2010 IEEE International Conference on Power and Energy (PECon2010), Nov 29 - Dec 1, 2010, Kuala Lumpur, Malaysia
Sizing and Locating Distributed Generations
for Losses Minimization and Voltage
Stability Improvement
Karar Mahmoud , Mamdouh Abdel-Akher and Abdel-Fatah A.Ahmed
APEARC, Department of Electrical Engineering
Aswan Faculty of Engineering, South Valley University
81542 ASWAN, Egypt
Abstract— The paper presents analysis of distribution
system connected with distributed generations. The study
addresses aspects related to optimal sizing and location of
DG units for losses minimization and voltage stability
improvements. Many cases have investigated to highlight
the relationship between the optimum size and location for
losses minimization and the optimum size and location for
stability improvements. The student version of the AMPL
software is used in the proposed study. The objective
function is formulated with full consideration of both
quality and inequality constraints. On the other hand, the
stability index criterion is used for calculating the best
location and size for system stability improvements. The 90
bus test system from the literature is used for the different
studied cases. The results show that calculating minimum
system losses is not necessary to achieve coherence
improvement for the voltage stability problem.
Keywords – Distributed Generations; DG size; Optimization
technique; Stability index
I.
INTRODUCTION
Due to the rapid increase of distributed generations
(DGs), distribution systems can no longer be modeled as
passive networks receiving electric power from high
voltage transmission networks. The DGs are considered
one of the fastest-growing technologies in power
industries. They are currently being connected and
implemented in many distribution utilities worldwide. The
DGs installation is expected to increase accompanied with
a reduction in the cost of these technologies.
DG units are usually allocated and sized such that
benefit the distribution systems through enhancing the
system reliability. In this context, DGs are usually placed
close to the load centers. A common strategy to find the
size and location of DGs is to minimize the power loss of
the distribution network [1- 2]. In [3], the authors
developed a method “2/3 rule” to find the optimal size and
location of DGs into a distribution system for minimum
power losses. This rule is simple and easy to use, but it
cannot be applied directly to a feeder with distribution
loads or meshed networks. There are also some techniques
which use power-flow tools for the optimization problem
[4-5]. These methods are simple since they assume that
every load bus is connected to a distribution generator.
Figure 1. Distribution system connected with DG
With the rapid increase in electric power demand,
distribution systems are usually operates near their
boundary limits [6]. Consequently, the voltage stability
issue become not only of interest in transmission systems,
but also in distribution networks. The DGs can offer an
opportunity to increase the capacity of existing
distribution systems. However, the DGs placement for
power losses minimization may not guarantee similar
enhancement to the voltage stability phenomena. The
focus of this paper is to investigate the optimal size and
location for distributed generations and their impact on
both distribution system losses and voltage stability
problem. The AMPL optimization tool is used to conduct
many results to assess the correct location and size for
DGs. In addition, the stability index criterion is used to
find the best location for DGs. Many test cases are studied
using the 90 bus system from the literature.
II.
DISTRIBUTION LOSSES MINIMIZATION
A. Problem formulation
The problem is to determine allocation and size of DG
which minimizes the distribution power losses under the
condition that number of DGs and total capacity of DGs
are known.
Consider a distribution system connecting to as in Fig.
1. This system is supplying from the grid and a single
distribution generator.
Pgen − Ploss = PD
Pgen = PDG + Pgrid
600
(2)
From (1) and (2),
PDG + Pgrid − Ploss = PD
978-1-4244-8946-6/10/$26.00 ©2010 IEEE
(1)
Where
(3)
Figure 3. Two node test system
Subject to:
i − Pi −
PG
D
∑ Vi V j (Gij cos(θij ) + Bij sin(θij )) = 0
NDG
(9)
i =1
Qi − Qi − ∑ Vi V j ⎛⎜ Gij cos( θij ) − Bij sin(θij )⎞⎟ = 0
G D i =1
⎠
⎝
(10)
i = 1 2 A NB −1
(11)
N
Figure2. Total power losses variation with DG size for a
distribution system
The above equation can be rewritten as:
PDG + Pgrid + Ploss = PD + 2 Ploss
(4)
Since the right hand side for the above equation is
function of the total demand power and total power loss.
F = PD + 2 Ploss
(5)
Where the function F equals to the DG and grid
generated power, DG power, and the total power loss. The
demand power (PD) is constant under a certain loading
condition. Thus, the minimum value for the function F is
defined at the point of minimum real power loss. This
function can also be expressed by this formula:
F = PDG + Pgrid + Ploss
(6)
Thus, the minimum value for the function F is also at
the minimum value of the Ploss. The value of the generated
power of the grid and the DG has no effect into the
minimum value for this function. Figure (2) shows the
variation of total power losses and function F with the DG
power output for a distribution system. The curve shows
that. The minimum value for the power losses and the
minimum value for the function F is at the same optimal
DG size.
From the above detailed analysis we can consider that
the minimum value of the function F is at the minimum
value for the total power loss. However, this is true only
for equal Weighting factor of the DG generated power,
grid generated power and the power loss. Thus, for
different weighting factors of the three members, the
minimum value of the Function F will not at the minimum
value of the total power loss.
B. Objective Function
The mathematical formulation of objective function can
be written as:
Minimize
∑P
NL
i
PDG
+ KL
i =1
K DG + K L + K G = 1
j
∑
NDG
OF = K DG
loss
+ K G Pgrid
(7)
j =1
(8)
601
Vi
≤ V ≤ Vi
,i = 1,2,...NB -1
Min i
Max
⎛ i ⎞
≤ ⎛⎜ Pi ⎞⎟ ≤ ⎛⎜ Pi ⎞⎟
, i = 1,2,...NDG
⎜ PDG ⎟
⎠ Min ⎝ DG ⎠ ⎝ DG ⎠ Max
⎝
⎞
⎞ ⎛
⎛
⎞
⎛
⎟
⎟ ≤ ⎜P
⎟
⎜P
≤ ⎜⎜ P
⎟
⎟ ⎜
⎜ grid ⎟
⎠Min ⎝ grid ⎠ ⎝ grid ⎠Max
⎝
(12)
(13)
(14)
Where KDG ,KL and KG is the weighting factors of
equality constraint. It is noticeable that, when the values
of the three weighting factors is equal, the minimum value
of this function OF is minimum at the same DG size
which the minimum total power losses is found. However,
with different Weighting factors, the minimum value of
the function OF is not at minimum value of the total
power loss. The minimization problem (7) is subjected to
some technical constraints (9-14). The real and reactive
power injection in buses are treated as equality
constraints. The voltage limits at buses and real and
reactive generation limits of generators are treated as
inequality constraints.
III. VOLTAGE STABILITY INDEX
If we considered a two node test system shown in Fig.
3, the static voltage stability index L from a simple power
system can be calculated as in [6]
[(
L j = 4 X Pj − R Q j
)2 + (X Q j − R Pj )Vi 2 ] Vi4
Where:
X:
reactance of branch j
R:
resistance of branch j
active power at the receiving bus j
Pj:
Qj:
reactive power at the receiving bus j
Vi: voltage at receiving bus
(15)
13
Slack Bus
1
2
3
4
21
5
7
9
11
12
17
19
20
28
6
8
10
14
16
22
24
26
27
23
25
74
76
58
73
75
77
69
70
71
15
29
35
30
31
18
32
34
33
56
57
36
80
37
72
38
40
39
59
41
79
81
60
44
46
78
42
43
49
45
50
82
64
47
61
48
62
63
66
65
63
68
67
53
54
86
83
85
51
88
86
84
89
55
90
.
Figure 4. The 90 bus distribution system
OF
P loss
6.8
6.78
6.76
Optimal
Point
0.6
0.5
0.4
OF
6.74
6.72
0.3
6.7
0.2
6.68
6.66
Power Loss (MW)
6.82
0.1
6.64
6.62
2
5
8
11
14
17
20
23
26
29
32
35
38
41
44
47
50
53
56
59
62
65
68
71
74
77
80
83
86
89
0
Bus
Figure 5. The objectives function and total power loss values for each optimal DG size for 90 bus distribution system
Lj stands for the voltage stability index of branch j.
This formula has an advantage that the effect of node
voltages is taken into account in the expression. The
voltage stability index of total distribution system is
defined by
L = Max{L1 L2 .... Ln −1}
(16)
602
The branch corresponding to the index value L is called
the weakest branch. The voltage collapse must start from
the weakest branch. Therefore, the margin of voltage
stability can be obtained according to the deviation
between L and the critical value 1.0.
the optimum size of the DG at bus 10 is of 10.99 MW.
The system total loss in this case is 0.3 MW. However,
there is no any improvements in terms of the stability has
been made due to this optimal sizing and placement for
losses reduction. The stability index is found to be of a
0.24 pu at bus 42 which is similar to that of the base case
Table I Losses for different case studies
Power
loss (MW)
Grid
Power (MW)
DG
Power (MW)
Weakest
Branch
Index L
Case 2
Case 3
Case 4
-
10
42
42
0.50
0.30
0.42
0.41
19.95
8.77
16.87
17.75
-
10.99
3.00
2.11
0.25
Case 1
0.2
42
42
43
43
0.24
0.24
0.11
0.11
0.15
L
DG Bus
Case 1
0.1
0.05
87
82
77
72
67
62
57
52
47
42
37
32
27
22
17
7
12
2
0
0.25
Case 2
0.2
L
0.15
0.1
0.05
87
82
77
72
67
62
57
52
47
42
37
32
27
22
17
12
2
0.25
Case 3
0.2
0.15
L
IV.
RESULTS AND DISCUSSIONS
The 90 bus distribution system in ref. [6] is used to
study different case studies (fig. 4). The system under
study contains 90 buses and 89 branches. It is a radial
system with the total load of 19.45MW and 9.72 MVAR.
The nonlinear programming problem is solved by using
the student version of the AMPL solver [7]. The following
different cases are simulated and studied:
7
0
Figure 6 the stability index variation at different DG size
0.1
0.05
Case 4:
87
82
77
72
67
62
57
52
47
42
37
32
27
22
17
12
7
Without DG.
Connecting a DG with optimal size and
location for minimum power loss.
Connecting a DG at the weakest branch of
the test system.
Connecting a DG with optimal size at the
weakest branch of the test system.
0.25
Case 4
0.2
0.15
L
Case 3:
2
0
Case 1:
Case 2:
0.1
0.05
603
87
82
77
72
67
62
57
52
47
42
37
32
27
22
17
7
12
0
2
Table 1 shows a summary for results calculated from
different studied cases. In case 1, the classical power-flow
analysis is performed and the total losses are calculated.
Also for the stability index is calculated using (15). The
power supplied from the grid in Case 1 equals to 19.95
MW with the absence of any DGs. The losses in this case
equals to 0.5 MW and the highest stability index is of 0.24
pu at bus 42.
In case 2, the objective is to find the correct size of the
DG such that the losses are minimum. In this test, the
optimum size of the DG is obtained at all buses of the
feeder. Fig. 5 shows the variation for both the value of the
objective function FO as well as the value of the losses at
each bus. The figure shows that the best location of the
DG is at bus 11 at which minimum losses occurs whereas
Bus ID
Figure 7 the Stability index for different cases
study.
In case 3, the DG is allocated at the weakest at bus 42
witout any considerstion to the system losses. The size of
the DG has been changed in steps to find the best value of
the stability index, this is shown in Fig. 6. The stability
index value in this case found to be 0.11 as given in Table
1. Altough, the objective function in (6) does not
considered but a reduction in losses has been made due to
1.02
Case 1
Case 3
1
Voltage (Pu)
0.98
Case 2
Case 4
0.96
0.94
0.92
0.9
0.88
86
81
76
71
66
61
56
51
46
41
36
31
26
21
16
11
6
1
0.86
Bus
Figure 8 Voltage profile for different studied cases
allocating the DG at bus 42. The total losses in this case
found to be of 0.43 MW.
Case 4 is the final case study. This case kept to choose
the location of the DG at the weakest or near by the
weakest branch. Then, the size of the DG is determined
using the optimization solver basedon the objective
function given by (6). The results for this test is also has
been recorded in Table 1. In this case study, the losses has
been decresed with a small value and the stability index
have the same value as case 3. The stability index at all
busses for different studied cases is exhibited in Fig. 7.
The fgiure indicates that the recution in system losses does
not gurantee oveall system reliabity improvement
condidering the system voltage stability. Finally, Figure 8
shows the volatge profile for different cases, the figure
exhibts a volatge improvemnet when the DG is installed at
bus 42 in Case 3 and Case 4.
VII. CONCLUSION
The paper has presented investigation when more than
one aspect is considered for DG sizing and allocation. In
this paper, both losses reduction as well as voltage
stability are considered. The initial results shows that
more comprehensive studies are required for that topic to
allocate the DG with considering other important aspects
rather than considering only the distribution system losses.
The initial results obtained here shows that the allocation
of the DG is not necessary be based on the system losses
amount, however the size should be computed based on
minimum loses as have been demonstrated by different
studied cases. The allocation and sizing of the DG
according the system did not improve the system voltage
stability problem. On the other hand, when the DG is
located at the weakest branch and sized according to the
minimum losses, improvements has been made in both
system voltage stability as well as system losses.
604
ACKNOWLEDGMENT
The authors gratefully acknowledge the contribution of
the Science and Technology Development Fund (STDF)
under project no. 346 and the US Egypt Joint Science and
Technolgy Fund Under the project no. 839 for providing
research funding to the work reported in this paper.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Mithulananthan N, Oo Than, Van Phu Le. Distributed generator
placement in power distribution system using genetic algorithm to
reduce losses. TIJSAT 2004;9(3):55–62.
Griffin T, Tomosovic K, Secrest D, Law A. Placement of
dispersed generations systems for reduced losses. In: Proceedings
of the 33rdHawaii international conference on sciences, Hawaii,
2000.
H. L. Willis, “Analytical methods and rules of thumb for modeling
DG-distribution interaction,” in Proc. 2000 IEEE Power
Engineering Society Summer Meeting, vol. 3, Seattle,WA, July
2000, pp. 1643–1644.
N. S. Rau and Y.-H.Wan, “Optimum location of resources in
distributed planning,” IEEE Trans. Power Syst., vol. 9, pp. 2014–
2020, Nov. 1994.
J. O. Kim, S. W. Nam, S. K. Park, and C. Singh, “Dispersed
generation planning using improved hereford ranch algorithm,”
Elect. Power Syst. Res. , vol. 47, no. 1, pp. 47–55, Oct. 1998.
H. Chen, J. Chen, D. Shi, and X. Duan, “Power flow study and
voltage stability analysis for distribution systems with distributed
generation,” in Proc. IEEE PES General Meeting, Jun. 2006, pp.
1–8.
A Modeling Language for Mathematical Programming, the
website can be : http://www.ampl.com/