Electrical Power Systems

Optimal System Operation 409
2. The lower limit on P gi is due to boiler and/or other thermodynamic considerations and
upper limit is set by thermal limits on the turbine generator unit.
3. The voltage constraint will keep the system voltages near their rated or nominal values.
The voltage should be neither too high nor too low and the objective is to help maintain
the consumer’s voltage.
4. Constraints on transmission line powers relate to stability and thermal limits.
5. The minimization of cost function C T subject to equality and inequality constraints is
treated by a branch of applied mathematics called Nonlinear Programming.
Nonlinear programming methods do not easily give the insights into the nature of the
optimal solutions and computationally they are very expensive. Hence, some approximations
will be made to simplify the problem formulation and give physical insights into the problem of
economic dispatch.
16.4 CLASSICAL ECONOMIC DISPATCH NEGLECTING LOSSES
Let us assume that which generators are to run to meet a particular load demand are known
a priori.
Total fuel cost is given by
such that
and
m
å
CT = C ( P )
i=1
i gi
m
n
å Pgi = PD = å Pdi i=1
i=1
...(16.7)
...(16.8)
P gi min < Pgi < P gi max , i = 1, 2, ..., m ...(16.9)
rom eqn. (16.8), we get
HG
I
m
åPgi
i=1 KJ – PD = 0 ...(16.10)
Also for the time being, do not consider the generator power limits given by eqn. (16.9).
The problem can be solved by the classical method using Lagrange multipliers, for minimizing
(or maximizing) a function with equality constraints as side conditions.
Using the method, we replace the cost function CT by an augmented cost function
~
(Lagrangian) as
C T
~
= CT – l å Pgi - PD
...(16.11)
C T
HG
m
i=1
where l is the Lagrangian multiplier.
Minimization can be achieved by the condition
~
CT
= 0 ...(16.12)
P
gi
I
KJ