• Economic  distribution  of  load  between  power  stations

When transmission distances are large, the transmission losses are a significant part of the generation and have to be considered in the generation schedule for economic operation.

Consider a system with n number of generating plants supplying the total demand PR. The problem is to determine the P1, P2, …, Pn dispatch levels to be able to serve PR in the most economical way with the consideration of transmission losses PL. Assume total power loss, PL = f (P1,P2,…..,Pn).

If Ci is the cost of plant i in Rs/h and the power output of the ith unit is Pi. The input cost can be expressed in terms of the power output as

Ci = αi + βi Pi + ϒi (Pi )2 = f(Pi )

The mathematical formulation of the problem of economic scheduling can be stated as follows:

Minimize  C = sum(Ci) ;  i = 1, 2, ....n.

Such that   f(Pi) = PR + PL - sum(Pi) = 0

where, C  = total cost, Pi = generation cost of ith plant, PD or PR = total demand.

This is a constrained optimization problem, which can be solved by Lagrange’s method. Such problem can be solved by using Lagrange multiplier (λ) and Lagrange cost function (C*).

C* = C + λ f

Since Ci is function of Pi, partial derivative becomes full derivatives. So,

Where,

Li = Penalty factor of ith unit

When losses are considered & there is no generator limits, the product of IFCi and Li for all plants must be equal for most economic operation.