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Load Flow Analysis Study Notes for GATE, ESE & Electrical Exams
By BYJU'S Exam Prep
Updated on: September 25th, 2023
In this article, you will find the study notes on Load Flow Analysis, which will cover the topics such as Bus Classification, Formation Zbus & Ybus, Advantage of Ybus over Zbus, Development of Load Flow Equations, Acceleration of convergence in Load Flow Studies etc. Load Flow analysis carry 12 marks weightage in GATE Examination and 23 marks weightage in ESE 2022 examination.
Load Flow Analysis is an essential part of topics in Electrical Engineering, especially if you are preparing for the electrical branch exams. The article contains fundamental study notes and every tiny detail on the “Load Flow Analysis” topic of the “Power Systems” subject.
The topicwise Load Flow Analysis study notes are useful for preparing for various upcoming exams like GATE, IES, BARC Scientific Officer, ISRO Scientist Engineer and other important electrical engineering exams.
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Table of content
 1. Load Flow Analysis
 2. Bus Classification
 3. Bus Admittance Matrix (Ybus)
 4. Advantage of Ybus over Zbus
 5. Development of Load Flow Equations
 6. Acceleration of Convergence in Load Flow Studies
 7. GaussSeidel Method
 8. Algorithm for Load Flow Solution
 9. NewtonRaphson Method
 10. Comparison of Load Flow Methods
 11. Economic Operation of Power System
Load Flow Analysis
Load flow analysis is the steadystate analysis of the power system network. The load flow study defines the system’s operating state for shared loading. Load flow decodes a set of simultaneous nonlinear algebraic power equations for the two unidentified variables (V and ∠δ ) at each node in a system.
We already know that the voltage determines the Power ﬂow in a network at each bus of the network and the impedances of the lines between buses. Power ﬂow into and out of each of the buses that are network terminals is the sum of the power flow of all of the lines connected to that bus.
The load ﬂow problem consists of ﬁnding the set of voltages: magnitude and angle, which, together with the network impedances, produces the load ﬂows known to be correct at the system terminals. To start, we view the power system as being a collection of buses, connected together by lines. At each bus we may regard as nodes, we may connect equipment that will supply power to or remove power from the system.
Load flow (or power flow) solution is the determination of current, voltage, active power and reactive voltamperes at various points in a power system operating under normal steadystate or static conditions.
Short circuits occur in power systems due to various reasons like equipment failure, lightning strikes, falling of branches or trees on the transmission lines, switching surges, insulation failures and other electrical or mechanical causes. All these are collectively called faults in power systems.
Bus Classification
Depending upon which two variables are specified a priori, the buses are classified into three categories.
 Load Bus or PQ Bus
At this type of bus, the net powers P_{i} and Q_{i} are known (P_{Di} and Q_{Di} are known from Load forecasting and P_{Gi} and Q_{Gi} are specified). The unknowns are V_{i} and δ_{i}. A pure load bus (no generating facility at the bus, i.e., P_{Gi} = Q_{Gi} = 0) is a PQ bus.
 PV Bus/Generator Bus/Voltage Controlled Bus
At this type of bus P_{Di}, and Q_{Di}, are known a priori and V_{i} and P_{i} (hence P_{Gi}) are specified. The unknowns are Q_{i} (hence Q_{Gi}) and δ_{i}.
 Slack Bus/Swing Bus/Reference Bus
This bus is distinguished from the other two types by the fact that real and reactive powers at this bus are not specified. Instead, voltage magnitude and phase angle (normally set equal to zero) are specified.
Note:
 Because in a load flow study, real and reactive powers (i.e. complex power) cannot be fixed a priori at all the buses as the net complex power flow into the network is not known in advance, the system power loss being unknown till the load flow study is complete.
 It is, therefore, necessary to have one bus (i.e. the slack bus) at which complex power is unspecified so that it supplies the difference in the total system load plus losses and the sum of the complex powers specified at the remaining buses.
where V = Magnitude of voltage
δ = Phase angle of the voltage
P = Active power
Q = Reactive voltampere
Bus Admittance Matrix (Ybus)
Load Flow Problem– The complex power injected by the source into the i^{th} bus of a power system is
where V_{i} is the voltage at the i^{th} bus with respect to ground and J_{i} is the source current injected into the bus.
The Load Flow analysis is handled more conveniently by the use of J_{i} rather than J^{*}_{i}. Therefore, taking the complex conjugate of the last equation, we have
Equating real and imaginary parts we get
In Polar Form
V_{i} = V_{i}e^{jδ1}
Y_{ik} = Y_{ik}e^{jθ1k}
Real and reactive powers can now be expressed as
where i going from i = 1,2…..n
Equations for P_{i} & Q_{i} are referred to as static load flow equations (SLFE). By transposing all the variables on one side, these equations can be written in the vector form
f (x, y) = 0
where
f = vector function of dimension 2n
x = dependent or state vector of dimension 2n (2n unspecified variables)
y = vector of independent variables of dimension 2n
(2n independent variables which are specified a priori)
Some of the independent variables in y can be used to manipulate some of the state variables. These adjustable independent variables are called control parameters. Vector y can then be partitioned into a vector u of control parameters and a vector p of fixed parameters.
To study SLFE solution to have practical significance, all the state and control variables must lie within specified practical limits These limits, which are dictated by specifications of power system hardware and operating constraints, are described below
 Voltage magnitude V_{i} must satisfy the inequality
The power system equipment is designed to operate at fixed voltages with allowable variations of ± (5 —10)% of the rated values.
 Certain of the δ_{i}_{s} (state variables) must satisfy the inequality constraint
δ_{i} – δ_{k}≤ δ_{i} – δ_{k}_{max}
 Owing to the physical limitations of P and/or Q generation sources, P_{Gi} and Q_{Gi} are constrained as follows
It is, of course, obvious that the total generation of real and reactive power must equal the total load demand plus losses, i.e.
where P_{L} and Q_{L} are system real and reactive power loss, respectively.
Formation Z_{bus} or Y_{bus}
Consider a power system network, consisting of two generating stations which is shown in the figure below
The bus admittance matrix Y_{bus} relates the bus voltage V_{bus} current I_{bus} through the relation
I_{bus} = Y_{bus} V_{bus}
These equations can be written in matrix form as
The compact form of these equations can be written as
(where p = 1,2, … , n)
Bus Impedance Matrix (Z_{bus})
Bus impedance matrix (Z_{bus}) is the inverse matrix of Y_{bus} (bus admittance matrix)
And the relation between Z_{bus} and V_{bus} is as follows
V_{bus} = Z_{bus} I_{bus}
or
Advantage of Ybus over Zbus
 Y_{BUS} is a sparse matrix
 Diagonal elements are dominating Off diagonal elements are symmetric.
 Offdiagonal elements are symmetric, the diagonal elements of each node are the sum of the admittances connected to it.
 The diagonal elements of each node are the sum of the admittances connected to it.
 The offdiagonal element is negated admittance.
Development of Load Flow Equations
For the nbus system, the nodal current equation is
(where p = 1,2, … , n)
where, P = Active power
Q = Reactive power
Substituting for I_{p} in Eq. (iii),
(where p = 1,2, …,n)
I_{p} has been substituted by the real and reactive powers because normally in a power system, these quantities are specified.
Acceleration of Convergence in Load Flow Studies
Convergence in the GaussSeidel Method can sometimes be speeded up by the use of the acceleration factor. For the i^{th} bus, the accelerated value of the voltage at the (r + 1)^{th} iteration is given by

where α is a real number called the acceleration factor. A suitable value of α for any system can be obtained by trial load flow studies. A generally recommended value is α = 1.6. A wrong choice of cc may indeed slow down convergence or even cause the method to diverge.

This concludes the load flow analysis for the case of PQ buses only.
GaussSeidel Method
 The GaussSeidel Method (GS) is an iterative algorithm for solving a set of nonlinear algebraic equations. To start with, a solution vector is assumed based on guidance from practical experience in a physical situation.
 One of the equations is then used to obtain the revised value of a particular variable by substituting in it the present values of the remaining variables. The solution vector is immediately updated in respect of this variable. The process is then repeated for all the variables, thereby completing one iteration.
 The iterative process is then repeated till the solution vector converges within the prescribed accuracy. The convergence is quite sensitive to the starting values assumed. Fortunately, in a load flow study, a starting vector close to the final solution can be easily identified with previous experience.
To explain how the GaussSeidel Method is applied to obtain the load flow solution, let it be assumed that all buses other than the slack bus are PQ buses. We shall see later that the method can be easily adapted to include PV buses. The slack bus voltage being specified has (n — 1) bus voltage starting values whose magnitudes and angles are assumed. These values are then updated through an iterative process.
Algorithm for Load Flow Solution
We shall continue to consider the case where all buses other than the Slack are PQ buses. The steps of a computational algorithm are given below.
 With the load profile known at each bus (i.e. P_{Di }and Q_{Di }known), allocate P_{Gi} and Q_{Gi} to all generating stations. While active and reactive generations are allocated to the slack bus, these are permitted to vary during iterative computation.
 Assembly of bus admittance matrix Y_{BUS}: With the line and shunt admittance data stored in the computer, Y_{BUS} is assembled by using the rule for self and mutual admittances.
 Iterative computation of bus voltages (V_{i}; i = 2, 3,…, n): To start the iterations a set of initial voltage values is assumed. Since, in a power system, the voltage spread is not too wide, it is normal practice to use a flat voltage start, e., initially all voltages are set equal to (1 + j0) except the voltage of the slack bus, which is fixed.
A significant reduction in the computer time can be achieved by performing in advance all the arithmetic operations that do not change with the iterations.
Now for the (r + 1)^{th} iteration, the voltage Equation becomes
The iterative process is continued till the change in magnitude of bus voltage, ΔV_{i}^{(r+1)} between two consecutive iterations is less than a certain tolerance for all bus voltages, i.e.
 Computation of slack bus power: Substitution of all bus voltages computed in step 3 along with V_{1 }it yields S_{1}^{*} = P_{1} — jQ_{1}.
 Computation of line flows: This is the last step in the load flow analysis wherein the power flows on the various lines of the network are computed. Consider the line connecting buses I and The line and transformers at each end can be represented by a circuit with series admittance y_{ik} and two shunt admittances y_{ik0} and y_{ki0} as shown in Figure below.
The power loss in the (i — k)^{th} line is the sum of the power flows determined from Eqs. S_{ik} & S_{ki}. Total transmission loss can be computed by summing all the line flows
(i.e. S_{ik} + S_{ki} for all i, k).
It may be noted that the slack bus power can also be found by summing the flows on the lines terminating at the slack bus.
Algorithm Modification when PV Buses are Also Present
At the PV buses, P and V are specified and Q and δ are the unknowns to be determined. Therefore, the values of Q and δ are to be updated in every GS iteration through appropriate bus equations. This is accomplished in the following steps for the ith PV bus.
From equation
The revised value of Q_{i} is obtained from the above equation by substituting the most updated values of voltages on the righthand side. In fact, for the (r + 1)^{th} iteration one can write from the above equation
The revised value of δ_{i }is obtained immediately following step 1. Thus
where
Note: The algorithm for PQ buses remains unchanged.
NewtonRaphson Method
The NewtonRaphson Method is a powerful method of solving nonlinear algebraic equations. It works faster and is sure to converge in most cases as compared to the GS method. It is indeed the practical method of load flow solution of large power networks.
Its only drawback is the large requirement of computer memory which has been overcome through a compact storage scheme. Convergence can be considerably speeded up by performing the first iteration through the GS method and using the values so obtained for starting the NR iterations.
Consider a set of n nonlinear algebraic equations
Assume initial values of unknowns as
be the corrections, which on being added to the initial guess, give the actual solution. Therefore
Expanding these equations in the Taylor series around the initial guess, we have
Neglecting higher order terms we can write Eqn. in matrix form
or in vectormatrix form
J^{o} is known as the Jacobian matrix (obtained by differentiating the function vector f with respect to x and evaluating it at X^{o}).
Updated values of x are then
or, in general, for the (r+1)^{th} iteration
Iterations are continued it is satisfied to any desired accuracy, i.e.
Comparison of Load Flow Methods
 GaussSeidel Method and Newton Raphson methods are compared when both use Y_{BUS} as the network model. It is experienced that the GaussSeidel Method works well when programmed using rectangular coordinates, whereas Newton Raphson requires more memory when rectangular coordinates are used. Hence, polar coordinates are preferred for the NewtonRaphson methods.
 The GaussSeidel Method requires the fewest number of arithmetic operations to complete an iteration. This is because of the sparsity of the network matrix and the simplicity of the solution techniques. Consequently, this method requires less time per iteration.
 With the NR method, the elements of the Jacobian are to be computed in each iteration, so the time is considerably longer. For typical large systems, the time per iteration in the NR method is roughly equivalent to 7 times that of the GaussSeidel Method. The time per iteration in both these methods increases almost directly as the number of buses in the network.
 The rate of convergence of the GaussSeidel Method is slow (linear convergence characteristic), requiring a considerably greater number of iterations to obtain a solution than the NR method which has quadratic convergence characteristics and is the best among all methods from the standpoint of convergence.
 In addition, the number of iterations for the GaussSeidel Method increases directly as the number of buses of the network, whereas the number of iterations for the NewtonRaphson method remains practically constant, independent of system size.
 The NewtonRaphson method needs 3 to 5 iterations to reach an acceptable solution for a large system. In the GaussSeidel Method and other methods, convergence is affected by the choice of slack bus and the presence of a series capacitor, but the sensitivity of the Newton Raphson methods is minimal to these factors which causes poor convergence.
 The chief advantage of the GaussSeidel Method is the ease of programming and the most efficient utilization of core memory. It is, however, restricted in the use of smallsize systems because of its doubtful convergence and the longer time needed for the solution of large power networks.
 Thus the NewtonRaphson method is decidedly more suitable than the GaussSeidel Method for all but very small systems.
 For FDLF, the convergence is geometric, two to five iterations are normally required for practical accuracies, and it is more reliable than the formal NewtonRaphson methods. This is due to the fact that the elements of [B’] and [B”] are fixed approximations to the tangents of the defining functions ΔP/V and ΔQ/V, and are not sensitive to any ‘humps’ in the defining functions.
 If ΔP/V and ΔQ/V are calculated efficiently, then the speed for iterations of the FDLF is nearly five times that of the formal Newton Raphson or about twothirds that of the GaussSeidel Method. Storage requirements are around 60 per cent of the formal Newton Raphson, but slightly more than the decoupled Newton Raphson methods.
 The FDLF can be employed in optimization studies and is specially used for accurate information of both real and reactive power for multiple load flow studies, as in contingency evaluation for system security assessment and enhancement analysis.
Economic Operation of Power System
Economic Loading Neglecting Transmission Losses: To formulate economic loading into a mathematical problem, the fuel input curve is modeled as a quadratic equation.
Let the fuel input curve is
₹/h
where, F_{i} = Fuel input cost of unit i in ₹/h
p_{i} = Power output of the unit / in MW
Then, the incremental operating cost of each unit is computed as,
₹/MWh
For the k unit,
For Two Generating Units
Total cost
Total output
Economic Loading Including Transmission Losses: For 2 plants Transmission loss
where, B_{11}, B_{22} and B_{12} are called loss coefficients or –coefficients.
For three plant systems:
P_{Losses }= P_{1}^{2}B_{11} + P_{2}^{2}B_{22} + P_{3}^{2}B_{33} + 2P_{1}P_{2}B_{12} + 2P_{2}P_{3}B_{23} + 2P_{3}P_{1}B_{31}
Penalty Factor:
where, L_{n} = Penalty factor of plant n.
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