International Journal of Electrical and Electronics Engineering Research (IJEEER) ISSN 2250-155X Vol.2, Issue 3 Sep 2012 37-55 Š TJPRC Pvt. Ltd.,
LOAD FLOW SOLUTION FOR UNBALANCED RADIAL DISTRIBUTION SYSTEMS 1
K.JITHENDRA GOWD, 2CH.SAI BABU& 3S.SIVANAGARAJU 1
JNTUA College of Engineering, Anantapur, Andhra Pradesh, India
2
University College of Engineering, JNT University, Kakinada, Andhra Pradesh, India
ABSTRACT This paper presents a simple three phase load flow method to solve three-phase unbalanced radial distribution system (RDS). A three phase load flow solution with considering most of load modeling is presented which has good convergence property for any practical distribution networks with practical R/X ratio. It solves a simple algebraic recursive expression of voltage magnitude, and all the data are stored in vector form. The algorithm uses basic principles of circuit theory and can be easily understood. Mutual coupling between the phases has been included in the mathematical model. The proposed algorithm has been tested with several unbalanced distribution networks and the result of an unbalanced RDS is presented in the article. The application of the proposed method is also extended to find optimum location for reactive power compensation and network reconfiguration for planning and day-today operation of distribution networks.
KEYWORDS: Radial Distribution Networks, Load Flow, Circuit Model, Three-Phase Four-Wire, Unbalance.
INTRODUCTION For any electrical system, the determination of the steady state behavior is the one of the most fundamental calculation. In power systems, this calculation is the steady state power flow problem, also called load flow. The majority of power flow algorithms in wide use in industry today, most notably, the Newton-Raphson method and its variants [1,2] have been developed specifically for transmission systems which have a meshed structure, with parallel lines and many redundant paths from the generation points to the load points. The focus of this paper is on the solution of the power flow problem for the distribution system. Typically, a distribution system originates at a substation where the electric power is converted from the high voltage transmission system to a lower voltage for delivery to the customers. Unlike a transmission system, a distribution system typically has a radial topological structure. Unfortunately, this radial structure, along with the higher resistance/reactance (R/X) ratio of the lines, makes the fast-decoupled Newton method unsuitable for most distribution power flow problems. Various efficient distribution
38
K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
power flow algorithms which exploit the radial structure have been proposed in the literature. These algorithms can be classified into three groups: • Network reduction methods [3] • Backward/ forward sweep methods [4-9] • Fast decoupled methods [10-12] All of the proposed methods have some limitations. Recently, many researchers presented techniques, especially to obtain the load flow solution for distribution networks. Das et al. [13] have proposed a load flow solution method by writing an algebraic equation for bus voltage magnitude. However, this method is suitable for single-phase analysis. A few researchers have proposed a load flow solution techniques to analyze unbalanced distribution systems. Zimmerman et al. [12] have formulated load flow problem as a set of non-linear power mismatch equations as a function of the bus voltages. These equations have been solved by Newton’s method. Thukaram et al. [14] have proposed three phase power flow algorithm based on the forward backward walk along the network. The method considers some aspects of three phase modeling of branches and detailed load modeling. In recent years the threephase current injection method (TCIM) has been proposed [15]. TCIM is based on the current injection equations written in rectangular coordinates and is a full Newton method. As such, it presents quadratic convergence properties and convergence is obtained for all but some extremely ill-conditioned cases. Further TCIM developments led to the representation of control devices [16], [17]. Miu et al., [18] have also proposed method for solving three-phase radial distribution networks. The objective of this work is to develop a formulation and an efficient solution algorithm for the distribution power flow problem which takes into account the detailed and extensive modeling necessary for use in the distribution automation environment of a real world power system which is based on basic systems analysis method and circuit theory. The proposed method requires lesser computer memory, computationally fast and involves only recursive algebraic equations to be solved. The algorithm has been developed considering that all loads draw constant power. However, the algorithm can easily accommodate composite load modeling, if the composition of load is known. The algorithm has good convergence property for practical unbalanced radial distribution networks.
MODELLING OF COMPONENTS OF UNBALACED RADIAL DISTRIBUTION SYSTEM Unbalanced Three Phase Line Model A three phase line section model between bus p and q is shown in Fig 1.
39
Load Flow Solution for Unbalanced Radial Distribution Systems
Fig 1 Three phase line section model A 4X4 matrix, which takes into account the self and mutual coupling effects of the unbalanced three-phase line section, can be expressed as
=
(1)
After Kron’s reduction is applied, the effects of the neutral or ground wire are still included in this model and (1) can then be rewritten as
=
(2)
The relation between branchy voltages and branch currents in Fig.1 can be expressed as
(3)
For any phases fail to present, the corresponding row and column in this matrix will contain null entries. The general forms of showing the branch voltage and branch current are
40
K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
Line Shunt Admittance Model
Fig 2 Shunt admittances of the line sections
In the Fig.2, the shunt admittances (capacitances) are modeled. It represents the shunt admittances connected to each phase and the admittances connected between the phase and ground. Since the currents are injected into the line, the directions of the injected currents are as represented in the figure. These current injections for representing the line charging, which should be added to the respective compensation current injections at nodes p and q are given by
=
(4)
Shunt Capacitor Model
Shunt capacitors, often used for reactive power compensation in a distribution network, are modeled as constant capacitance devices. Capacitors are often placed in distribution networks to regulate voltage levels and to reduce real power loss. As with loads, they can be connected in a grounded wyes configuration or an ungrounded delta configuration. In fact, they are treated in exactly the same way as a purely reactive constant impedance load. It is assumed that shunt capacitors in grounded sections of the network are wye connected and those in ungrounded sections are three phase and delta connected. The constant model parameter, in this case, is the admittance which is computed from the given nominal reactive power injection. The Fig. 3 and Fig 4 represent the capacitors placement in star and delta connections
41
Load Flow Solution for Unbalanced Radial Distribution Systems
Fig 3 Capacitors connected in wye connection. Ia = -YaaVa Ib= -YbbVb
(5)
cc
Ic= -Y Vc
Fig 3 Capacitors connected in delta connection. Ia = Yab/3 (-2Va+Vb+Vc) Ia = Yab/3 (-2Va+Vb+Vc) (6) Ia = Yab/3 (-2Va+Vb+Vc) Load Model All the loads are assumed to draw constant complex power (S = P+jQ). It is further assumed that all three-phase loads are star connected and all double and single-phase loads are connected between line and neutral. A node in a radial system is connected to several other nodes. However, owing to the structure, in a radial system, it is obvious that a node is connected to the substation through only one line that feeds the node. The equations (7) to (9) can be written refer to the power at the receiving end node q.
=
(7)
=
(8)
=
(9)
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K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
Star-Connected Loads
In the case of loads which are connected in star is single phase loads connected line-to-neutral, the load current injections at the Kth bus can be given by =
,m â‚Ź [a,b,c ] Where
and
(10 ) denote real power, reactive power and complex conjugate of the voltage
phasor of each phase, respectively. For simplicity for a constant power consumption at each bus was assumed during the simulation tests.
Fig 4 Loads connected in star connection
Delta-Connected Loads The current injection at the Kth bus for three-phase load connected in Delta are single-phase load connected line-to-line can be expressed by =
-
(11)
=
-
(12)
Ic=
-
(13)
Fig 5. Loads connected in delta connection
Load Flow Solution for Unbalanced Radial Distribution Systems
43
Transformer Model The impact of the numerous transformers in a distribution system is significant. Transformers affect system loss, zero sequence current, grounding method, and protection strategy. Although the transformer is one of the most important components of modern electric power systems, highly developed transformer models are not employed in system studies. In this thesis a transformer model and its implementation method are shown, so that large-scale unbalanced distribution system problems such as power flow, short circuit, system loss, and contingency studies, can be solved. Recent interest in unbalanced system phenomena has also produced a transformer model adaptable to the unbalanced problem which is well outlined in [19]. The model developed thus far can be applied directly to distribution power flow and short-circuit analysis. However, it is still not accurate for system loss analysis because the transformer core loss contribution to total system loss is significant [20, 21]. To calculate total system loss, the core loss of the transformer must be included in the model. The complete transformer model combines the unbalanced and loss models from [20] and [21] in order to integrate system loss analysis in power flow or short-circuit studies. It is important to note that the unbalanced transformer model derived by Dillon in reference [20] cannot be applied directly to either the factorized YBus or direct inverse YBus method because of numerical considerations. For some connections such as grounded wye-delta, delta- grounded wye, these models make the system YBus singular. Therefore, the application of the factorized or direct inverse methods becomes impossible. To solve this problem, this thesis introduces an implementation method in which artificial injection currents are used to make the system YBus nonsingular. \Derivation of Transformer Models A three-phase transformer is presented by two blocks shown in Fig. 6.One block represents the per unit leakage admittance matrix YTabc, and the other block models the core loss as a function of voltage on the secondary side of the transformer.
Fig 6 Overall Proposed Transformer Model The presence of the admittance matrix block is the major distinction between the proposed model and the model used in [14]. In the proposed model, Dillon's model is integrated with the admittance matrix part. As a result, the copper loss, core loss, system imbalance, and phase shift
44
K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
characteristics are taken into account. The implementation method is introduced in the following sections. Core Loss The core loss of a transformer is approximated by shunt core loss functions on each phase of the secondary terminal of the transformer. These core loss approximate functions are based on the results of EPRI load modeling research which state that real and reactive power losses in the transformer core can be expressed as functions of the terminal voltage of the transformer. Transformer core loss functions represented in per unit at the system power base.
Where A = 0.00267 D = 0.00167
B=0.734 x 10-9 C=13.5 E= 0.268 x 10-1
F= 22.7
is the voltage magnitude in per unit. It must be noted that the coefficients A, B, C, D, E and F are machine dependent constants. For this work, core losses are represented by the functions and typical constants shown above. Admittance Matrix The admittance matrix part of the proposed three-phase transformer models follows the methodology derived by Dillon, but a novel implementation is introduced here in. For simplification, a single three-phase transformer is approximated by three identical single-phase transformers connected appropriately. This assumption is not essential; however, it simplifies the ensuing derivation and explanation. Based upon this assumption, the characteristic sub matrices used in forming the three- phase transformer admittance matrices can be developed.
=
Where
=
=
=
Load Flow Solution for Unbalanced Radial Distribution Systems
45
SOLUTION METHODOLOGY The proposed power-flow technique is used for solving radial distribution networks by calculating the total real and reactive power fed through any node. A unique node, branch and lateral numbering scheme, which help to evaluate exact real and reactive power loads fed through any node and receiving-end voltages is proposed. It is assumed that the three-phase radial distribution networks are balanced and can be represented by their equivalent single-line diagrams. Identification of Nodes Beyond All The Branches The detailed flowchart for identifying the nodes beyond all branches is presented in Fig 2.12. This procedure is very easy in finding the nodes of the system, to find the number of branches and to find the exact current flowing through all the branches.
Fig 2.12 Flowchart for identifying nodes
46
K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
Load Flow Calculation The load flow solution is carried out by considering a branch which consists of three phases in the network as shown below.
The receiving-end node voltage can be written as
Where
=
The equation (1) can be evaluated for p = 1,2 . . . ln. where ln is the total number of branches. Current through branch i is equal to the sum of the load currents of all the nodes beyond branch i plus the sum of the charging currents of all the nodes beyond branch i plus the sum of all injected capacitor currents of all the nodes, i.e.
(14) The real and reactive power losses of pth node is given by
(15)
Load Flow Solution for Unbalanced Radial Distribution Systems
47
Initially, a constant voltage of all the nodes is assumed and load currents, charging currents and capacitor currents are computed. After currents have been calculated, the voltage of each node is then calculated. The real and reactive power losses are calculated. Once the new values of voltages of all the nodes are computed, convergence of the solution is checked. If it does not converge, then the load and charging currents are computed using the recent values of the voltages and the whole process is repeated. The convergence criterion of the proposed method is that if, in successive iterations the maximum difference in voltage magnitude (Dvmax) is less than 0.0001 p.u., the solution has then converged. This solution method is known recursive voltage computation method The power flow calculation explained can be easily understood by representing in the form of flow chart shown in Fig 2.13
2.13 Flowchart for load flow solution
RESULTS AND ANALYSIS The proposed method is illustrated with two IEEE test systems consisting of 13 bus and 37 bus Unbalanced Radial Distribution Systems on on P-IV computer with 2.4 GHz frequency
48
K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
Example 1: The proposed algorithm is tested on 13 bus URDS. For the load flow, the base voltage and base kVA are chosen as 4.16 kV and 100 kVA respectively. Table 2.2 Voltages and phase angles of a 13 node radial distribution system Bus No
Phase A Va
deg
1
1
Phase B
Phase C
Vb
Vc
deg 1
0.99655
0 0.14 0.88 0.88 0.16 0.16
0
0
0.99
0
0 0.88 0.94 0.89
0.99
0 0.87
2 0.99734 3 0.99172 4 0.99172 5 0.99655 6
1
-120 120.2 120.2 120.2 120.2 120.2 120.3 120.4 120.2 120.2
0
0
0.9882
119.24
0
0
0.9877
119.2
0
0
0
0
1 1 1 1 1
7 8 9 0.99172 10 0.99013 11 0.99121 12
0
13 0.98979
deg
1
1
120
0.9984
119.84
0.9887
119.26
0.9887
119.26
0.9978
119.82
0.9978
119.82
0.9997
119.84
1.0003
119.84
0.9887
119.26
0.9882
119.27
Table 2.3 Power losses of 13 node radial distribution system Description
Phase A
Phase B
Phase C
Total Active Power Loss (kW)
13.4527
18.816
16.4604
Total Reactive Power Loss (kVar)
12.7443
35.0432
32.3409
49
Load Flow Solution for Unbalanced Radial Distribution Systems
Table 2.4 Summary of test results of 13 bus radial distribution system Proposed method Branch
Phase A
Phase B
Phase C
kW
kVar
kW
kVar
kW
kVar
1
2.2522
0.8919
0.9955
8.954
3.96
8.8135
2
1.5601
0.9494
7.9322
17.69
0.28
16.941
3
0
0
0
0
0
0
4
0.1966
0.0324
0.2581
0.75
0.24
0.094
5
7.8779
5.8727
7.6779
5.872
7.98
6
0
3.449
0
0
2.48
5.8727 -1E04
7
0
0.9825
0
0
0.9
0
8
0
0
0
0
0
0
9
0.9846
0.5664
0.7641
0.842
0.61
-0.003
10
0.2712
0
0.2923
0.459
0
0.3965
11
0
0
0.8959
0
0
0.2262
12
0.3101
0
0
0.48
0
0
The Total Active Power losses are 48.72911 kW The Total Reactive Power losses are 80.1294 kVar For the proposed method, the maximum deviation of voltage and its phase angle from the Forward backward sweep method is 0.0001 p.u and 0.01 deg. The load flow is converged in 2 iterations for the tolerance of 0.001 p.u.. When the tolerance limit is set as 0.0001, the number of iterations required for the convergence is 3 for Forward backward sweep method and 2 for proposed method. The execution time is 0.048 seconds for the Forward backward sweep method and 0.016 seconds. Example – 2:The proposed algorithm is also tested on IEEE 37 bus URDS . The feeder consists of three-wire delta operating at a nominal voltage of 4.8 kV.
50
K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
Table 2.5 Voltages and phase angles of a 37 node radial distribution system Bus No.
Proposed method Phase A Va
Phase B deg
Phase C
Vb
deg
1
1
0
1
2
0.99862
-0.026
3
0.99703
-0.126
0.99656
4
0.99489
0.268
0.99325
5
0.9924
-0.314
0.99185
6
0.99148
-0.325
0.99138
7
0.99148
-0.325
0.99138
8
0.99808
-0.037
0.9983
9
0.99721
-0.049
0.99729
10
0.99732
-0.025
0.99556
11
0.9986
-0.011
0.99784
12
0.99853
0.002
0.99692
13
0.99445
-0.274
0.99317
14
0.99412
-0.281
0.99315
15
0.9939
-0.286
0.9932
16
0.99148
-0.3
0.99039
17
0.99055
-0.361
0.99149
18
0.99058
-0.362
0.99149
19
0.9973
0.003
0.99244
20
0.99728
0.021
0.9914
21
0.99731
-0.019
0.99538
22
0.99731
-0.017
0.9953
23
0.99698
-0.054
0.9973
24
0.99557
-0.092
0.99766
25
0.99865
-0.013
0.99784
0.999
120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120
Vc
deg 1
120
1
119.95
1
119.87
0.99
119.76
0.99
119.71
0.99
119.71
0.99
119.71
1
119.95
1
119.96
1
119.95
1
119.93
1
119.92
0.99
119.77
0.99
119.77
0.99
119.77
0.99
119.7
0.99
119.71
0.99
119.7
0.99
120
1
119.99
1
119.94
1
119.94
1
119.96
1
119.98
1
119.92
51
Load Flow Solution for Unbalanced Radial Distribution Systems
26
0.99399
-0.28
0.99302
27
0.98961
-0.396
0.9916
28
0.9885
-0.439
0.99161
29
0.98711
-0.499
0.99216
30
0.9973
0.03
0.9922
31
0.98858
-0.426
0.99088
32
0.98688
-0.514
0.99229
33
0.987
-0.519
0.99229
34
0.98706
-0.524
0.99229
35
0.98854
-0.393
0.98913
36
0.98862
-0.428
0.99088
37
0.98704
-0.52
0.99229
120 120 120 120 120 120 120 120 120 120 120 120
0.99
119.77
0.99
119.7
0.99
119.69
0.99
119.7
0.99
120
0.99
119.67
0.99
119.69
0.99
119.68
0.99
119.67
0.99
119.66
0.99
119.67
0.99
119.67
Table 2.6 Power losses of 37 node radial distribution system Branch
Proposed method Phase B
Phase A
Phase C
kW
kVar
kW
kVar
kW
kVar
1.2301
0.8762
1.2541
0.9863
0.9989
1.4117
0.6697 0.9163
0.8688 1.1762
0.7869 1.0888
1.3336 1.639
0.7275 1.0071
0.8825 1.2059
1.6254
0.4763
1.5154
0.7072
0.2596
0.5594
0.4339 0.6433
0.1498 13.9297
0.1259 0.6233
0.2107 13.9297
0.1372 0.6433
0.0568 13.9297
0.096 0.1546
0.2098 0.3033
0.136 0.1241
0.0394 0.0461
0.1268 0.2184
0.0752 0.0479
10
0.0071 0.0011
0.4697 0.1979
0.1599 0.0055
-0.0017 -0.0039
0.1739 0.0562
0.1682 0.0169
11
0.0011
0.1573
0
-0.0036
0.0586
0
12 13
0.0491
0.0036
0.0184
0.0213
0.0011
0.0016
0.0356 0.0106
0 0
0.005 0
0.0206 0.0039
0.0028 0
-0.0011 0
0.0008 0.3015
0.1677 0.0083
0 0.1071
0 0.2098
0.0618 0.0117
0 0.008
0
0
0.0041
0
0
0.0015
0.0031 0.0006
0.754 0.086
0.2824 0
0.0001 0
0.2191 0.0305
0.101 0
1 2 3 4 5 6 7 8 9
14 15 16 17 18 19
52
K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
20
0.0003
0.0049
0
0
0.0025
0
21
0
0.0022
0
0
0.0011
0
22 23
0.0438
0.0007
0
0.0144
0.0009
0
0.2353 0
0 0
0 0.0122
0.0871 0
0 0
0 0.0045
0.0061 0.3106
0.0058 0.0082
0.0056 0.0768
0.0019 0.2035
0.0017 0.0145
0.002 -0.0021
0.2578 0.3378
0.0004 0
0.1428 0.055
0.1937 0.2387
0.2143 0
0.0338 -0.023
0 0.2019
0.0381 0.0592
0.0446 0.2156
0 -0.0006
0.0092 0.0108
0.0164 0.0163
0.2188 0.0004
0 0
0.2402 0.0404
0.2227 -0.0005
0 0
0.0102 0.0231
24 25 26 27 28 29 30 31 32 33 34 35 36
0
0
0.0093
-0.0001
0
0.0049
0.0019 0
0.1446 0
0 0.0103
0.0001 0
0.0517 0
0 0.0038
0
0
0.0103
0
0
0.0038
Table 2.7 Summary of test results of 37 bus radial distribution system Description
Phase A
Phase B
Phase C
Total Active Power Loss (kW)
7.7946
7.1
5.0412
Total Reactive Power Loss (kVar)
20.0987
20.0994
18.5589
The Total Active Power losses are 19.9358 kW The Total Reactive Power losses are 58.7570 kVar Table 2.5 shows comparison of the voltage magnitudes obtained by Forward backward sweep method [21] and proposed method. For proposed method, the maximum deviation of voltage and its phase angle from the Forward backward sweep method is 0.0001 p.u and 0.01 deg. Thus, the two discussed methods are quite accurate. For both the methods, load flow converged in 2 iterations for the tolerance of 0.001 p.u. When the tolerance limit is set as 0.0001, the number of iterations required for the convergence is 4 for Forward backward sweep method and 3 for proposed method. The summary of test results is given in Table 2. The execution time is 0.016 seconds for the Forward backward sweep method and 0.00659 seconds for the proposed method on P-IV computer with 2.4 GHz frequency.
CONCLUSIONS
Load Flow Solution for Unbalanced Radial Distribution Systems
53
In this chapter, a simple and efficient computer algorithm has been presented to solve unbalanced radial distribution networks. A three phase load flow solution is proposed considering most of load modeling. The proposed method has good convergence property for any practical distribution networks with practical R/X ratio. Computationally, this method is extremely efficient, as it solves simple algebraic recursive equations for voltage phasors. Using the proposed method, the node identification will be easy where as it is difficult for existing method. Another advantage of the proposed method is all the data is stored in vector form, thus saving enormous amount of computer memory when tested for large systems.
REFERENCES [1] B. Stott, “Review of Load-Flow Calculation Methods”, Proceedings of the IEEE, Vol. 62, No. 7, July 1974, pp. 916-929. [2] W. F. Tinney and C. E. Hart, “Power Flow Solution by Newton’s Method”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-86, No. 11, November 1967, pp. 1449-1460. [3] R. Berg, Jr., E. S. Hawkins, and W. W. Pleines, “Mechanized Calculation of Unbalanced Load Flow on Radial Distribution Circuits”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-86, No. 4, April 1967, pp. 415-421. [4] M. E. Baran and F. F. Wu, “Optimal Sizing of Capacitors Placed on a Radial Distribution System”, IEEE Transactions on Power Delivery, Vol. 4, No. 1, January 1989, pp. 735-742. [5] C. S. Cheng and D. Shirmohammadi, “A Three-Phase Power Flow Method for Real-Time Distribution System Analysis”, IEEE/PES 1994 Summer Meeting, San Francisco, CA, July 1994, 94 SM 603-1 PWRS. 181 [6] W. H. Kersting and W. H. Phillips, “A Radial Three-phase Power Flow Program for the PC”, Conference paper, presented at 1987 Frontiers Power Conference, Stillwater, OK, October 1987. [7] G.X. Luo and A. Semlyen, “Efficient Load Flow for Large Weakly Meshed Networks”, IEEE Transactions on Power Systems, Vol. 5, No. 4, November 1990, pp. 1309-1316. [8] D. Raji i , R. A kovski, R. Taleski, “Voltage Correction Power Flow”, IEEE/PES 1993 Summer Meeting, Vancouver, B.C., Canada, July 1993, 93 SM 570-2. [9] D. Shirmohammadi, H. W. Hong, A. Semlyen, and G. X. Luo, “A Compensation-based Power Flow Method for Weakly Meshed Distribution and Transmission Networks”, IEEE Transactions on Power Systems, Vol. 3, No. 2, May 1988, pp. 753-762. [10] H. D. Chiang, “A Decoupled Load Flow Method for Distribution Power Networks: Algorithms, Analysis and Convergence Study”, Electrical Power & Energy Systems, Vol. 13, No. 3, June 1991, pp. 130-138.
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K. Jithendra Gowd, CH. Sai Babu & S. Sivanagaraju
[11] W. H. Kersting and D. L. Mendive, “An Application of Ladder Network Theory to the Solution of Three-Phase Radial Load-Flow Problems”, IEEE/PES 1976 Winter Meeting, New York, NY, January 1976. [12] R. D. Zimmerman and H. D. Chiang, “Fast Decoupled Power Flow for Unbalanced Radial Distribution Systems”, IEEE/PES 1995 Winter Meeting, New York, NY, January 1995, 95 WM 219-6 PWRS. [13] D. Das, P. Kothari and A. Kalam, "Simple and efficient method for load flow solution of radial distribution systems”, Electrical Power and Energy Systems, vol. 17, no. 5, Oct. 1995, pp. 335-346. [14]. D. Thukarm, H. M. Wijekoon Banda and J. Jerome, "A Robust three phase power flow algorithm for radial distributionsystems”, Electric Power System Research, vol. 50, no. 3, Jun.1999, pp. 227-236. [15] P. A. N. Garcia, J. L. R. Pereira, S. Carnerio, V. M. da Costa, and N.Martins, 2000. Three-Phase power flow calculations using the currentinjection method, IEEE Trans. on Power Systems, 15( 2):508514. [16] P. A. N. Garcia, J. L. R. Pereira, and S. Carneiro, Jr., 2001. Voltage control devices models for distribution power flow analysis, IEEE Trans. Power Syst., 16(4):586–594. [17] P. A. N. Garcia, J. L. R. Pereira, and S. Carneiro, Jr., 2004. Improvements in the representation of PV buses on three-phase distribution power flow, IEEE Trans. Power Del., 19(2):894–896. [18] K. N. Mui and H. D. Chiang, 2000. Existence, uniqueness and monotonic properties of the feasible power flow solution for radial three phase distribution networks, IEEE Trans. Circuit and Syst., 47(10):1502-1514. [19] W. E. Dillon, "Modeling and Analysis of an Electrically Propelled Transportation System," The University of Texas at Arlington, May 1972. [20] R. Ranjan, B.Venkatesh, A.Chaturvedi, and D.Das, “Power flow solution of Three-phase unbalanced Radial Distribution Network.”, Electrical Power Components and systems, Vol 3, issue 4, 2004 pp 421-433 [21] D. Shirmohammadi, S. Carol and A. Cheng, “A Three phase power flow method for real time distribution system analysis”, IEEE Trans. on Power System, Vol. 10, No. 2, May 1995. pp.671-679
AUTHOR BIBLIOGRAPHY
Load Flow Solution for Unbalanced Radial Distribution Systems
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K. Jithendra Gowd completed is B.Tech from JNT University, Hyderabad in 2002, M.Tech from JNT University, Kakinada in 2006 and pursuing his Ph.D. Presently working as Assistant Professor in Dept. of EEE, JNTUA College of Engineering, Anantapur . His areas of interest are Distribution Systems, HVDC Transmission.
Dr. Ch. Sai Babu received the B.E from Andhra University (Electrical & Electronics Engineering), M.Tech in Electrical Machines and Industrial Drives from REC, Warangal and Ph.D in Reliability Studies of HVDC Converters from JNTU, Hyderabad. Currently he is working as a Professor in Dept. of EEE in University College of Engineering, JNT University, Kakinada. He has published several National and International Journals and Conferences. His area of interest is Power Electronics and Drives, Power System Reliability, HVDC Converter Reliability, Optimization of Electrical Systems and Real Time Energy Management.
Dr. S.Sivanagaraju is graduated in 1998, Masters in 2000 fron IIT Kharaghpur and did his Ph.D in JNT University, Hyderabad in 2004 and working as a Associate Professor in Department of Electrical Engineering, University College of Engineering, Kakinada. He received two national awards (Pandit Madhan Mohan memorial Prize and best paper) for the year 2003-04. His areas of interest are Distribution and Automation.