Transmission loss equation
Generally, in a power system, several plants are situated at different places. They are interconnected by long transmission lines. The entire system load along with transmission loss shall be met by the power plants in the system. Transmission loss depends on i) line parameters ii) bus voltages and iii) power flow. Determination of transmission loss requires complex computations. However, with reasonable approximations, for a power system with N number of power plants, transmission loss can be represented as
P_{L} = P^{T}BP
Thus P_{L }can be written as
When the powers are in MW, the B_{mn }coefficients are of dimension 1/ MW. If powers are in per-unit, then B_{mn }coefficients are also in per-unit. Loss coefficient matrix of a power system shall be determined before hand and made available for economic dispatch.
For a two plant system, the expression for the transmission loss is
Since B_{mn }coefficient matrix is symmetric, for two plant system
Consider the simple case of two generating plants connected to an arbitrary number of loads through a transmission network as shown in figure.
Let, I_{1} = Current through line AC
I_{2} = Current through line BC
I_{1} + I_{2} = Current through line CD
R_{AC} = Resistance of line AC
R_{BC} = Resistance of line BC
R_{CD} = Resistance of line CD
IV_{1}I= Voltage magnitude at bus A
IV_{2}I= Voltage magnitude at bus B
P_{1} = Power generated by plant - 1
P_{2} = Power generated by plant - 2
cosØ1 = power factor at bus A
cosØ2 = power factor at bus B
Transmission loss for the given system is
P_{L} = 3*II_{1}I^{2}*R_{AC }+ 3*II_{2}I^{2}*R_{BC }+ 3*II_{1} + I_{2}I^{2}*R_{CD}
Since
II_{1}I = P_{1}/(1.732 *IV_{1}I* cosØ1)
II_{2}I = P_{2}/(1.732 *IV_{2}I* cosØ2) and assume that I_{1} & I_{2} are in phase.
So,
P_{L} = (P_{1})^{2}B_{11 }+ 2(P_{1})(P_{2})B_{12} + (P_{2})^{2}B_{22 }
Where,
The term B_{mn} are called loss coefficient or B-coefficient.
cosØ1 = power factor at bus AcosØ1 = power factor at bus A