A Decomposition Aggregation Method for Solving Electrical Power Dispatch Problems.pdf

A Decomposition/ Aggregation Method for Solving
Electrical Power Dispatch Problems
M. H. Mansor
Universiti Tenaga Nasional
(UNITEN), Malaysia
E-mail:mhelmi@uniten.edu.my
M. R. Irving
Brunel Institute of Power Systems,
UK
E-mail: Malcolm.Irving@brunel.ac.uk
G. A. Taylor
Brunel Institute of Power Systems,
UK
E-mail:Gareth.Taylor@brunel.ac.uk
Abstract-This paper presents a new approach to solving the
Economic Dispatch (ED) Problem for a large number of
generators using a decomposition / aggregation method. A
program has been developed to demonstrate the algorithm using
the MATLAB programming language. A 5-bus test system and
the IEEE 26-bus test system are used as demonstration systems.
Each test system is decomposed into small areas and each area
has been solved for Economic Dispatch (locally) using an
Evolutionary Programming (EP) technique. It was ensured that
each area contains at least one generating unit and one supplied
load. The EP will minimise the objective funtion for each area,
minimising the local operating cost including the effects of real
power losses in each area. The optimisation problem for each
area can be regarded as a sub-problem of the decomposition
scheme. Subsequently, the solutions from the areas are
combined (aggregated) to solve the overall system problem. The
results obtained using the decomposition / aggregation method
are compared with the results found when the ED Problem was
solved using a centralised EP approach and the base-case results
found from solving a (non-optimal) load flow. It was found that
applying the aggregation method is a prospective approach for
solving economic dispatch problems with a large numbers of
generators in a power system.
Index Terms--Economic Dispatch (ED), Decompositon,
Aggregation, Evolutionary Programming (EP).
I. INTRODUCTION
Demand for electricity in homes and business has increased
rapidly. The available supply may not be enough to dispatch
sufficient amount of electricity to cater for the demand in 5
years time. This situation will result in unreliable energy for
the consumers [1]. Therefore, more generating units or power
plants need to be built in order to handle the situation. In ten
years time there would be many more than 200 generating
units in the grid system in the UK to serve the high demand.
Today, there are various computer methods existing for the
dispatch of generator power outputs. These include two types
of optimisation techniques: mathematical optimisation
methods and heuristics optimisation. Papers that apply
mathematical optimisation methods such as linear
programming, quadratic programming, integer and mixed-
integer programming, dynamic programming, Lagrangian
relaxation method include [2-10]. Many of the latest papers
for solving the ED Problem use heuristics optimisation
methods such as evolutionary algorithms, simulated
annealing, tabu search, neural networks, fuzzy programming,
hybrid techniques, etc [11-18].
Unfortunately, all the above methods are only efficient for
problems involving up to about 200 generators. For systems
that have many hundreds or thousands of generators, a new
approach is required. This project presents a new approach
for dispatching generator power outputs and shceduling of
generator on / off status for a large scale problerm. The new
approach to dispatching generator power outputs, that is
presented in this paper, is decomposition/ aggregation
method.
This paper presents an investigation into applying the
decomposition/ aggregation methods in conjunction with the
Evolutionary Programming (EP) optimisation technique to
solve the ED problem. EP is one of the Evolutionary
Algorithm (EA) Optimisation Techniques [19]. This approach
is suitable for a general Economic Dispatch problem
formulation.
II. ECONOMIC DISPATCH
Economic Dispatch (ED) is one of the electrical power
systems problems. ED also known as electrical power
dispatch. The definition of economic dispatch (ED) provided
by FERC (E P Act section 1234) is stated as follows: “The
operation of generation facilities to produce energy at the
lowest cost, to reliably serve consumers, recognizing any
operational limits of generation and transmission
facilities.”[20]. Likewise, Ross Baldick (2004) gave the
following definition: “Economic dispatch is sharing
generation between generators to minimize the total fuel
costs” [21]. From above definitions, ED can generally be
identified as a method of determining the lowest cost of
dispatching electrical power from the generating units to the
demand on the system with respect to the system constraints
and the unit constraints [20]. The systems constraints can be
the real power balance between the generation and the
demand, reserve generation capacity, transmission network
limits, network security, etc. Furthermore, the unit constraints
are the operating limits of generators, ramp rate limits,
minimum ‘up time’, etc.
Economic Dispatch plays an important role in improving
the economy of generating unit operation effectively and
hence minimise the total generation cost. This is because the
latest optimisation techniques for solving the ED Problem
give reliable and convincing solutions, as all constraints (the
systems constraints and the unit constraints) are considered in
the formulation of the ED problem. Furthermore, the latest
ED problems may also consider some of the the dynamic

characteristics of the system. Today, power systems operators
(dispatchers) rely on the ED solution as a reference in making
judgements and decisions for daily operation of a grid system
at the lowest feasible operating cost.
The objective function of the ED problem (i.e., total
production cost) can be written as follows:
Minimise ( )
∑
=
=
n
i
i
i
total P
C
C
1
(1)
Where total
C is the total production cost.
( )
i
i P
C is the fuel cost function of unit i in terms of real
power output, Pi. The fuel cost function can be expressed in
the form of a quadratic equation as follows:
( ) ∑
=
+
+
=
n
i
i
i
i
i
i
i
i c
P
b
P
a
P
C
1
2
Where ai, bi and ci are cost coefficients of generating unit i,
subject to the constraints:
i. Real power balance
∑
=
=
+
n
i
i
loss
demand P
P
P
1
(2)
Where demand
P is the total system load demand and loss
P is
the total system loss.
ii. Generator operating limits
max
min i
i
i P
P
P ≤
≤ , i = 1, …, n (3)
Where min
i
P and max
i
P are the minimum and the maximum
real power outputs of ith generator, respectively.
III. PROPOSED APPROACH
The proposed approach is illustrated in Figure 1.
The process of the proposed approach starts with
decomposing an electrical power system in to a set of areas
that contain a number of generators and loads. The number of
areas depends on the size of the system. This paper mainly
concerns the application of the decomposition / aggregation
method in solving the economic dispatch problem, rather than
the choice of how to subdivide the network. Each area must
have at least one generating unit and one supplied load. The
individual areas will be solved for ED by using the EP
technique. The EP will minimise the objective function for
each area, minimising the local operating cost in each area.
The optimisation problem for each area can be regarded as a
sub-problem of the decomposition scheme. Subsequently, the
solutions from the areas are combined (aggregated) to solve
the co-ordination optimisation problem.
The co-ordination optimisation problem can be regarded as
the ‘master problem’ of the decomposition / aggregation
scheme. The master objective function is to minimise the sum
of the costs in area 1, in area 2 and upto area (n), subject to
proper interaction balance of the real power (P) and the
reactive power (Q) flows in and out of each sub-problem.
Hence, the local cost and the real and reactive power
produced by each area, calculated by the EP technique will be
added to obtain the total cost and total loss of the whole
system. The variables and control parameters for the dispatch
problem are the real power generated by each generating unit.
The real power balance and the generator operating limits are
considered as contraints.
A. 5-Bus Test System
A program has been developed to demonstrate the
algorithm using the MATLAB programming language. A
small 5-bus test system and a 26-bus test system are used as
demonstration systems. The 5-bus test system data are
tabulated in Table 1 and Table 2.
TABLE 1: GENERATOR DATA AND COST COEFFICIENTS OF 5-
BUS TEST SYSTEM.
Bus No. Generator
1 (G1)
Generator
2 (G2)
Generator
3 (G3)
Size (MW) 10 to 85 10 to 80 10 to 80
Generator
Cost
Coefficients
a ($/MW2
H) 200 180 140
b ($/MWH) 7.0 6.3 6.8
c ($/H) 0.008 0.009 0.007
TABLE 2: THE SYSTEM LOAD DEMAND OF 5-BUS TEST SYSTEM.
Bus No. Bus 2 (L2) Bus 3 (L3) Bus 4 (L4) Bus 5 (L5)
Real Power
(MW)
20 20 50 60
Reactive
Power (Mvar)
10 15 30 40
The 5-bus system was decomposed into two areas. As
stated earlier, it was ensured that each area contains at least
one generating unit and one supplied load. The decomposed
areas are as shown in Figure 2.
There are two areas in the decomposed electrical system,
separated via the dotted line shown in Figure 2. The top part
of the figure is Area 1 and the bottom part of the figure is
Area 2. These two areas linked together with four
transmission lines f1, f3, f4 and f7. Each area was solved for
Economic Dispatch (locally) using an Evolutionary
Programming (EP) technique. Only transmission lines f2 and
f6 are considered for solving the ED problem in Area 1. For
Area 2 transmission line f5 is considered. The transmission
lines f1, f3, f4 and f7 are paralled together to be approximated
by one equivalent transmission line when solving the master
problem. Since the system is divided into two areas, two sub-
problem need to be solved within each master problem
iteration.

Figure 1: Overview of the decomposition/ aggregation method for solving ED problem.
Figure 2: Decomposed 5-Bus Test System.

There are two areas in the decomposed electrical
system, separated via the dotted line shown in Figure 2.
The top part of the figure is Area 1 and the bottom part of
the figure is Area 2. These two areas linked together with
four transmission lines f1, f3, f4 and f7. Each area was
solved for Economic Dispatch (locally) using an
Evolutionary Programming (EP) technique. Only
transmission lines f2 and f6 are considered for solving the
ED problem in Area 1. For Area 2 transmission line f5 is
considered. The transmission lines f1, f3, f4 and f7 are
paralled together to be approximated by one equivalent
transmission line when solving the master problem. Since
the system is divided into two areas, two sub-problem
need to be solved within each master problem iteration.
Sub-problem 1 (Area 1)
The cost functions for generating units in Area 1 are as
follows:
Generator 1:
(4)
Generator 2:
(5)
The total operating cost in Area 1 is
Cost3
Cost1
1
Area
of
Cost
Total +
= (6)
Hence, the objective function of ED problem of Area 1 can
be written as
Minimise Cost3
Cost1
1
Area
of
Cost
Total +
= (7)
There is only one generating unit in Area 2 which is
generator 2. the cost function for generator 2 is:
Generator 1:
2
g2
0.009P
g2
6.3P
180
Cost2 +
+
= (8)
The objective function of ED problem of Area 2 can be
written as
Minimise Cost2
2
Area
of
Cost
Total = (9)
Master problem
The solutions from solving the ED problem in Area 1
and Area 2 are used to solve the master problem. The
solutions include the total operating cost and the real and
reactive power available in each area. The master objective
function is to minimise the sum of the costs in area 1 and
in area 2.The master problem objective function can be
written as:
Minimise =
Cost
Overall
2
Area
of
Cost
Total
1
Area
of
Cost
Total +
(10)
The transmission lines that connected between these
two areas are paralleled and assumed it connected between
two busbars. The real and reactive power of each area
found from the EP optimisation technique will be used as
the parameters for these two buses for solving the master
problem. Area 1 will be the slack bus for the master
problem. Therefore, only the real power of area 1 will be
varied to solve the master problem. While minimising the
overall cost of the system as shown in equation 10, the
master problem will keep updating the real power
generation required for each generating units in the 5-bus
system until the best set of real power generation to supply
the demand, while minimising the overall operating cost
and also the system losses, is found.
B. 26-Bus Test System
The aggregation/ decomposition program also
demonstrated on 26-bus system. The 26-bus test system
data are tabulated in Table 1 and Table 2.
TABLE 3: GENERATOR DATA AND COST
COEFFICIENTS OF 26-BUS TEST SYSTEM.
The 26-bus system was decomposed into two areas. As
stated earlier, it was ensured that each area contains at
least one generating unit and one supplied load. The
decomposed areas are as shown in Figure 2.
Bus No. Size
(MW)
Generator Cost Coefficients
a ($/MW2
H) b ($/MWH) c ($/H)
Generator 1
(G1)
10 to 500 240 7 0.007
Generator 2
(G2)
50 to 200 200 10 0.0095
Generator 3
(G3)
80 to 300 220 8.5 0.009
Generator 4
(G4)
50 to150 200 11 0.009
Generator 5
(G5)
50 to 200 220 10.5 0.008
Generator 26
(G26)
50 to 120 190 12 0.0075
2
g1
0.008P
g1
7.0P
200
Cost1 +
+
=
2
g3
0.007P
g3
6.8P
140
Cost3 +
+
=

Figure 3: Decomposed 26-Bus Test System.
There are two areas in the decomposed electrical
system, separated via the dotted line shown in Figure 3.
The left part of the figure is Area 1 and the right part of the
figure is Area 2. Each area was solved for Economic
Dispatch (locally) using an Evolutionary Programming
(EP) technique. The transmission lines connecting these
two areas are paralled together to be approximated by one
equivalent transmission line when solving the master
problem. Since the system is divided into two areas, two
sub-problem need to be solved within each master problem
iteration.
follows:
Generator 1:
2
g1
0.007P
g1
7.0P
240
Cost1 +
+
= (11)
Generator 5:
2
g3
0.008P
g3
10.5P
220
Cost5 +
+
= (12)
Generator 26:
2
g3
0.0075P
g3
12.0P
190
Cost26 +
+
= (13)
The total operating cost in Area 1 is
Cos26
Cost5
Cost1
1
Area
of
Cost
Total +
+
= (14)
Hence, the objective function of ED problem of Area 1
can be written as
Minimise
6
Cost5_Cos2
Cost1
1
Area
of
Cost
Total +
= (15)

follows:
Generator 2:
2
g2
0.0095P
g2
10.0P
200
Cost2 +
+
= (16)
Generator 3:
2
g2
0.009P
g2
8.5P
220
Cost3 +
+
= (17)
Generator 4:
2
g2
0.009P
g2
11.0P
200
Cost4 +
+
= (18)
The objective function of ED problem of Area 2 can be
written as
Minimise
Cos4
Cos3
Cost2
2
Area
of
Cost
Total +
+
= (19)
Master problem
The master objective function is to minimise the sum
of the costs in area 1 and in area 2.The master problem
objective function for 26-bus test system can be written as:
Minimise
=
Cost
Overall
2
Area
of
Cost
Total
1
Area
of
Cost
Total + (20)
As mentioned previously, the transmission lines that
connected between these two areas are paralleled and
assumed it connected between two busbars.
IV. RESULTS AND DISCUSSION
Prior to the decomposition / aggregation scheme, the
ED problem of 5-bus system and 26-bus system were
solved using a centralised EP approach and also was
solved for a load flow (non-optimal) solution. These tests
are for the purpose of comparison. The comparison of the
results found from the three methods are shown in Table 4
and Table 5. It is important to compare the results found
using decomposition / aggregation method with the other
approach in order to know the advantage of this approach
over the standard methods.
TABLE 4: COMPARED RESULTS OF LOAD FLOW,
CENTRALISED EP AND THE PROPOSED
DECOMPOSITION/ AGGREGATION METHOD FOR
5-BUS SYSTEM
Representation Load Flow
(Non-Optimal)
Centralised EP Decomposition/
Aggregation Method
Pg1 (MW) 83.051 77.3216 26.3368
Pg2 (MW) 40.000 37.2568 43.3350
Pg3 (MW) 30.000 52.6737 82.1591
Total Generation
Cost ($/h)
2028.2 1608.3 1606.0
Total System
Loss (MW)
3.0526 2.5081 2.6731
TABLE 5: COMPARED RESULTS OF LOAD FLOW,
CENTRALISED EP AND THE PROPOSED
DECOMPOSITION/ AGGREGATION METHOD FOR
26-BUS SYSTEM
Representation Load Flow
(Non-Optimal)
Centralised EP Decomposition/
Aggregation Method
Pg1 (MW) 719.5341 473.8551 477.4357
Pg2 (MW) 79 172.3794 200.316
Pg3 (MW) 20 246.5948 223.6499
Pg4 (MW) 100 113.1528 146.6501
Pg5 (MW) 300 179.8036 166.1310
Pg26 (MW) 60 92.7484 54.2673
Total Generation
Cost ($/h)
17289 15457.9 13505.4
Total System
Loss (MW)
15.534 13.0190 10.0759
From Table 4 and Table 5, it is found that the total
generation cost obtained using decomposition /
aggregation method for 5-bus system and 26-bus system
are 16060 $/h and 13505.4 $/h respectively. The cost is
lesser than the total generation cost obtained using the
centralized EP and load flow (non-optimal) solution.
V. CONCLUSION
An approach of using a decompostion / aggregation
method has been presented in this paper. The method were
implemented on a 5-bus system and 26-bus system. It has
been found that applying the decomposition / aggregation
method is a suitable prospective approach for solving
economic dispatch problems with a large numbers of
generators in a power system. As stated earlier, the main
concern of this paper is not selection of the number and
definition of decomposed areas, but to begin to investigate
the the advantage of implementing the decomposition /
aggregation method in solving ED problem. The number
of decomposed area will be higher as the number of buses
is increased. It is hoped that this approach can be
developed further to allow the electrical power dispatch
problem to be expanded to cope with increasing numbers
of generators in the future.

ACKNOWLEDGEMENTS
The work presented in the present paper is sponsored by
Universiti Tenaga Nasional (UNITEN), Malaysia.
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