DOI : 10.17577/IJERTCONV4IS07010

Text Only Version

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

M. Tech. Power Systems Engineering Associate Professor

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 Ã— (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.

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 0.789 0.779 31 0.497 0.579 32 0.0525 0.0816

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 0.623766 30 0.608571 31 0.607975 32 0.608449

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):335Â±46.

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

1998;13(1):115Â±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):725Â±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