This document summarizes and compares economic dispatch solutions with and without transmission losses. Economic dispatch is described as an optimization problem that determines generator outputs to minimize total production costs while satisfying demand. Without losses, the problem is solved using Lagrange multipliers to minimize costs subject to the demand constraint. With losses modeled as a quadratic function of outputs, the same approach is used but includes transmission losses in the objective function and calculations. The conclusion is that accounting for transmission losses is important to obtain the most economic dispatch solution.

2. Comparison of the Economic Dispatch
Solutions with and without Transmission
Losses
(Implementation in MATLAB)
By:
Muhammad Abdullah Adnan Farooqi
(NUST201261151MCEME35012F)

3. What is Economic Dispatch?
 Economic Dispatch is a constraint optimization
problem, it is the process of deciding what are the
power outputs of generators so that the total cost of
production is minimum and on the other hand the
power demand constraint should be satisfied.

4.  Every generator has its own cost function, which
provides a relationship b/w output power and fuel
cost.
 The fuel cost of ith generator is shown above.
 The derivative of the above function w.r.t power gives
us the following result.
 It is called incremental fuel cost and is equal to λ.

5. Economic Dispatch without Losses
 The simplest economic dispatch problem is without
Transmission losses. Since transmission losses are
neglected, the total demand PD is the sum of all
generation. A cost function Ci is assumed to known for
each unit. The problem is to find the power generation
for each plant so that the objective function is
minimized
 Subject to the constraint

6.  Where Ctotal is the total production cost and Ci is the
production cost of ith plant, ng is number of
generating plants.
 A typical approach to solve this constraint
optimization problem is by using the Lagrange
multiplier. Lagrange function is obtained by adding
the constraint function to the objective function, after
the constraint function has been multiplied by an
unknown multiplier (λ).

7.  The minimum of above function is found at the point
where the partials of the function to its variables are
zero.
which gives:

8.  After again doing some simplification we can write:
 Expanding Taylor’s series:
 The new lambda is written as:
 where
 To get started take any initial guess of λ and the
iterative procedure is continued until a certain level of
accuracy is achieved such as (∆λ<0.0001).

9. Economic Dispatch with Losses
 When transmission distances are small, transmission
losses may be neglected. However in a large
interconnected network where power is transmitted
over long distances, transmission losses are a major
factor and affect the economic dispatch. One common
practice is to express the total transmission loss as a
quadratic function of generator power outputs. The
approximated loss formula is expressed as:
 Where Bii are called loss coefficients or B-coefficients.

10.  Now from above derivation steps the Lagrange
function is same, except for the addition of
transmission loss which will now be taken into
account.
 Now to find minimum we find partials of above
function and put them to zero. We get:

11.  After performing the similar derivations steps (as we
did for economic dispatch solution without losses), we
can lead to another iterative method known as
Lambda Iterative Method for Economic Dispatch
Solution with Losses.

12. Results & Conclusion

13.  So we see that the transmission loss plays an important
role in deciding the economic dispatch solution. If the
transmission losses are more we have to pay heavy
price for that. Therefore the transmission loss should
be kept under control using various different available
techniques.

14. THANK
YOU