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Load flow analysis is the steady-state 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 non-linear 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 volt-amperes at various points in a power system operating under normal steady-state or static conditions.

Short circuits occur in power system 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.

Depending upon which two variables are specified a priori, the buses are classified into three categories.

At this type of bus, the net powers Pi and Qi are known (PDi and QDi are known from Load forecasting and PGi and QGi are specified). The unknowns are |Vi| and δi. A pure load bus (no generating facility at the bus, i.e., PGi = QGi = 0) is a PQ bus.

At this type of bus PDi, and QDi, are known a priori and |Vi| and Pi (hence PGi) are specified. The unknowns are Qi (hence QGi) and δi.

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:

where |V| = Magnitude of voltage

δ = Phase angle of the voltage

P = Active power

Q = Reactive volt-ampere

Load Flow Problem– The complex power injected by the source into the ith bus of a power system is

where Vi is the voltage at the ith bus with respect to ground and Ji is the source current injected into the bus.

The Load Flow analysis is handled more conveniently by use of Ji rather than J*i. Therefore, taking the complex conjugate of last equation, we have

Equating real and imaginary parts we get

In Polar Form

Vi = |Vi|ejδ1

Yik = |Yik|ejθ1k

Real and reactive powers can now be expressed as

where i going from i = 1,2.....n

Equations for Pi & Qi 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

The power system equipment is designed to operate at fixed voltages with allowable variations of ± (5 —10)% of the rated values.

|δi - δk|≤ |δi - δkmax|

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 PL and QL are system real and reactive power loss, respectively.

Consider a power system network, consisting of two generating stations which is shown in figure below

The bus admittance matrix Ybus relates the bus voltage Vbus current Ibus through the relation

Ibus = Ybus Vbus

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 (Zbus)

Bus impedance matrix (Zbus) is the inverse matrix of Ybus (bus admittance matrix)

And the relation between Zbus and Vbus is as follows

Vbus = Zbus Ibus

or

For n-bus system, the nodal current equation is

(where p = 1,2, … , n)

where, P = Active power

Q = Reactive power

Substituting for Ip in Eq. (iii),

(where p = 1,2, …,n)

Ip has been substituted by the real and reactive powers because normally in a power system these quantities are specified.

Convergence in the Gauss-Seidel Method can sometimes be speeded up by the use of the acceleration factor. For the ith bus, the accelerated value of voltage at the (r + 1)th iteration is given by

To explain how the Gauss-Seidel 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.

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.

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, |ΔVi(r+1)| between two consecutive iterations is less than a certain tolerance for all bus voltages, i.e.

The power loss in the (i — k)th line is the sum of the power flows determined from Eqs. Sik & Ski. Total transmission loss can be computed by summing all the line flows

(i.e. Sik + Ski 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 Qi is obtained from the above equation by substituting most updated values of voltages on the right-hand side. In fact, for the (r + 1)th iteration one can write from the above equation

The revised value of δi is obtained immediately following step 1. Thus

where

Note: The algorithm for PQ buses remains unchanged.

The Newton-Raphson Method is a powerful method of solving non-linear 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 non-linear 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 Taylor series around the initial guess, we have

Neglecting higher order terms we can write Eqn. in matrix form

or in vector-matrix form

Jo is known as the Jacobian matrix (obtained by differentiating the function vector f with respect to x and evaluating it at Xo).

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.

Economic Loading Neglecting Transmission Losses: To formulate economic loading into mathematical problem, the fuel input curve is modeled as a quadratic equation.

Let the fuel input curve is

₹/h

where, Fi = Fuel input cost of an unit i in ₹/h

pi = Power output of the unit / in MW

Then, the incremental operating cost of each unit is computed as,

₹/MW-h

For k unit,

For Two Generating Units

Total cost

Total output

Economic Loading Including Transmission Losses: For 2 plants Transmission loss

where, B11, B22 and B12 are called loss coefficients or –coefficients.

For three plant system:

PLosses = P12B11 + P22B22 + P32B33 + 2P1P2B12 + 2P2P3B23 + 2P3P1B31

Penalty Factor:

where, Ln = Penalty factor of plant n.

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