
Offdiagonal term of Lmatrix
Multiplying above equation by V_{k}
The following example demonstrates the development of Jacobian elements.
Example 5.3
Figure 5.2 represents a 3bus power system. Develop the network equations for power flow study according to NR Method. Bus 1 is the slack bus and buses 2 and 3 are PQ type.
Fig 5.2 A 3Bus Power System Network
Solution:
The dimensions of the Jacobian for an nbus power system with one slack bus and the remaining PQ buses are (2n – 2 × 2n – 2). The set of equations according to NR method is given by:
and is the expanded form of the above equation is:
The dimensions of the matrices are determined as shown:
 The size of power mismatch matrix is 2n – 2 × 1. For this example its size is 4 × 1.
 The size of Jacobian matrix is 2n – 2 × 2n – 2. For this example its size is 4 × 4.
 The size of increment matrix is 2n – 2 × 1. For this example its size is 4 × 1.
Jacobian elements
Using Equation (5.32) the diagonal elements of the H submatrix are:
Using Equation (5.33) the offdiagonal elements of the H submatrix are:
Using Equation (5.36) the diagonal elements of the N submatrix are:
Using Equation (5.37) the offdiagonal elements of the N submatrix are:
Using Equation (5.39) the diagonal elements of the J submatrix are:
Using Equation (5.40) the offdiagonal elements of J submatrix are:
Using Equation (5.43) the diagonal elements of the L submatrix are:
Using Equation (5.44) the offdiagonal elements of L submatrix are:
Consideration of PV Buses
The set of equations described by Equation (5.26) need to be modified for the case where PV buses are present. Let there be ‘x’ number of PV buses.
 Since the magnitude of voltage is specified for PV, generator and voltage controlled buses, their increment ∆Vs do not exist. Hence, in the increment matrix the elements corresponding to PV buses should be eliminated. With this, the dimension of the increment matrix reduces to (2n – 2 – x × 1).
 In the power mismatch matrix, the reactive power mismatch ∆Q corresponding to PV buses cannot be calculated, as reactive powers for these buses are not specified. Hence the size of the matrix reduces to (2n – 2 – x × 1).
 Due to modifications in other matrices the size of Jacobian now reduces to (2n – 2 – x × 2n – 2 – x).
Example 5.4 explains all these modifications.
Example 5.4
Consider the 3bus power system given in Example 5.3. The second bus is the PV bus. Show effect of the PV bus in the power mismatch, Jacobian matrix and increment matrices.
Solution:
 Since Q_{2} is not specified
∆Q_{2} = Q_{2, specified} – Q_{2, calculated} cannot be determined.
 Since V_{2} is specified, ∆V_{2} does not exist.
Now, the matrices are modified considering above effects and are shown as below.
NOTE: In this case, we need to determine less number of Jacobian elements.
Algorithm for Newton–Raphson Method
Let the power system consist of total nnumber of buses.
Bus 1 is slack bus.
Buses 2, 3,…, x + 1 are x number of PV buses and the remaining
Buses x + 2, x + 3,…, n are PQ buses.
Algorithm for the NR method is as follows:
The flow chart for power flow solution using the NR method is given in Figure 5.3
Fig 5.3 Flow Chart for NR Method
5.3.4 NR Method when Bus Voltages are Expressed in the Rectangular Form
In this method bus voltages are expressed in the rectangular form as:
The injected P and Q of each bus is a function of all bus voltages and can be written as:
The power mismatch equations can be written as:
Equations (5.47) and (5.48) can be represented in the form of a matrix as:
The submatrices H^{1}, N^{1}, J^{1} and L^{1} are similar to H, N, J and L as described earlier. Also, the algorithm for this method is similar to the algorithm for Newton–Raphson method, except that the Jacobian elements are evaluated differently. The rectangular version is less reliable as compared to the polar version, though it is slightly faster in convergence. Hence, the rectangular version is rarely used.
Example 5.5
Consider a 3bus power system shown in Figure 5.4. The line data and bus data are given. The reactive power limits for Bus2 are Q_{2, min} = 0 and Q_{2, max} = 0.8 p. u.
Update the voltages and phase angles using the NR Method. Perform one iteration. Neglect line changing admittances. All the numerical values are given in p u.
Table: Line Data
Line  Series Impedance 

L_{1}  0.025 + j0.1 
L_{2}  0.025 + j0.1 
L_{3}  0.025 + j0.1 
Fig 5.4 A 3Bus Power System Network
Table: Bus Data
Solution:
The series admittance of each line is:
For a 3bus power system, the size of the Y_{Bus} is 3 × 3.
The elements of the Y_{Bus} matrix are:
Y_{11} = Y_{22} = Y_{33} = 4.7059 – j18.8235 p. u.
Y_{12} = Y_{21} = Y_{23} = Y_{32} = Y_{13} = Y_{31} = –2.3529 + j9.418 p. u.
The real and imaginary parts of Y_{Bus} are given below
G = Real {Y_{Bus}}
B = Imaginary {Y_{Bus}}
In polar form, Y_{Bus} is:
Step2: Computation of powers
The summary of specified bus quantities is as given below:
Bus1: V_{1} = 1.02 p. u; δ_{1} = 0 radians;
Bus2: P_{2} = 1.4 p. u; V_{2} = 1.03 p. u;
Bus3: P_{3} = –1.1 p. u; Q_{3} = –0.4 p. u;
Assume flat start for bus voltages and phase angles
The injected active powers can be computed by using Equation. (5.27) as
The injected reactive powers can be computed by using Equation (5.28)
Step3: Check reactive power limits for PV buses
It may be seen that Q_{2}^{0} is more than Q_{2, min} and less than Q_{2, max} as:
0 ≤ 0.3877 ≤ 0.8 p. u
Step4: Compute power mismatches
Power mismatch is the difference between specified power and computed power.
Step5: Compute Jacobian elements
The power flow matrices for NR method is given below
Hmatrix elements may be computed by using Equations. (5.32) and (5.33)
Nmatrix elements may be computed by using Equation (5.36) and (5.37)
Jmatrix elements may be computed by using Equation (5.39) and (5.40)
The Lmatrix elements may be computed by using Equations (5.43) and (5.44)
Step6 Compute the increment matrix
The Vvalues and modified values of δ are as shown:
5.3.5 Comparison of Gauss–Seidel and Newton–Raphson Method
5.4 Decoupled Newton Method
The complexities in performing calculations using the NR method are simplified by considering the practical behaviour of the power system. It is understood that P–δ and Q–V are strongly coupled and P–V and Q–δ are weakly coupled. In other words, P is insensitive for variations in V and Q is insensitive for variations in δ. Mathematically,
Considering the above effect, the set of equations described in Equation (5.26) modifies to:
Equation (5.50) is the linearised form of Equation (5.26)
The diagonal and offdiagonal elements of H and L submatrices can be obtained by using Equations (5.32, 5.33, 5.43 and 5.44). Equation (5.51a) can be used to find ∆δ. The updated δ values are used in Equation (5.51b) to compute ∆V.
5.4.1 Algorithm for Decoupled Power Flow Method
Example 5.6
Solve the power flow problem given in Example 55 using the decoupled power flow method.
Solution:
The matrices of the NR method are simplified in the decoupled method as:
Using the numerical results obtained in Example 55, the matrices can be written as:
Substituting the values,
Solving the above and using the results obtained in Example 5.5, the increments in δ and [v] can be obtained as follows:
Solving the above,
and the increments in the voltage are:
Solving the above,
At the end of the first iteration, the updated values of δ and vare:
5.5 Fast Decoupled Power Flow Method
Decoupled Newton method is a simplified version of the NR method, while fastdecoupled method is a simplified version of the decoupled method. In this method the power flow calculations can be made faster by making suitable assumptions.
Assumption1: Neglect the angle differences (δ_{i} – δ_{k}) such that, cos (δ_{i} – δ_{k}) 1 and
Assumption2: Power systems generally consist of lengthy transmission network where the ratio X/R is very high. Hence, the resistance of individual elements is neglected against the reactance values. In other words, G_{ik} can be ignored
In view of the above assumption, observe the following simplifications:
In general the value of Q_{i} is much smaller than
Considering the above simplifications, the Jacobian elements modify as:
From the above equations, the following relations can be shown amongst the Jacobian elements:
Recalling power flow equations of the decoupled method:
From Equation (5.53), the above matrices can be written as:
Equation (5.54) can be written as:
Setting V_{k} = 1 p. u. in Equation (5.55), it can be written as:
Also, from
The generalized term can be written as:
Equations (5.56) and (5.57) can be written in the condensed form as:
In Equations (5.58) and (5.59),
 B′ is the susceptance matrix having the elements – B_{ik} (for i = 2, 3,…,n and k = 2,3,…,n)
 B″ is part of the susceptance matrix having the elements – B_{ik} (for i = x + 2, x + 3…, n and k = x + 2, x + 3,…,n) corresponding to PQ buses.
Note: The student is advised to go through numerical problems for better understanding of the extraction of B’ and B” matrices from Y_{Bus}.
5.5.1 Algorithm for FastDecoupled Power Flow Method
The algorithm for power flow solution by the fastdecoupled power flow method is presented below:
Some more assumptions made in the FDLF method for further simplifications
Assumption3:  Omit the elements of B′ that affect the MVAR but not the MW value such as shunt reactance, offnominal inphase taps etc. 
Assumption4:  Omit the elements of B” such as the angle shifting effect that predominantly affects MVAR flow. Example: phase shift transformer reactance etc. 
Assumption5:  Neglect the series reactances in calculating the elements of [B]. 
Power flow solutions can be obtained faster through the assumptions made above. The matrices B′ and B” are real and sparse. These matrices have constant values that need to be evaluated at the beginning of the study.
A flow chart for fastdecoupled power flow method is given in Fig 5.5
Example 5.7
Solve the power flow problem given in Example 5.5 by the fastdecoupled method.
Solution:
The susceptance matrix B, computed in Example 5.6 is rewritten below:
The values for matrices B' and B” in equations (5.58) and (5.59) are extracted from the above B matrix.
B′ matrix corresponding PV and PQ buses (except slack bus)
Using the results obtained in Example 5.5, increments for ∆δ and ∆V can be obtained as follows:
Using Equation (5.58),
From Equation (5.59),
From the above,
The updated values of δ and V are:
5.5.2 Comparison of NR, Decoupled and Fast Decoupled Power Flow Methods
Fig 5.5 Flow Chart for Fast Decoupled Power Flow Method
Example 5.8
A typical 4bus power system is shown in Figure 5.6
Fig 5.6 A 4Bus Power System Network
The line data and bus data are given in the following tables. Neglect charging admittances.
All the values are in p.u.
Table: Line Data
Line No.  Between Buses  Series Impedance of line in P.U. 

1  1–2  0.07 + j 0.15 
2  1–3  0.06 + j 0.1 
3  1–4  0.08 + j 0.25 
4  2–4  0.04 + j 0.1 
5  3–4  0.04 + j 0.2 
Table: Bus Data
Update the bus voltages and phase angles by performing one iteration and by using
 NR method
 Fast decoupled method
Solution:
a) NR method:
Step1: Obtain Y_{Bus} by direct inspection method separating the real and imaginary matrices of Y_{Bus}'
Step2: The summary of specified bus quantities are:
V_{1} = 1.05 p.u; δ_{1} = 0 rad;
P_{2} = –0.4; P_{3} = –0.5; P_{4} = –0.7;
Q_{3} = –0.4; Q_{4} = –0.2
Step3: Compute injected powers using Equation (5.27) and (5.28)
P_{2}^{(c)} = –0.1767 p.u; P_{3}^{(c)} = –0.21955 p.u
P_{4}^{(c)} = –0.121155 p.u;
Q_{3}^{(c)} = 0.3666 p.u
Q_{4}^{(c)} = –0.18082 p.u
Step4: Calculate power mismatches
∆P_{2} = P_{2} – P_{2}^{(c)} = –0.2233 p.u
∆P_{3} = P_{3} – P_{3}^{(c)} = –0.28045 p.u
∆P_{4} = P_{4} – P_{4}^{(c)} = –0.578845 p.u
∆Q_{3} = Q_{3} – Q_{3}^{(c)} = –0.0334 p.u
∆Q_{4} = Q_{4} – Q_{4}^{(c)} = –0.01918 p.u
Step5: Compute Jacobian elements
The network matrices for NR Method for the power system shown are given below:
The procedure for calculating the Jacobian elements can be referred to from the previous examples. The Jacobian matrix is given below.
The increment matrix can be obtained by multiplying the [J]^{−1} with the power mismatch matrix. The increment matrix is given below.
Updated values of δ and V are
b) FastDecoupled Method:
The imaginary component B matrix of the Y_{Bus} was given earlier. The B' and B” matrices are given below:
Increments for phase angles can be calculated as:
Increments for bus voltage magnitudes can be calculated as:
Substituting the numerical values in the matrices, increments are computed as shown below:
The updated phase angles and voltages are:
Questions from Previous Question Papers
 Derive the power balance equation in a power system and explain the N–R method of load flow analysis. Draw the flow chart giving the sequence of analysis. Show that the polar coordinate representation is advantageous over the rectangular coordinates.
 Explain the advantages of using the bus admittance matrix in load flow studies.
 Consider the single line diagram of a power system shown in Figure Q1: Take Bus1 as the slack bus. The Y_{Bus} matrix is given below
The scheduled generation and loads are as follows:
Using the Newton–Raphson method, obtain the bus voltages at the end of the first iteration.
Fig Q1
 With the data given below, obtain V_{3} using N.R method after the first iteration.
Fig Q2
Bus code pq Impedance Z_{pq}p.u 1–2 0.08 + j0.24 1–3 0.02 + j0.06 2–3 0.06 + j0.18  Develop from basics, the equations for determining the elements of the H and L matrices in the fast decoupled method. State the assumptions that are made for faster convergence.

 Describe the Newton–Raphson method for the solution of power flow equations in power systems.
 What are P–V Buses? How are they handled in the above method?
 For the network shown in Figure Q3, obtain the complex bus bar voltages at Bus2 at the end of the first iteration, using the fastdecoupled method. Line impedances are in p. u. Given that Bus1 is a slack bus with
Fig Q3
 A sample power system is shown in Figure Q7. Determine V_{2} and V_{3} by the N.R method after one iteration. The p. u. values of line impedances are as shown:
Fig Q4
 Carry out one iteration of load flow solution for the system shown in Figure Q1, using the fastdecoupled method. Take Q limits of Generator2 as
 Find δ_{2} and Q_{2} for the system shown in Figure Q5. Use the N.R method up to one iteration.
Fig Q5
 Derive the algorithm for fast decoupled power flow analysis and give the steps for implementation of this algorithm.

 Obtain the decoupled load flow model starting from the Newton–Raphson method.
 What are the assumptions made in fastdecoupled method to speed up the rate of convergence?
 For the system shown in Figure Q6, find the bus voltage at the receiving end at the end of the first iteration. The load is 2 + j0.8 p.u. Voltage at the sending end (slack) is 1 + j0 p.u. The line admittance is 1.0 – j4.0 p.u. and the transformer reactance is j0.4 p.u. Use the decoupled load flow method. Assume V_{r} = 10^{0} (Nov2006)
Fig Q6
 Consider the given threebus system. The p.u. line reactances are as indicated in Figure Q7. The line resistances are negligible.
Fig Q7
The data of bus voltages and powers are given below
Determine the load flow solution to be solved using the decoupled method for one iteration.
 Give the general form of load flow equation to be solved in the Newton–Rapshon method. Explain in detail, the approximations in Newton–Raphson method to arrive at decoupled methods.
 State merits and demerits of the fastdecoupled method.
 Derive the power balance equations in a power system and explain the NR method of load flow analysis. Draw the flow chart giving the sequence of analysis. Show that the polar coordinate representation is advantageous over the rectangular coordinates (Or) Describe the Newton–Raphson method for the solution of power flow equations in power systems deriving necessary equations.
 Explain briefly what do you understand by load flow solution. Obtain the mathematical model for the above study using NR Method. Use the polar coordinate Method.
 Give a neat flow chart for NR Method of solving load flow equations using rectangular coordinates. Explain clearly the major steps involved in the solution.
 When PV buses are not present
 When PV buses are present.
 Using data given below, obtain V_{3} using NR Method after first iteration.
Fig Q8
Line Data:
Bus Code P – q Impedance Z_{pq} (p. 4) 1–2 0.08 + j0.24 1–3 0.02 + j0.06 2–3 0.06 + j0.18 Bus Data:
Take base MVA as 100
 A sample power system is shown in Figure Q9. Determine V_{2} and V_{3} by NR Method after one iteration. The per unit values of line impedances are shown in figure. Bus data is given below.
Fig Q9
Bus Data
V_{1} = 1.04 p.u.; Injected powers S_{2} = 0.5 + j1.0 p.u. and S_{3} = 1.5 + j 0.6 p.u
 Consider the single line diagram of a power system shown in Figure Q6. Take bus 1 as slack bus and the Y_{Bus} is given below
Scheduled generation and loads are as follows:
Take base power as 100 MVA.
Using the NR Method, obtain the bus voltages at the end of 1^{st} iteration.
Fig Q10
 Explain the necessary equations for the load flow solution using the NR Method. What is the Jacobian Matrix? Derive necessary equations for computing all the elements of the above matrix using rectangular coordinates.
 Find V_{2} and S_{2} for the system shown in Figure Q11. Use the NR Method upto one iteration.
Fig Q11
 Develop the power flow model using decoupled method and explain the assumptions made to derive at the fast decoupled load flow method. Draw the flow chart and explain.
 Explain with a flow chart, the computational procedure for load flow solution using fast decoupled method, deriving necessary equations.
 Develop the equations for determining the elements of H and L matrices in the fast decoupled load flow method, from the basics. State the assumptions that are made for faster convergence.
 State merits and demerits of fast decoupled load flow method.
 Compare the GS, NR and FDLF methods.
 For the system shown in Figure Q12, find the voltage at the receiving end bus at the end of first iteration. Load is 2 + j0.8 p.u. Voltage at the sending end (slack) is (1 + j0) p.u. Line admittance is 1  j4 p.u. Transformer reactance is j0.4 p.u. Use the decoupled load flow method. Assume V_{R} = 1 p.u.
Fig Q12
 For the network shown in Figure Q13, obtain the complex bus bar voltages at bus (2) at the end of first iteration using fast decoupled Method. Line impedances are in p.u. Given bus (1) is slack bus with V_{1}= 1 ,
Fig Q13
 Carry out one iteration of load flow solution for the system shown by fast decoupled method for the network shown in Figure Q14. Take Q limits of generator  as : Q _{2}, min = 0; Q _{2},max = 5
Bus 1: slack bus V_{specified} = 1.05 p.u.
Bus 2: PV bus V_{specified}  = 1.00 p.u, P_{G2} = 3 p.u.
Bus 3: PQ bus P_{D3} = 4 p.u.; Q_{D3} = 2 p.u.
Fig Q14
 Consider the three bus system shown in Figure Q17. The p.u. line reactances are indicated on the figure. The line resistances are negligible.
Fig Q15
The data of bus voltages and powers are given below.
Determine the load flow solution to be solved using the decoupled method for one iteration.
Competitive Examination Questions
 Load flow studies involve solving simultaneous
 linear algebraic equations
 nonlinear algebraic equations
 linear differential equations
 nonlinear differential equations
 The principal information obtained from load flow studies in a P.S pertains to the
 magnitude and phase angle of the voltage at each bus
 reactive and real power flows in each of the lines
 total power loss in the network
 transient stability limit of the system
 1 and 2
 3 and 4
 1, 2 and 3
 2 and 4
 A power system consists of 300 buses, out of which 20 are generator buses, 25 are ones with reactive power support and 15 are ones with fixed shunt capacitors. All the other buses are load buses. It is proposed to perform a load flow analysis for the system using NR method. The size of the NewtonRaphson Jacobian matrix is
 553 × 553
 540 × 540
 555 × 555
 554 × 554
[GATE 2003 Q.NO 12]
 For a 15bus power system with a 3voltage controlled bus, the size of the Jacobian matrix is
 11 × 11
 12 × 12
 24 × 24
 25 × 25
[IES 1996 Q.NO 113]
 In the solution of a load flow equation, the Newton–Raphson method is superior to the Guass–Siedel method, because the
 time taken to perform one iteration in the NR method is less than that in the GS method
 number of iterations required in the NR method is more when compared to that in the GS method
 number of iterations required is not independent of the size of the system in the NR method
 convergence characteristics of the NR method are not affected by the selection of slack bus.
[IES 1997 Q.NO 40]
 Compared to the Guass–Siedel method, the Newton–Raphson method takes
 lesser number of iterations and more time per iteration
 lesser number of iterations and less time per iteration
 more number of iterations and more time per iteration
 more number of iterations and less time per iteration
[IES 1999 Q.NO 49]
 A 12bus power system has three voltagecontrolled buses. The dimensions of the Jacobian matrix will be
 21 × 21
 21 × 19
 19 × 19
 19 × 21
[IES 2000 Q.NO 67]
 Match ListI with ListII.
ListI ListII (Load flow methods) (System environment) A. GuassSiedel load flow 1. Guass elimination B. NewtonRaphson load flow 2. I–V factors C. Fast decoupled load flow 3. Contingency studies D. Real time load flow 4. Offline solution [IES 2002 Q.NO 104]
 Consider the two power system shown in the figure A below, which are initially not interconnected, and are operating in steady state at the same frequency. Separate load flow solutions are computed individually for the two systems, corresponding to this scenario. The bus voltage phasors so obtained are indicated on Figure A. These two isolated systems are now interconnected by a short transmission line as shown in Figure B, and it is found that P_{1} = P_{2} = Q_{1} = Q_{2} = 0.
[GATE 2006 Q.No. 11]
Figure A
Figure B
The bus voltage phase angular difference between generator bus X and generator bus Y after the interconnection is:
 10°
 25°
 –30°
 30°