Load Flow analysis for Radial Distribution Network with Network Topology Method

DOI : 10.17577/IJERTCONV4IS07010

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Load Flow analysis for Radial Distribution Network with Network Topology Method

M. V. S. Ram Dr. G. Srinivasa Rao

M. Tech. Power Systems Engineering Associate Professor

    1. Siddhartha Engineering College (Autonomous) V.R. Siddhartha Engineering College (Autonomous) Vijayawada, Andhra Pradesh, India Vijayawada, Andhra Pradesh, India

      Abstract- Voltage stability has become an important issue of power system stability. This paper work concentrated on simple a n d e f f i c i e n t load flow analysis of radial distribution system for finding voltage magnitudes without trigonometric equations. Various methods are identified in literature survey for indication of voltage magnitude. The proposed analysis will be implemented on any IEEE standard test systems for composite load modeling. This feature enables us to set an index threshold to monitor and predict system stability online so that a proper action can be taken to prevent the system from collapse.

      Keywords: Distribution load flow, Branch Injection and Branch Current (BIBC), mathematical techniques, stability index.

      1. INTRODUCTION

The transmission system is distinctly different, in both its operation and characteristics, from the distribution system. Whereas the latter draws power from a single source and transmits it to individual loads, the transmission system not only handles the largest blocks of power but also the system.

The main difference between the transmission system and the distribution system shows up in the

network structure. The former tends to be a loop structure and the latter generally, a radial structure. The modern power distribution network is constantly being faced with an ever-growing load demand. Distribution networks e x p e r i e n c e a distinct change from low to high load level every day.

Literature survey shows that a lot of work has been done on the voltage stability analysis of transmission systems [2].

Power transmission systems may include sub transmission stages to supply intermediate voltage levels. Sub-transmission stages are used to enable a more practical or economical transition between transmission and distribution systems. It operates at the highest voltage levels (typically, 230 kV and above). The generator voltages are usually in the range of 11 kV to 35 kV. There are also a few transmission networks operating in the extremely high voltage class (345 kV to 765 kV). As compared to transmission system sub-transmission system transmits energy at a lower voltage level to the distribution substations. Generally, sub-transmission systems supply power directly to the industrial customers. The distribution system is the final link in the transfer of electrical energy to the individual customers. Between 30 to 40% of total investment in the electrical sector goes to distribution systems, but nevertheless, they havent received the technological improvement in the same manner as the generation

and transmission systems. The distribution network differs from its two of siblings in topological structure as well as its associated voltage levels. The distribution networks are generally of radial or tree structure and hence referred as Radial Distribution Networks (RDNs). Its primary voltage level is typically between 4.0 to 35 kV, while the secondary distribution feeders supply residential and commercial customers at 120/240/440 volts. It generally consists of feeders, laterals (circuit- breakers) and the service mains.

According to these studies, power flow analysis of RDNs may be divided into two categories. The first group of methods includes: ladder network methods for radial structure distribution systems using basic laws of circuit theories like Kirchhoffs Current Law (KCL) and Kirchhoffs Voltage Law (KVL ) [1]. On the other hand, the second category includes Gauss- Seidel, Newton- Raphson and Decoupled Newton- Raphson methods for transmission systems and is usually based on nodal analysis method. The characteristics of RDNs are dynamic in nature and general features of RDS are

  1. Uncertainties and Imperfection of network parameters.

  2. High R/X ratio

  3. Extremely large number of nodes and branches.

  4. Dynamic change in imposed load.

    line shunt capacitances at distribution voltage level are very small and thus can be neglected. The base values taken for calculation of voltages in the system are

    11 KV and 200 KVA. The simplified mathematical model of a section of a RDN is shown in Fig.1.

    Fig 1: Single Line of an Existing Distribution Feeder

    I(i)=S(J)/V*(J) ———————-(1)

    I(i)=P(J)-Q(J)/V*(J) ———————–(2)

    Fig 2: 33-bus Radial Distribution Network single line diagram

    and also

    I(i)=|V(i)| (i)-|V(j)| (j)/(R(i)+X(j))—–(3)

    |V(2)|={[(P(j)*R(i)+Q(j)*X(i)-0.5|V(i)|2)2-

    (R2(i)+X2(i))(P2(j)+Q2(j))]1/2-(P(j)*R(i)+Q(j)*X(i)-

    2 1/2

    1. MATHEMATICALMODEL OF A RDN 0.5|V(i)| )} ———–(4)

      The real and reactive power losses in branch ij are given by:

      In RDNs, the large R/X ratio causes problems in convergence of conventional load flow algorithms. For a balanced RDN, the network can be represented by an equivalent single-line diagram. The

      LP(j)=R(j)*[P2(m2)+Q2(m2)]/|V(m2)|2 ——-(5)

      QP(j)=X(j)*[P2(m2)+Q2(m2)]/|V(m2)|2 ——(6)

      Fig 3: Voltage magnitudes of 33-bus Radial Distribution Network

      Initially, if LP(j) and LQ(j) are set to zero for all j, then initial estimates of P(m2) and Q(m2) will be the sum of the loads of all the nodes beyond node m2 plus the load of the node m2 itself.

    2. NETWORK TOPOLOGY FOR MATRIX FORMATION

      The branch current B is calculated with the help of Bus-Injection to Branch-Current matrix (BIBC)[4]. The BIBC matrix is the result of the relationship between the bus current injections and branch currents.The e l e m e n t s of B I B C m a t r i x consists of 0s or 1s. [B]Nnb*1=[BIBC]nb*(n-1)*[I](n-1)*1———————-(7)

      Where nb is the number of branches, [I] is the vector of the equivalent current injection for each bus except the reference bus. This produced matrix is used for

      calculation of voltage magnitudes. The building Step for the BIBC matrix is shown in Fig .1.

      Fig 4: Graphical view of BIBC matrix

      Step (1): For a distribution system with nb branch sections and n buses, the dimension of the BIBC matrix is nb × (n1) Step (2): If a line section (Bk) is located between Bus i and Bus j, copy the column of the ith bus of the BIBC matrix to the column of the jth bus and fill +1 in the position of the jth bus column as shown below. Step (3): Repeat Step (2) until all the line sections are

      Fig 5: stability index of 33-bus Radial Distribution Network

      included in the BIBC matrix. The building Step (2) for the BIBC matrix is shown in Fig. 4.

      The BIBC matrix is responsible for the relations between the bus current injections and branch currents. (iii) The corresponding variation of the branch currents, which is generated by the variation at the current injection buses, can be found directly by using the BIBC matrix.

    3. MATHEMATICALMODEL OF STABILITY INDEX For a distribution line model, given in fig.1, the

      quadratic equation which is mostly used for the calculation of the line sending end voltages in load flow analysis can be written in general form as

      2 2 2 2 2

      Vr4+2 V 2(PR+QX)- Vr V +(P +Q )|Z| =0——-(8)

      and from this equation a feasible solution has considered , line receiving end active and reactive power can e written

      2 4 2 2 2 2

      2 4 2 2 2 2

      2Vr2 Vr – Vr -2Vr (PR+QX)-|Z| (P +Q )0——–(9)

      The above equation is most feasible of Eq (8).From the last equation, it is clearly seen that the value of the Eq.9 is decrease with the increase of the transferred power and impedance of the line, and it can be used as a bus stability index for a distribution networks as

      SI(m2)= V(m1)4 4{P(m2)r(jj) Q(m2)x(jj)}2

      4{P(m2)r(jj)+ Q(m2)x(jj)}V(m1)2 ——————(10)

      In this study the above simple stability criterion, given in eq. 10, is used to find the stability index for

      each line receiving end bus in radial distribution

      r r networks. After the load flow study, the voltages of all nodes and the branch currents are known, therefore P and Q at the receiving end of each line

      can easily be calculated and hence using Eq. 5 the voltage stability index of each node can easily be calculated. The node, at which the value of the stability index is at minimum, is the most sensitive to the voltage collapse.

    4. LOAD MODELING

      For the purpose of voltage stability analysis of radial distribution networks, composite load modeling is considered. The real and reactive power loads of node `i' is given as:

      PL (i) =PLo(i)(c1+c2|V(i)|+c3|V(i)|2) —– (11)

      QL (i) =QLo(i)(d1+d2|V(i)|+d3|V(i)|2) —– (12)

      In the above equations, loads are gradually increased at every node. Constants (c1, d1), (c2, d2) and (c3, d3) are the compositions of constant power, constant current and constant impedance loads, respectively.

      To demonstrate the effectiveness of the proposed method, a 33-bus radial distribution network [3] is considered. Fig.2 shows a 33-node radial distribution network. Line data and nominal load data (i.e. r, x, PLo and QLo) are given in Appendix A and Appendix B. In the present work, a composition of

      40% constant power=c1=d1= 0:4; 30% of constant current=c2

      =d2= 0.3; and 30% of constant impedance

      =c3 =d3 =0.3; are considered.

      Table 1: Voltages at different nodes

      branch

      Voltage(pu)

      1

      1

      2

      0.995894

      3

      0.976348

      4

      0.96592

      5

      0.955601

      6

      0.929896

      7

      0.92505

      8

      0.918269

      9

      0.90949

      10

      0.901343

      11

      0.900134

      12

      0.898013

      13

      0.889443

      14

      0.886274

      15

      0.884297

      16

      0.882379

      17

      0.879547

      18

      0.878696

      19

      0.995177

      20

      0.990319

      21

      0.989363

      22

      0.988498

      23

      0.971436

      24

      0.962298

      25

      0.957741

      26

      0.927204

      27

      0.923625

      28

      0.907613

      29

      0.896112

      30

      0.891151

      31

      0.885285

      32

      0.883993

      33

      0.883593

      The above table shows the voltage magnitudes (pu) for 33-bus network .The first bus voltage is as obtained to be 1pu and further it continues with

      respect to algorithm of proposed method. The above voltages are obtained for composite loads which are in algebraic form which are shown in Eqns (8) & (9). The Fig.3. shows how voltages are changes with respect their busses, at 19th bus the a sudden increase had obtained as this is because of the reason that the bus is nearer to substation bus and it continues.

      Table 2: Line flows of P and Q of 33- bus radial

      distribution network

      30

      7.89E-01

      7.79E-01

      31

      4.97E-01

      5.79E-01

      32

      5.25E-02

      8.16E-02

      The above table shows the line flow of real and reactive power 33- bus radial distribution network. The total real power loss of the system is 44.80KW and reactive power loss is 33.82KVAR.The stability indices are shown in Fig 5.

      Branch number

      Real Power loss(KW)

      Reactive Power loss(KVAR)

      1

      3.29E-02

      1.68E-02

      2

      1.28E-01

      6.52E-02

      3

      2.06E-01

      1.05E-01

      4

      4.69E-02

      2.39E-02

      5

      9.21E-02

      7.95E-02

      6

      2.65E-01

      8.74E-01

      7

      1.01E+00

      3.35E-01

      8

      1.18E-01

      8.51E-02

      9

      1.21E-01

      8.59E-02

      10

      1.67E-02

      5.52E-03

      11

      5.26E-02

      1.82E-02

      12

      2.08E-01

      1.64E-01

      13

      3.32E-01

      4.37E-01

      14

      6.47E-02

      5.75E-02

      15

      8.85E-02

      6.46E-02

      16

      1.53E-01

      2.05E-01

      17

      2.11E-01

      1.66E-01

      18

      4.18E-02

      3.99E-02

      19

      3.85E-01

      3.47E-01

      20

      1.05E-01

      1.23E-01

      21

      1.82E-01

      2.40E-01

      22

      1.29E-01

      8.80E-02

      23

      5.28E+00

      4.17E+00

      24

      5.29E+00

      4.14E+00

      25

      2.42E-02

      1.23E-02

      26

      3.40E-02

      1.73E-02

      27

      1.22E-01

      1.08E-01

      28

      4.53E-01

      3.95E-01

      29

      5.96E+00

      3.03E+00

      Branch number

      Real Power loss(KW)

      Reactive Power loss(KVAR)

      1

      3.29E-02

      1.68E-02

      2

      1.28E-0

      6.52E-02

      3

      2.06E-01

      1.05E-01

      4

      4.69E-02

      2.39E-02

      5

      9.21E-02

      7.95E-02

      6

      2.65E-01

      8.74E-01

      7

      1.01E+00

      3.35E-01

      8

      1.18E-01

      8.51E-02

      9

      1.21E-01

      8.59E-02

      10

      1.67E-02

      5.52E-03

      11

      5.26E-02

      1.82E-02

      12

      2.08E-01

      1.64E-01

      13

      3.32E-01

      4.37E-01

      14

      6.47E-02

      5.75E-02

      15

      8.85E-02

      6.46E-02

      16

      1.53E-01

      2.05E-01

      17

      2.11E-01

      1.66E-01

      18

      4.18E-02

      3.99E-02

      19

      3.85E-01

      3.47E-01

      20

      1.05E-01

      1.23E-01

      21

      1.82E-01

      2.40E-01

      22

      1.29E-01

      8.80E-02

      23

      5.28E+00

      4.17E+00

      24

      5.29E+00

      4.14E+00

      25

      2.42E-02

      1.23E-02

      26

      3.40E-02

      1.73E-02

      27

      1.22E-01

      1.08E-01

      28

      4.53E-01

      3.95E-01

      29

      5.96E+00

      3.03E+00

      Table 3: Stability Index of 33- bus radial distribution

      network

      Branch number

      Stability Index

      1

      0.983272583

      2

      0.906940845

      3

      0.868636821

      4

      0.832999639

      5

      0.745868785

      6

      0.729374704

      7

      0.706288905

      8

      0.682070313

      9

      0.657896675

      10

      0.656196014

      11

      0.649589051

      12

      0.622415428

      13

      0.613741796

      14

      0.61041852

      15

      0.604740201

      16

      0.595537497

      17

      0.593828343

      18

      0.980141808

      19

      0.955548193

      20

      0.95627273

      21

      0.951432533

      22

      0.888761294

      23

      0.841258052

      24

      0.825358375

      25

      0.7386677

      26

      0.72715612

      27

      0.67629338

      28

      0.640883659

      29

      0.623765846

      30

      0.60857052

      31

      0.607974998

      32

      0.608448922

      The 17th bus in the system is nearer to collapse, so that we must take care of that node by optimal using of distributed generators or shunt capacitors [5].

    5. RESULTS

      The rate of convergence of the proposed approach is tested using IEEE 33 node radial distribution systems with varying load conditions ranging from 0.5 to 3.0 times of the given load condition. The voltages had been plotted in Fig 3.and Fig 5. Voltage magnitude and stability index values are obtained from RDS network which is shown in Table 1.The total real power loss of the system is 44.80KW and reactive power loss is 33.82KVAR.

    6. CONCLUSION

      It has been shown that the load flow solutions of radial distribution networks are unique. The power system issues Distributed Generation for optimally placed and sized at Radial Distribution Feeder where the voltage stability index value are minimum and most sensitive to voltage collapse. Optimal sizing of Distributed Generation can be calculated using

      analytical expression and an efficient approach is used to determine the optimum location for distributed generators. The effectiveness of the proposed technique has been demonstrated through a

      33- bus radial distribution network and it can be evaluated for any IEEE test system.

    7. REFERENCES

  1. Das D, Kothari DP, Kalam A. A simple and efficient method for load flow solution of radial distribution networks. International Journal of Electrical Power and Energy Systems 1995;17(5):335±46.

  2. Ajjarapu V, Lee B. Bibliography on voltage stability. IEEE Transactions on Power Systems

    1998;13(1):115±25

  3. S. Ghosh., D. Das, Method for load flow solution of radial distribution network, IEE Proc.- Gener. Transm. Distrib. Vol. 146, No. 6, pp.641-

    648, 1999

  4. W. H. Kersting, D. L. Mendive, An application of ladder network theory to the solution of three phase radial load flow problem IEEE PES winter meeting, New York, Jan. 1976

  5. Mesut EB, Wu FF. Optimal capacitor placement on radial distribution systems. IEEE Transactions on Power Delivery 1989;4(1):725±34.

Appendix A

Line data of 33-bus radial distribution network

16

BRANCHNUMBE

SENDINGNODE

RECEIVINGNODE

RESISITANCE

REACTANCE

1

1

2

0.000152388

7.77E-05

2

2

3

0.00081483

0.000415018

3

3

4

0.000604925

0.000308082

4

4

5

0.000629882

0.000320808

5

5

6

0.001353643

0.00116853

6

6

7

0.000309404

0.001022753

7

7

8

0.001175802

0.000388573

8

8

9

0.001702384

0.001223072

9

9

10

0.001725523

0.001223072

10

10

11

0.00032494

0.000107432

11

11

12

0.000618808

0.000214533

12

12

13

0.00242631

0.001908984

13

13

14

0.000895156

0.001178281

14

14

15

0.000976805

0.000869373

15

15

16

0.001233485

0.000900776

16

17

0.002130459

0.002844469

17

17

18

0.00120985

0.000948707

18

2

19

0.000271059

0.000258663

19

19

20

0.002486142

0.002240205

20

20

21

0.000676822

0.0007907

21

21

22

0.00117167

0.001549169

22

3

23

0.000745743

0.000509558

23

23

24

0.001484214

0.001171835

24

24

25

0.001480909

0.001158778

25

6

26

0.000335518

0.0001709

26

26

27

0.000469726

0.00023916

27

27

28

0.001750315

0.001543219

28

28

29

0.001329182

0.001157952

29

29

30

0.000838796

0.000427249

30

30

31

0.001610488

0.001591646

31

31

32

0.000513194

0.000598148

32

32

33

0.000563605

0.000876315

Appendix B

Bus data of 33-bus radial distribution network

Node number

PL(composite load)pu

QL(composite load)pu

1

0

0

2

0.511344

0.306806

3

0.460209

0.204538

4

0.613613

0.409075

5

0.306806

0.153403

6

0.306806

0.102269

7

1.022688

0.511344

8

1.022688

0.511344

9

0.306806

0.102269

10

0.306806

0.102269

11

0.230105

0.153403

12

0.306806

0.17897

13

0.306806

0.17897

14

0.613613

0.409075

15

0.306806

0.051134

16

0.306806

0.102269

17

0.306806

0.102269

18

0.460209

0.204538

19

0.460209

0.204538

20

0.460209

0.204538

21

0.460209

0.204538

22

0.460209

0.204538

23

0.460209

0.255672

24

2.147644

1.022688

25

2.147644

1.022688

26

0.306806

0.127836

27

0.306806

0.127836

28

0.306806

0.102269

29

0.613613

0.357941

30

1.022688

3.068063

31

0.767016

0.357941

32

1.073822

0.511344

33

0.306806

0.204538

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