Genel kültür kategorisine 12 Temmuz, 2007 tarihinde eklendi, 25 defa okundu


In certain power system studies (e.g., reliability studies ), a very large number of load-flow runs may be needed . Therefore,a very fast (and not necessarily accurate ,due to the linear approximation involved) method can be used for such studies. The method of calculating the real power flows by solving first for the bus angles is known as the “dc load-flow method” , in contrast with the exact nonlinear solution ,which is known as the ac solution.

2)DC Load Flow Method

Because the resistances of transmission lines are rather small compared with reactances of those lines ,for certain types of operations (e.g., fault studies ) the resistances are neglected to simplify the solution.

Let’s consider the nominal p model of a medium line. The circuit parameters are :

A=1+ [ (1/2)YZ ]

B=Z+ [ (1/4)YZ 2 ]


D=1+ [ (1/2)YZ ]

Neglecting line resistance and shunt admittance , we can represent the line with its inductive reactance only, as shown in Figure 1.

Figure 1. Simplified transmission line model

The transmission (ABCD) parameters for this representation are:

A=D=1 0

B= jXik

C= 0

And the power flows through the line ,

PS = 2VS VR cos(90 + d) / X = (VS VR sind) / X

PR = 2 VS VR cos(90 + d) / X= (VS VR sind) / X (1)

For d << 1 radian, sind d.Hence, in general, the line flows become

Pik=(Vi Vk dik) / Xik (2)


dik = di 2dk (3)

Since all the bus voltages of a power system are around 1 pu, then let

Vi = Vk =1 pu

bik =21 / Xik (4)

Then equation (3) becomes

Pik = 2bik dik = (di 2dk) / Xik (5)

Now, the bus power at any bus is the sum of the power flows in the lines connected to that bus. .Hence ,

Pi = Pik = (2bik dik) i = 1, 2, …., N (6)

or in matrix form ,

P1 b11 b12 … b1N d1

P2 b21 b22 … b2N d2

. = . . … . . (7)

. . . … . .

PN bN1 bN2 … b NN d N

which can be abbreviated as

[P] = [b] [ d ] (8)


bik = 21 / Xik (9)


bii = (2bik) (10)

The matrix [b] is the imaginary part of YBUS . The solution for [d] is

[d] = [b] –1 [P] (11)

[b] matrix is an (n –1) 3 (n21) matrix dimensionally for an n-bus system. The diagonal and off – diagonal elements of the [b] matrix can be found by adding the series susceptances of the branches connected to bus i and by setting them equal to the negated series susceptance of branch ik , respectively.

Until now , we have kept the system ground as the reference bus. However since we have dropped all shunt brunches in simplifying things , we have lost our reference. This means that the matrix [b] of equation (7) is obtained by equation (9) and (10) will be a singular matrix. Hence , [b]21 of equation (11) does not in fact exist. To overcome this difficulty , we shall select one of the buses as reference and assign zero radians to its angle (as the swing bus for AC power flow ). Then ,all the calculated angles will be referred to this bus and the row and column corresponding to the reference bus in [b] will be dropped to produce a nonsingular [b] which will result in an inverse to be used in eq. (11) .

Note that since the system is linearized , the solution is direct and this means there is no need for an iterative procedure.


It is possible with the dc load – flow method to carry out the thousands of load-flow runs that required for comprehensive contincency analysis on large-scale systems.Of course ,the adequacy assesment provided by this representation is restricted to overload related system problems.

In summary , the choice of a load-flow method is a matter of choice between speed and accuracy. For a given degree of accuracy, the speed depends on the size, complexity, and configuration of the power system and on the numerical approach chosen.



2)DC Load Flow Method



[1] Gonen ,T ., ‘Modern Power System Analysis’, John Wiley & Sons, 1988

[2] Goren , B ., ‘Power System Analysis’ , John Wiley & Sons, 1989


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