Download Fast Decoupled Power Flow to Emerging Distribution Systems via

1
Fast Decoupled Power Flow to Emerging
Distribution Systems via
Complex pu Normalization
O. L. Tortelli, E. M. Lourenço, Member, IEEE, A. V. Garcia, and B. C. Pal, Fellow, IEEE

Abstract-- This paper proposes a generalized approach of the
per unit normalization, named complex per unit normalization
(cpu), to improve the performance of fast decoupled power flow
methods applied to emerging distribution networks. The
proposed approach takes into account the changes envisaged and
also already faced by distribution systems, such as high
penetration of generation sources and more interconnection
between feeders, while considering the typical characteristics of
distribution systems, as the high R/X ratios. These characteristics
impose difficulties on the performance of both backwardforward sweep and decoupled-based power flow methods. The
cpu concept is centred on the use of a complex volt-ampere base,
which overcomes the numerical problems raised by the high R/X
ratios of distribution feeders. As a consequence, decoupled power
flow methods can be efficiently applied to distribution system
analysis. The performance of the proposed technique and the
simplicity of adapting it to existing power flow programs are
addressed in the paper. Different distribution network
configurations and load conditions have been used to illustrate
and evaluate the use of cpu.
n
Pcpu
Qcpu
RΩ
Rcpu
Sbase
Scpu
Spu
SVA
Vbase
Vcpu
Vpu
XΩ
Xcpu
ZΩ
Zbase
Zcpu
Zpu
Index Terms— Distribution System, Complex Normalization,
Decoupled Power Flow Analysis.
I. NOMENCLATURE
avg
γavg




base
IA
Ibase
Icpu
Ipu
l
Average R/X ratio, rad
Average of the maximum and minimum R/X ratio, rad
Power factor index
Current angle, degree
Volt-ampere power angle, degree
Impedance angle, degree
Base angle, degree
Current, A
Base current, A
Complex normalized current, pu
Conventional normalized current, pu
Number of branches
O. L. Tortelli, and E. M. Lourenço acknowledge the financial support of
the Brazilian government provided through the Science without Borders –
CAPES (Brazilian Post-Graduate Federal Agency).
O. L. Tortelli, and E. M. Lourenço are with the Department of Electrical
Engineering, Federal University of Paraná (UFPR), Curitiba, PR, Brazil (emails: [email protected], [email protected]).
A. V. Garcia is an independent consultant in the area of real-time control
of power systems (email: [email protected]).
B. C. Pal is with the Electrical and Electronics Engineering Department of
Imperial College London, London, UK (email: [email protected]).
Number of buses
Complex normalized active power, pu
Complex normalized reactive power, pu
Resistance, Ω
Complex normalized resistance, pu
Power base, MVA
Complex normalized power, pu
Conventional normalized power, pu
Complex power, MVA
Voltage base, kV
Complex normalized voltage, pu
Conventional normalized voltage, kV
Reactance, Ω
Complex normalized reactance, pu
Impedance, Ω
Impedance base, Ω
Complex normalized impedance, pu
Conventional normalized impedance, pu
II. INTRODUCTION
T
HE Distribution System (DS) is widely impacted by the
new technologies oriented by the Smart Grid concepts. In
order to improve the system reliability, combined with the
increasing addition of generation sources closer to the load
(Distributed Generation concept), the distribution feeders tend
to evolve towards a more meshed topology and to operate as
an active network, with bidirectional power flows.
Distribution System used to be operated as a passive
network with radial topology, and thereby subjected to
unidirectional power flows, i.e., from the feeding substation to
the loads. These assumptions guided the development of
analytical tools dedicated to support operational and planning
studies of DS.
Power flow calculation is the most frequently performed
analysis in electrical power systems. The improvement of this
essential tool taking into account of the evolution of
distribution systems, but considering their peculiarities, such
as the high R/X ratio of its lines, is relevant in the envisioned
context of the emerging power system.
This work is oriented by this context, and proposes a new
approach based on a complex per unit normalization (cpu),
which generalizes the conventional per unit normalization.
The novelty lies in the definition of a complex volt-ampere
basis to circumvent the problems caused by high R/X ratio
2
lines faced by distribution systems. The new technique is
simple to demonstrate and to apply and provides the same
state variables (complex bus voltages) as that obtained through
the conventional pu normalization. Initial efforts reported in
[1, 2] have stimulated the present research, aiming to
consolidate the cpu approach and to demonstrate its
application and advantages when applied to power flow
calculations in emerging distribution systems.
Simulations carried out with different test systems and
distinct topologies and arrangements, including Distributed
Generation (DG), are used to evaluate the performance of the
proposed approach.
III. DS POWER FLOW ANALYSIS – A BRIEF REVIEW
In order to ensure secure operating conditions, two features
are expected from a power flow analysis routine: speed and
solution accuracy. Numerical methods based on NewtonRaphson (NR) [3] are the traditional tools to solve the power
flow, due to its quadratic convergence characteristic that
provides great accuracy with few iteration steps. However, the
full NR method is computationally expensive, even when
techniques to improve efficiency are applied [4]. Several
techniques to overcome the computational burden of full-NR
method have been proposed, usually taking advantage of the
weak coupling between voltage magnitude and active power,
and also between voltage phase angles and reactive power.
Among these, the so-called Fast Decoupled Power Flow
(FDPF) formulated by Stott and Alsac [5] became a standard,
being considered the most efficient and fast method to solve
the power flow problem.
Though the fast-decoupled method is very effective when
applied to transmission systems, it cannot be readily used to
the load flow analysis of distribution systems. Decoupled
load-flow algorithms face convergence problems when
network branches have high R/X ratios [6], since much of the
coupling between the active and reactive power in a circuit is
due to the resistive component of the line impedance. Several
attempts were presented in the 1980’s to overcome this
limitation. An artificial series compensation method was
proposed by Dy Liacco and Ramarao [7] and a parallel one by
Deckmann et al. [8]. There are two main weak points in both
compensation approaches: the structure of the network is
altered with an expansion of the matrix size, and also the
determination of a suitable compensation level to promote a
consistent convergence rate is not an easy task.
Rotation techniques to convert the branch impedances [9]
and admittance matrix elements [10] into nearly
reactive/susceptive values were also proposed as an alternative
to overcome the high R/X ratios of the lines. Rajicic and Bose
[11] followed by Wang et al. [12] presented a modification on
the FDPF equations, but retaining the original structure of the
network. The matrix of the active sub problem (B’) is
modified by an empirical criterion and the equations of the
reactive sub problem are also altered in order to improve the
convergence. Nanda et al. [13] and Van Amerongen [14] have
also reported variations in the FDPF sub matrices formulation
and its consequences in the convergence properties. Results
presented in [14] shown the superior performance on the socalled BX version of the FDPF for systems with high R/X
ratio lines. Another approach, regarding slack bus selection to
improve the convergence, was proposed by Singh et al. [15].
On the other hand and despite of the above efforts, the wellknown Backward-Forward Sweep (BFS) based algorithms,
oriented by a passive and radial DS network, became the most
popular approach for distribution power flow calculation. The
first version of BFS was developed by Berg et al. in 1967
[16]. Since then, several variations and improvements on the
method have been proposed. A solution for weakly meshed
topologies where loops are converted into radial configuration
by choosing appropriate break points was presented in [17].
However, in the emerging distribution systems, meshed
and/or looped networks are expected even between more than
one feeding bus and also with multiple DG sources, making
DS an active network. In this scenario, the Backward-Forward
approach is not the best option since it was not designed to
solve meshed networks.
This paper brings the power flow solution for networks with
high R/X ratio lines into discussion under the impact of
meshed topologies, DG and the increased automation level
expected to DS operation. The axis rotation technique [9] is
recalled to achieve this goal. A new interpretation of this
technique, based on the well-known per unit normalization, is
presented and discussed next.
IV. COMPLEX PER UNIT NORMALIZATION
Traditionally, the parameters and variables of the power
system are normalized in per unit (pu) basis, where real values
are chosen for the voltage and power bases. The complex
normalization (cpu) proposed in this paper can be seen as an
extension of the conventional one, in which a complex voltampere base is adopted, that is:
S base  S base .e  jbase
(1)
The voltage bases, however, are maintained real and
defined just as in the conventional pu normalization, that is, a
different (but related) magnitude value is adopted for each
voltage level:
j0


V
 Vbase
base  Vbase .e
(2)
According to (1) and (2), the impedance base will be also
complex, with the same base angle applied to the power base,
that is:
2
Vbase 

Z
 Z base .e  jbase
base   *
S base
(3)
The complex normalized values of impedances can be
easily determined by:
 .e j
Z

Z  R  jX  
 .e j ( base)
 Z
cpu
pu
 jbase
Z

Z
.
e
base
base
(4)
3
Equation (4) shows that the magnitude of the normalized
impedance is the same regardless of which normalization have
been adopted: conventional (pu) or complex (cpu). However,
cpu also normalizes the angle of the impedance, which varies
according to the base angle, as expressed in (4).
The normalized values of resistances and reactances can be
easily determined from (4):
 . cos(   )
Rcpu  Z
pu
base
(5)
 .sin (   )
X cpu  Z
pu
base
(6)
which leads to:
X cpu
Rcpu
 tan  base 
(7)
The relation stated in (7), clearly shows that R/X ratios can
be adjusted by simply adopting an adequate value for the base
angle, ϕbase. The technique is fully exploited to allow
Newton’s decoupled methods to preserve their typical
convergence rates even when applied to distribution networks.
Once cpu normalization is employed, new values for
network parameters and power injections are obtained in the
same fashion as in conventional pu normalization.
The complex normalized values of volt-ampere power can
be obtained by using the complex power base, that is:
S cpu 
S VA .e j
S base .e  jbase
 S pu .e j ( base)
(9)
 .I* leads to (15) assuring
Substituting (9) and (14) into S  V
that the state variables are the same regardless the
normalization.
 
V
cpu
S pu .e j ( base )

 Vpu .e j (   )  V
pu
I .e j (  base )
Although cpu produces the same results of the impedance
rotation technique [9], it brings the adjustment approach of
R/X ratio for the new context of the DS network. It is
important to emphasize that cpu consists of a generalization of
the per unit normalization concept, in which the conventional
pu is a particular case (when a zero base angle is adopted).
Consequently, the same advantages obtained with the
application of conventional per unit normalization holds when
cpu is used, including the implications with transformers, such
as the equivalence between the impedance seen from the high
and low voltage level after normalization.
This generalization also allows further application of FDPF,
such as an efficient unified power flow analysis over power
networks with large differences in its R/X ratios and voltage
levels [18].
A. Determination of the base angle
The strategy proposed in this paper is based on the evidence
that, the lower the R/X ratio, the better the performance of the
decoupled PF methods (since stronger is the P-QV
decoupling).
Therefore, the choice of the base angle should be guided by
the original R/X ratio of the distribution lines. The first step of
the proposed strategy is to find the average X/R ratio of the
DS, as shown by (16):
∑
S cpu  ( Ppu  j.Q pu ).e jbase
Qcpu
 S pu .sin  base 
(11)
V base
 I base .e jbase
Z .e  jbase
base
( ⁄ )
(13)
Therefore,
I 
cpu
I .e j
A
I .e jbase
base
 I pu .e j (  base )
However, in some DSs, large differences are observed in
the R/X characteristic. In such cases, the average value
between the maximum and minimum X/R ratios of the DS can
be computed by (17) and included in (18), as described next.
( ⁄ )
(12)
In order to show that the solution of the power flow, that is,
the bus complex voltages, are not affected by the cpu
normalization, the corresponding current base is presented:
I
base 
(16)
(10)
so that:
Pcpu  S pu . cos  base 
(15)
pu
(14)
(17)
The average of the two angles given by (16) and (17)
provides a better condition to determine the base angle. As
mentioned before, the determination of the base angle is
guided by the perfect decoupling, which is implicit in (18).
(
)
(18)
Another influence in the base angle’s determination is the
power factor of the loads. In order to counterbalance the effect
of low power factor loads, an additional correction can be
applied to the base angle, provided by (19):
∑
(
( ⁄ ))
(19)
4
Then, the base angle can be achieved by applying (16)-(19)
as follows:
(
)
(20)
Equation (20) leads to a simple, but effective, way to
determine an adequate base angle to cpu normalization. It is
important to notice that for networks with low variation of
R/X ratios and high power factor loads, the determination of
the base angle becomes even simpler, as αavg ≈ avg and ε ≈ 0.
Moreover, small deviations in the configuration (or loading)
of the network usually do not produce large impacts on the
base angle. Considering this, an exhaustive power flow
evaluation of the network can be performed once, looking for
the perfect base angle. Then, it can be kept constant as the
base angle of that network.
B. cpu application to power flow algorithm
The proposed approach can be readily applied to existing
power flow programs or even be incorporated into a dedicated
power flow routine.
It is important to emphasize that, as shown in Fig.1, when
applied to existing programs the proposed approach does not
affect the main power flow algorithm, since cpu normalization
procedure just takes part in a pre- and post-processing stage.
In this case, the final values of the active and reactive power
flows can be easily obtained via reverse normalization, as
shown in the flowchart.
A. Two-bus test feeder
The single line system in Fig. 2 is used here to illustrate the
performance of the proposed normalization regarding the R/X
ratio. With the line impedance magnitude kept constant, the
impedance angle is changed in order to increment the R/X
ratio.
G
z = 0.1 pu
1 + j0.6 pu
Fig. 2. 2-Bus Test Feeder
The plots in Fig. 3 compares the number of iterations
verified by three different PF methods: the full NewtonRaphson (N-R), the XB version of the FDPF (FD-XB), the BX
version of the FDPF (FD-BX), and the proposed FDPF with
cpu approach (FD-CPU). It is straightforward to see from
Fig.3 that the cpu normalization provides the FDPF method a
NR similar behavior, allowing convergence even with very
high R/X ratios, where the other versions of the fast decoupled
method fails. All tests were performed using a 10-6 pu
tolerance to power mismatches.
Original line
and bus data
cpu
normalization
Fig. 3. 2-Bus Test Feeder results
PF program
Vcpu = Vpu
(State variables)
Scpu ≠ Spu
(Power Flows)
Reverse cpu
normalization
B. Base angle impact
The impact of the base angle of cpu on the performance of
the FDPF method (in terms of convergence) are evaluated
here. The simulations, considering the BX version, are related
to two different distribution systems: a small 12-bus test
feeder, depicted in Fig.4 (whose data can be found in [19]) and
the 95-bus feeder, shown in Fig.5, which is based on a real
distribution network model of an UK utility [20]. Both test
feeders considers the presence of DG sources, as indicated in
the Figs. 4 and 5.
Power Flows
Fig. 1. Application of Complex Normalization
V. SIMULATIONS AND RESULTS
Several simulations were performed using different
distribution networks, showing different aspects of the
implemented methodology.
Fig. 4. 12-Bus Test Feeder
Fig. 6 illustrates the graphical evaluation of the number of
iterations versus base angle of the complex normalization
procedure applied to the two test systems. Apart from the base
5
case, two more R/X conditions are considered, by multiplying
the resistance of the lines by a constant factor. It can be
noticed that, for both cases, a significant improvement in the
performance of the FDPF can be achieved with cpu, with at
least 50% of reduction in the number of iterations when
compared with the conventional pu normalization (zero base
angle). Also, with a multiplier factor of 3, the conventional
normalization is not able to reach convergence.
In addition, the convergence performance is tested
regarding different load conditions, as nominal load is
increased by constant factors. The results, summarized
graphically in Fig. 7, reinforce the positive impact of cpu on
the FDPF convergence process.
(b)
Fig. 6. Convergence performance versus base angles considering different
R/X ratios
(a) 12-bus feeder; (b) 95-bus feeder
7
11
12
14
8
10
13
9
15
29
16
17
27
30
25
6
39
26
40
23
3
85
Transformer
33/11kV
84
31
32
37 38
47
42
41 43
44
45
46 48
56
58
50
55
57
52
22
20
53
21
Power
System
33
34
35 36
24
5
28
1
2
4
49
51
54
19
60
18
61
DG
64
59
62
63
65
75
76 77
80
86
78
79
83
90 91
92
93
94
81
(a)
66
68
67
74
69
70
73
72
87
82
88
95
89
DG
71
Fig. 5. 95-bus UK Distribution Feeder
(b)
Fig. 7. Convergence performance versus base angles considering different
load conditions
(a) 12-bus feeder; (b) 95-bus feeder
(a)
Aiming to demonstrate the effectiveness of the proposed
approach to base angle’s determination, presented in Section
IV.A, Table I shows a comparison with the simulation results
presented in Fig.6. As one can see, the 12-bus test system has
low power factor loads (high ε), while the 95-bus feeder has
lines with large variation of R/X ratio (αavg significantly larger
than γavg).
6
TABLE I
BASE ANGLE’S DETERMINATION
TABLE III
BRANCH IMPEDANCES
12-bus test feeder
Branch
1x
2x
3x
71.2o
80.3o
83.5o
70.8o
80.1o
83.3o
R (pu)
0.3254
best
range
75o to 90o
95o to 105o
94.1o
110o to 120o
106.3o
110.5o
95-bus test feeder
1x
2x
3x
64.8o
76.8o
81.1o
49.0o
58.1o
64.3o
0.0315
best
range
Conventional Normalization
50o to 60o
65o to 75o
70o to 90o
58.7o
69.5o
75.0o
The results in Table I clearly show that the base angle value
obtained with (19) is within (or very close) to the best angle
range indicated by the exhaustive simulation.
In order to illustrate the effects of cpu application in the
feeder’s data, the active and reactive power injections and the
branch impedances of the 12-bus test system, along with their
normalized values are shown in Tables II and III. Also, the
resulting state variables, both, with conventional pu and cpu,
are presented in Table III, demonstrating the absolute
coherence between them, as expected.
The application of cpu on a fictitious network may contain
negative values of resistance and/or active loads, as shown in
Tables II and III. It is inherent to the normalization procedure,
but with no physical meaning.
1-2
2-3
3-4
4-5
5-6
6-7
7-8
8-9
9-10
10-11
11-12
X(pu)
|R/X|
Complex Normalization
R (pu)
X(pu)
0.038
2.37
-0.0442
0.0871
0.51
0.098
0.041
2.39
-0.0477
0.0949
0.50
0.173
0.072
2.40
-0.0839
0.1676
0.50
0.263
0.110
2.39
-0.1281
0.2547
0.50
0.090
0.038
2.37
-0.0442
0.0871
0.51
0.083
0.031
2.68
-0.0367
0.0806
0.46
0.361
0.100
3.61
-0.1249
0.3531
0.35
0.466
0.132
3.53
-0.1642
0.4557
0.36
0.239
0.068
3.51
-0.0845
0.2337
0.36
0.125
0.035
3.57
-0.0436
0.1223
0.36
0.102
0.029
3.52
-0.0360
0.0997
0.36
C. Topological considerations
In order to demonstrate the robustness and efficiency of the
proposed approach regarding topology aspects, additional
arrangements of test feeders are considered in this subsection.
In all of them, the selected feeder is replicated and the two
resulting feeders are connected in three different ways.
As presented schematically in Fig. 8(a), the radial
arrangement consists of the series connection of the selected
feeder by a transformer, emulating a two level distribution
system. In turn, weakly-meshed topology is created by
connecting the end of each feeder as a closed loop, as
illustrated in Fig. 8(b). The meshed arrangement, in Fig. 8(c),
arises from the inclusion of additional branches connecting the
two feeders.
(a)
TABLE II
ACTIVE AND REACTIVE POWER INJECTIONS
Conventional
Normalization
Bus
Complex
Normalization
1
Pd (MW)
0.000
Qd(MVAr)
0.000
Pd (pu)
0.00
Qd(pu)
0.00
Pd (cpu)
0.000
Qd(cpu)
0.000
2
0.060
0.060
6.00
6.00
-6.404
5.567
3
0.040
0.030
4.00
3.00
-3.272
3.781
4
0.055
0.055
5.50
5.50
-5.870
5.103
5
0.030
0.030
3.00
3.00
-3.202
2.783
6
-0.025
-0.005
-2.50
-0.50
0.673
-2.459
7
0.055
0.055
5.50
5.50
-5.870
5.103
8
0.045
0.045
4.50
4.50
-4.803
4.175
9
0.040
0.040
4.00
4.00
-4.269
3.711
10
0.035
0.030
3.50
3.00
-3.237
3.282
11
0.040
0.030
4.00
3.00
-3.272
3.781
12
-0.023
0.005
-2.30
0.50
-0.338
-2.329
|R/X|
0.090
(b)
(c)
Fig. 8. Schematic arrangements applied to test feeders:
(a) radial; (b) weakly meshed; (c) fully meshed
7
The performance of FDPF methods was evaluated by
applying the proposed approach to the 12- and 95-bus test
feeders, regarding the topological arrangements depicted in
Fig. 8. The number of iterations when conventional pu
normalization is applied in comparison with the adoption of
cpu is presented in Tables IV and V, considering two load
conditions: the nominal load and a 50% increased load.
TABLE IV
12-BUS: CONVERGENCE PERFORMANCE
Meshed
Looped
Radial
pu
Nominal
Load
Increased
Load
Nominal
Load
Increased
Load
Nominal
Load
Increased
Load
cpu
XB
BX
XB
BX
20
13.5
6
6
29
19.5
8
8
18
13.5
6.5
6.5
26
18.5
8.5
8.5
18
14
6.5
6.5
25
17.5
8
8
TABLE V
95-BUS: CONVERGENCE PERFORMANCE
Meshed
Looped
Radial
pu
Nominal
Load
Increased
Load
Nominal
Load
Increased
Load
Nominal
Load
Increased
Load
cpu
XB
BX
XB
BX
12
12.5
7
76
18
20.5
9.5
10.5
11
11.5
6
5.5
13
15
6.5
7.5
9
10.5
5
5
13
12.5
6.5
6.5
Simulation results in Tables IV to VI clearly show that the
number of iteration required by the proposed FDPF is
significantly reduced when compared to conventional one (at
least 50% less iterations). One can also notice that in some
cases cpu is crucial to reach convergence. Moreover, unlike
the conventional pu normalization, the cpu ensures a very
good performance of the method when the system operates in
a more stressed load condition.
Such a comparison reaffirms the consistent improvement in
the convergence process of the fast decoupled method by the
cpu normalization.
All results presented in this section demonstrates the
positive impact that complex per unit normalization promoted
in the FDPF, assuring a consistent improvement in its
convergence process.
VI. CONCLUSIONS
This paper presents the concept of complex per unit
normalization as a generalization of the conventional
procedure, through the adoption of a complex power base. The
influence of the R/X ratio and the load power factor in the
power base angle is discussed and incorporated in the base
angle’s determination. The resulting technique enables the
application of Fast Decoupled power flow methods to
distribution system analysis.
The application of the proposed approach is fully explored
in the likely scenario faced by distribution systems, including
looped and meshed topologies, where backward-forward
based algorithms are not a suitable choice.
The implementation of cpu is very simple and effective,
promoting a significant reduction in the number of iterations
and computational effort of FDPF routines when compared
with the conventional one.
From the results and bearing in mind that the proposed
approach can be coupled to existing power flow programs, cpu
can be seen as a promising tool for active distribution system
analysis, since it meets the new challenges brought by the
pursuit of a smarter grid.
Three additional test-feeders [18] are also considered in this
simulation section. Table VI summarizes the obtained results.
VII. REFERENCES
[1]
pu
[2]
126-bus
69-bus
33-bus
TABLE VI
ADDITIONAL TEST FEEDERS: CONVERGENCE PERFORMANCE
Nominal
Load
Increased
Load
Nominal
Load
Increased
Load
Nominal
Load
Increased
Load
cpu
XB
BX
XB
BX
14
10.5
4.5
5
27
14.5
6
6.5
14
11.5
4.5
4.5
nc
17.5
5.5
6
17.5
24
6.5
6
29.5
nc
10
9
[3]
[4]
[5]
[6]
[7]
C. C. Durce, O. L. Tortelli, E. M. Lourenço, T. Loddi. “Complex
Normalization to Perform Power Flow Analysis in Emerging
Distribution Systems”. IEEE PES ISGT Europe, Nov. 2012.
E. M. Lourenço, T. Loddi, O. L. Tortelli, “Unified Load Flow Analysis
for Emerging Distribution Systems”. IEEE PES ISGT Europe, Oct.
2010.
W. Tinney and C. Hart, “Power flow solution by Newton’s method,”
IEEE Trans. Power App. Syst., vol. PAS-86, no. 11, pp. 1449–1460,
Nov. 1967.
W. Tinney and J. Walker, “Direct solutions of sparse network equations
by optimally ordered triangular factorization,” Proc. IEEE, vol. 55, no.
11, pp. 1801–1809, Nov. 1967.
B. Stott and O. Alsac, “Fast decoupled load flow,” IEEE Trans. Power
App. Syst., vol. PAS-93, no. 3, pp. 859–869, May 1974.
F. Wu, “Theoretical study of the convergence of the fast decoupled load
flow,” IEEE Trans. Power App. Syst., vol. 96, no. 1, pp. 268–275, Jan.
1977.
T. E. DyLiacco and K.A. Ramarao, "Discussion of reference [6]”.
8
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott and O. Alsac,
"Numerical Testing of power system load flow equivalents", Trans. on
Power App. & Syst., Vol. PAS-99, No.6, pp. 2292-2300, 1980.
A. V. Garcia, A. Monticelli. “Simulation of Distribution Network
through Fast Decoupled Power Flow and Axis Rotation”. IX Nacional
Conference of Power Distribution Systems - SENDI. Bahia, Sep. 1984.
(In Portughese).
P.H. Haley and M. Ayres, ''Super decoupled load flow with distributed
slack bus" IEEE Trans. on Power App. & Syst, Vol - PAS - No.1, pp.
104-113. 1985.
D. Rajicic and A. Bose, "A modification to the fast decoupled power
flow for networks with high R/X ratios" IEEE Trans. On Power
Systems, Vo1.3, 1988.
L. Wang, N. Xiang S. Wang, B.Zhang and M. Huang, "Novel decoupled
power flow" IEE Proc. Pt. C, Vo1.137, No.1, pp.1-7, 1990.
J. Nanda, D. P. Kothari and S. C. Srivastava "Some important
observations on fast decoupled load f low algorithm", (Letter), Proc.
IEEE, Vo1.75, No.5, pp.732-733, 1987.
R. A. M. Van Amerongen, "A general purpose version of the fast
decoupled load flow", IEEE Trans. on Power Systems, Vo1.4, No.2, pp.
760-770, 1989.
S.P. Singh, G . S . Raju and V.S. Subba Rao. “A Technique to Improve
the Convergence of FDLF for Systems with High R/X Lines”.
TENCON'91 - IEEE Region 10 International Conference on Energy,
Computer, Communication and Control Systems.
R. Berg, E.S. Hawkins, and W.W. Pleines, “Mechanized calculation of
unbalanced load flow on radial distribution circuits,” IEEE Trans. On
Power Apparatus and Systems, vol. 86, no 4, pp. 415-421, Apr. 1967.
D. Shirmohammadi, H.W. Hong, A. Semlyen, and G.X. Luo, “A
compensation based power flow method for weakly meshed distribution
networks,” IEEE Trans. on Power Systems, Vol. 3, No. 2, pp. 753-762,
May 1988.
C. C. Durce, E. M. Lourenço, O. L. Tortelli. “Power flow analysis for
interconnected T&D networks with meshed topology”. IEEE PES ISGT
Europe, Dec. 2011.
U. Eminoglu, T. Gözel, M. H. Hocaoglu, “Distribution Systems Power
Flow Analysis Package Using Matlab Graphical User Interface (GUI)”,
2009 Wiley Periodicals Inc.
R. Singh, B.C. Pal, R.A. Jabr, “Distribution System State Estimation
Through Gaussian Mixture Model of the Load as Pseudo Measurement”,
IET Generation, Transmission and Distribution, Vol. 4, 2010.
VIII. BIOGRAPHIES
Odilon L. Tortelli received the degree of Electrical Engineer
in 1992 and the M.Sc. degree in 1994, both from the Federal
University of Santa Catarina (UFSC), Brazil, and the Ph.D.
degree in electrical engineering from the State University of
Campinas (UNICAMP), Brazil, in 2010. Since 1995, he has
been a Faculty member with the Department of Electrical
Engineering at the Federal University of Paraná (UFPR),
Curitiba, Brazil. His research interests are in the area of
computer methods for power systems operation and control.
Elizete M. Lourenço (M’02) received the Electrical
Engineering degree as well as the M.Sc. and Ph.D. degrees in
electrical engineering from the Federal University of Santa
Catarina (UFSC), Florianópolis, Brazil, in 1992, 1994, and
2001, respectively. Since 1995, she has been a Faculty
member with the Department of Electrical Engineering at the
Federal University of Paraná (UFPR), Curitiba, Brazil. Her
research interests are in the area of computer methods for
power systems operation.
Ariovaldo V. Garcia received the B.Sc., M.Sc., and Ph.D.
degrees in electrical engineering from the State University of
Campinas (UNICAMP), Brazil, in 1974, 1977, and 1981,
respectively. From 1975 to September, 2010, he was with the
Electrical Energy Systems Department at UNICAMP and now
is an independent consultant. He has served as a consultant for
a number of organizations. His general research interests are
in the areas of planning and control of electrical power
systems.
Bikash C. Pal (F'13) received the B.E.E. (with honors) degree
from Jadavpur University, Calcutta, India, the M.E. degree
from the Indian Institute of Science, Bangalore, India, and the
Ph.D. degree from Imperial College London, London, U.K., in
1990, 1992, and 1999, respectively, all in electrical
engineering. Currently, he is a Professor in the Department of
Electrical and Electronic Engineering, Imperial College
London. His current research interests include state estimation,
distribution system analysis, power system dynamics, and
flexible ac transmission system controllers.