Economic Dispatch of a Power System w/ Losses
Joseph Gregory E. Gumalal
Abdul Aziz Guevarra Mabaning
Department of Electrical Engineering and Technology
College of Engineering
Mindanao State University - Iligan Institute of Technology
Iligan City, Philippines
greggumz28@gmail.com
IEEE Member
Department of Electrical Engineering and Technology
College of Engineering and Technology
Mindanao State University - Iligan Institute of Technology
Iligan City, Philippines
mraaguevarra@gmail.com
Abstract—In optimization, economic dispatch is one of many methods
to ascertain the optimal output of generating facilities in the short term
to meet the system load. The researcher concisely explains the various
intricacies of Economic Dispatch and presents an example problem which
uses the various objective functions and constraints.
Index Terms—economic, dispatch, generator, cost, optimization
I. I NTRODUCTION
When trying to operate and run an electricity distribution business,
be it public or private, a general rule of thumb is always to maximize
profits while minimizing the cost. Optimization is a key factor in
these types of business and nominating the optimal power from the
generating facilities in the vicinity. One such method of optimization
is economic dispatch, wherein a short-term determination is done, to
ascertain the optimal output of a number of electricity generation
facilities, to meet the system load, at the lowest possible cost,
subject to transmission and operational constraints. In this paper, the
researcher will try to explain and give a sample problem to clearly
demonstrate the importance of Economic Dispatch.
II. O BJECTIVES
In performing economic dispatch operations, the following objectives must be met:
1) To be able to provide a clear and concise economic dispatch
problem
2) To define the various terms and intricacies of economic dispatch
3) To provide a satisfying and acceptable solution to the problem
The total Load demand L that is to be delivered to the
consumers is simply L = X1 + X2 [2].
3) INEQUALITY CONSTRAINT
In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several
types of constraints—primarily equality constraints, inequality
constraints, and integer constraints. [4]. In the case of power
systems, it is the maximum or minimum power that a generator
is allowed to output within the framework of the optimization
dilemma.
4) POWER LOSSES
Power losses refer simply to losses incurred within the
generator or in the delivery of power. Since not all power is to
be delivered to the consumers, some power drops are inevitable
and should be part of consideration when doing short-term
economic dispatches.
IV. STATEMENT OF THE PROBLEM
III. C ONCEPTS
1) COST FUNCTION
In performing basic economic dispatch manipulation,
there are a few key components that should be considered.
There is the cost or objective function, which is a mathematical
formula used to used to chart how production expenses will
change at different output levels. In other words, it estimates
the total cost of production given a specific quantity produced
[1]. For a case where there are two generators outputting load
P1 and P2 , the total cost function would equate to [2]:
Ctotal = α1 + α2 + β1 P1 + β2 P2 + γ1 P12 + γ2 P22
Fig. 1. Basic Line Diagram for Economic Dispatch
(1)
2) LOAD DEMAND
Load demand pertains to the total power your generators
are required to deliver to the consumers. In general, the total
power required by a facility [3]. This demand is also known
as the equality constraint for the function. In the case of the
following example, two generators G1 and G2 with power
payloads X1 and X2 ,
This problem is taken from a University of Nevada, Las Vegas
(UNLV) Lecture Portable Document Format (PDF) [5]. A certain
distribution company has calculated that the load demand is 150MW.
Nominations are in order, and three generators with power outputs
P1 ,P2 , and P3 are available to be used in this scenario. The fuel cost
and plant output limits are:
C1 = 200 + 7.0P1 + 0.008P12
10M W ≤ P1 ≤ 85M W (2)
C2 = 180 + 6.3P2 + 0.009P22
10M W ≤ P2 ≤ 80M W (3)
C3 = 140 + 6.8P3 + 0.007P32
10M W ≤ P1 ≤ 70M W (4)
with the total power losses defined by the equation:
Ploss = 0.000218P1 + 0.000228P2 + 0.000179P3
Find the optimal dispatch and total cost in
(5)
$
.
hr
Graphical representation of the three cost functions for each
generator is shown:
V. METHODOLOGY
•
Langrangian Function
To tackle this type of economic dispatch problem, a
Lagrangian function is formulated. The Lagrangian function
is a technique that combines the function being optimized
with functions describing the constraint or constraints into a
single equation. Solving the Lagrangian function allows you to
optimize the variable you choose, subject to the constraints you
can’t change. In an ideal scenario where losses are neglected,
the Lagrangian function is:
ngen
X L = Ctotal + λ Pdemand −
Pi
(6)
i=1
where Ctotal is the sum total of all generator cost functions
and Pdemand is the total load demand of the system that the
generators are trying to power. In the case of losses being
included, it is then added to the equation, thus becoming:
Fig. 2. Generator 1 Cost Function Curve
ngen
X L = Ctotal + λ Pdemand + Ploss −
Pi
(7)
i=1
•
Condition for optimality
By taking the derivative of the Langrangian function in
terms of Pi and λ, we can surmise the following equations:
dL
= βi + 2γi Pi − λ
dPi
dL
=
dλ
ngen
Pdemand −
X
(8)
Pi
(9)
i=1
rearranging and combining the equations to solve for λ, we now
have the equation
λ=
Pngen
Pdemand + i=1
Pngen 1
i=1
Fig. 3. Generator 2 Cost Function Curve
•
βi
2γi
(10)
2γi
Power Demand
After solving for the Lagrange multiplier λ, the power for
each generator can now then be solved by substituting λ to
equation (8), equating itP
to zero, and calculating for Pi . By
ngen
checking if Pdemand = i=1
Pi and that we are within the
constraints, we can then confirm that the solution is correct.
VI. RESULTS AND DISCUSSION
Following the rules and directions in solving the economic dispatch
for a three generator system with constraints and losses, we now have
the values for the power output of the three generators, along with
the losses associated with the problem.
P1 = 31.94M W
P2 = 67.278M W
P3 = 50.8118M W
Ploss = 29.8kW
Fig. 4. Generator 3 Cost Function Curve
Fortunately, the power output of the generators were within the
constraints, thus not requiring further iterative calculations.
VII. C ONCLUSION
Economic dispatch is a very important tool in making informed and
cost-effective decisions when it comes to meeting a load demand.
However, in cases where the computed power exceeds beyond the
constraints of the generator, high iteration calculations is not feasible
by just using manual or handwritten solving. By employing the help
of a program, most prominently used being MATLAB, we can lighten
the workload by allowing the program to optimize and calculate
efficiently.
R EFERENCES
[1] Anonymous, “What is a cost function?” online. [Online]. Available: https:
//www.myaccountingcourse.com/accounting-dictionary/cost-function
[2] A. A. G. Mabaning, “Introduction to Power System Economics,” EE 155,
2019.
[3] Anonymous, “Load demand,” 1996. [Online]. Available: https://www.its.
bldrdoc.gov/fs-1037/dir-011/ 1557.htm
[4] ——, “What are constraints,” online. [Online]. Available: https:
//en.wikipedia.org/wiki/Constraint (mathematics)
[5] UNLV, “Economic dispatch,” pDF. [Online]. Available: http://www.egr.
unlv.edu/∼eebag/Economic%20Generator%20Dispatch%20740.pdf