Derivation of DC Power Flow

Given the Power Flow Equations:

We can derive the equation for DC power flow using the following simplifying approximations;

1. Approximate the transmission resistance to zero.

The resistance of transmission circuits is significantly less than the reactance. Usually, it is the case that the x/r ratio is between 2 and 10. So any given transmission circuit with impedance of z = r + jx will have a,

and

If r is very small compared to x, then we observe that g will be very small compared to b, and it is reasonable to approximate g = 0 and b = -1/x

Now, if g = 0, then the real part of all of the Y-bus elements will also be zero. The power flow equation will now become,

2. Approximate the cosine term to zero and sine term to the radian angle.

For most typical operating conditions, the difference in angles of the voltage phasors at two buses k and j connected by a circuit, which is THETAk - THETAj for buses k and j, is less than 10 -15 degrees. It is extremely rare to ever see such angular separation exceed 30 degrees. Thus, we say that the angular separation across any transmission circuit is “small.”

It is clear that as the angle DELTA = THETAk - THETAj gets smaller and smaller, the cosine function approaches 1.0. The better approximation of the sine of a small angle is the angle itself in radian because for a very small angle, the sine of an angle is approximately equal to the angle itself. Thus we now have,

3. Approximate the product of the voltages to 1.0.

In the per-unit system, the numerical values of voltage magnitudes |Vk| and |Vj| are very close to 1.0. Typical range under most operating conditions is 0.95 to 1.05. Furthermore, its product is very much closer to 1.0.

Given the discussed practical approximations, the power flow in the transmission system can be approximated using the following equation.

It is the called the DC power flow equation. It is commonly used in the optimal power flow (OPF) and economic dispatch problems in the power systems. Since our point of interest is the power flow, we omit the power flow equation for reactive power.

Category Filed in: Transmission System and Load Flow Analysis -