Economic distribution of load between generating units within a plant
The simplest case of economic dispatch is the case when transmission losses are neglected. The model does not consider the system configuration or line impedances. Since losses are neglected, the total generation is equal to the total demand P_{D} (or P_{R}).
Consider a system with n number of generating plants supplying the total demand P_{R}. The problem is to determine the P_{1}, P_{2}, …, P_{n} dispatch levels to be able to serve P_{R} in the most economical way.
If C_{i} is the cost of plant i in Rs/h and the power output of the ith unit is P_{i}. The input cost can be expressed in terms of the power output as
C_{i} = α_{i} + β_{i }P_{i} + ϒ_{i} (P_{i} )^{2} = f(P_{i} )
The mathematical formulation of the problem of economic scheduling can be stated as follows:
Minimize C = sum(C_{i}) ; i = 1, 2, ....n.
Such that f(P_{i}) = P_{R} - sum(P_{i}) = 0
where, C = total cost, P_{i }= generation cost of i^{th} plant, P_{D} or P_{R} = 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
When losses are neglected & there is no generator limits, for most economic operation, all units must operate at equal incremental production cost.