Power Transmission System.

Single Phase and Three Phase Transmission

Power Flow Analysis

1-Phase Transmission

Let us consider an inductive circuit and let the instantaneous voltage be

                 v  =  V_m sin wt

Then the current will be i  = I_m sin (wt-ɸ) where ɸ is the angle by which the current lags the voltage.

Assume that the source is perfect sinusoidal with fundamental frequency component only.

The instantaneous power is given by

   p  =  vi  =  V_m sin wt . I_m sin(wt-ɸ)

                 =  V_m I_m  sin wt . sin(wt-ɸ)

                 = V_m I_m/2 [cos ɸ  - cos(2wt -ɸ)]

The value of p is positive when both v and i are either positive or negative and represents the rate at which the energy is being consumed by the load. In this case the current flows in the direction of voltage drop. On the other hand power is negative when the current flows in the direction of voltage rise which means that the energy is being transferred from the load into the network to which it is connected.

         

We have decomposed the instantaneous power into two components.

The component marked I defined the real power or active power P which physically means the useful power being transmitted. The power marked II contains sin ɸ which is negative for capacitive circuit and positive for inductive circuit. This component is known as reactive power.

  

                         p  =  P(1- cos 2wt)  -  Q sin 2wt.

 

3 Phase Transmission

Assume that the system is balanced which means that the 3 phase voltages and currents are balanced.

V_a  =  V_m sin wt

V_b  =  V_m sin(wt–120)

V_c  =  V_m sin(wt+120)

I_a  = I_m sin (wt-ɸ)

I_b  = I_m sin (wt-ɸ-120)

I_c  = I_m sin (wt-ɸ+120)

The total power transmitted equals to sum of the individual powers in each phase

 p  =  V_aI_a + V_bI_b + V_cI_c

         = 3VI cos ɸ

This shows that the total instantaneous 3-phase power is constant and is equal to three times the active power per phase i.e. p = 3P, where P is the power per phase.

 

In case of single phase transmission we noted that the instantaneous power expression contained both the active power and reactive power expression but here in case of 3-phase we find that the instantaneous power is constant. This does not mean that the reactive power is of no importance in a 3-phase system

For a 3-phase system the sum of three currents at any instant is zero, this does not mean that the current in each phase is zero. Similarly, even though the sum of reactive power instantaneously in 3-phase system is zero but in each phase it does exist and is equal to VI sin ɸ and, therefore, for 3-phase the reactive power is equal to Q3ɸ = 3VI sin ɸ = 3Q, here Q is the reactive power in each phase

 

Balance three phase system

A poly-phase system is said to be balanced if

The magnitude of corresponding quantity are equal in each phase and

The phase difference between the corresponding quantity is

            ɵ  =    360/n

n is the number of phases

n  =  4 for 2-phase system