_{min}
) and the associated lines having the most value of rate of change of real power loss with respect to effective reactive power
[19]
. The total load connected beyond the associated bus is called as the effective reactive power.
The above mentioned procedure is called sensitivity analysis and the relevant buses are called sensitivity buses. The sensitivity analysis is a conventional procedure practiced for many years for identifying the optimal location of capacitors. The mathematical equations related to formation of sensitivity analysis are described with the
Fig. 2
. The
Fig. 2
has a distribution line (m) connected between buses ‘i’ and ‘i+1’with a series impedance of R
_{m}
+jX
_{m}
and an effective load of P
_{eff}
+ jQ
_{eff}
at bus ‘i+1’.
Single line diagram of a distribution line
The real power loss of the distribution line (m) is calculated using the Eq. (9),
The loss sensitivity factor can be calculated using Eq. (10),
The Loss Sensitivity Factors (LSF) of all the lines can be calculated through conducting radial load flow. The calculated values of LSF are arranged in non-increasing order. The buses with most value LSF and lesser value (ie. < 1.01pu) of normalized voltage (|V|/0.95)
[19]
are selected as the candidate location for capacitor placement.
4. Search Strategy For Capacitor Sizing Through ODE Algorithm
Opposition based differential algorithm is a recent evolutionary algorithm with enhanced features such as self acceleration, self migration and assured optimal search with least population size. The efficiency of the algorithm can be well proven by applying into complex and/or large problems. In this paper, the purpose of introduction of ODE is to find the optimal capacitor size that need to be included at the optimal locations received at the end of sensitivity analysis. The number of variables for ODE searching is the number of identified locations. For instance, if the system has ‘n’ identified locations then ODE should have ‘n’ variables. The pseudocode of the ODE algorithm has been given below,
Proposed method combines Loss Sensitivity Factors (LSF) and Opposition based Differential Evolution (ODE) for optimization. LSF is used for identifying the capacitor location and ODE used for identifying the optimal size of the capacitors.
5. Test Results
The effectiveness of the proposed algorithm has been validated through applying on 10, IEEE 33, 34 and 85 bus radial distribution systems. The commercially available capacitors sizes and their cost in $/kVAR-year are given in
[19]
. The constants K
_{p}
and C
_{q,fixed}
are assumed as 168 $/(kW-year) and 1000 $ respectively
[19]
.
The proposed algorithm has been programmed using MATLAB programming and run on a P-IV processor with 266 MHz personal computer. The results obtained have been discussed in details in the following sub sections.
- 5.1 10-Bus RDS
This system is a balanced single feeder radial distribution system
[23]
with the base of 23kV. The initial operating cost of the system is 1, 31,674 $/year. The optimal locations for capacitor placement have been identified by doing sensitivity analysis.
The
Table 1
shows the values of loss sensitivity factor of each line along with end bus normalized voltage. The lines are arranged in descending order of the loss sensitivity factor and the end buses those have normalized voltage above 1.01 pu are eliminated from the list.
Initial configuration sensitivity factors of 10-Bus RDS
Initial configuration sensitivity factors of 10-Bus RDS
The first four buses from the sequence are selected as candidate nodes for the capacitor placement. The identified buses for capacitor location are 6, 5, 9 and 10. As a result, ODE takes four variables and search for the optimal size of capacitors. The ranges of the variables are from 150kVAR to 4050kVAR
[19]
.
The size of the capacitors at the locations 6, 5, 9 and 10 are 1200kVAR, 1200kVAR, 200kVAR and 407kVAR. The bus voltages before and after capacitor placement have been shown in the
Fig. 3
. It shows that bus voltages of the weaker buses 8, 9 and 10 are improved.
Bus voltages of 10-bus RDS before and after capacitor placement
The real power loss has been reduced from 783.77 kW to 694.93kW under initial load condition. The result of the proposed method has been compared with the other evolutionary algorithms and shown in
Table 2
. From the Table, all the algorithms bring global optimum and except SaHDE, all other evolutionary algorithms have taken larger population size even for the lesser number of variables.
Summary of results for 10-bus RDS
Summary of results for 10-bus RDS
Compared with SaHDE, the proposed ODE takes lesser number of load flow executions. The other methods in the literature are compared with the proposed method in
Table 3
. Proposed method ensures the global optimum. Also compared with PGSA, the proposed method solution methodology is simple.
Comparison with other methods in literature for 10-bus RDS
Comparison with other methods in literature for 10-bus RDS
- 5.2 IEEE 33-Bus RDS
The next system considered for implementation is the IEEE 33-bus radial distribution system
[6]
. This system is a 12.66kV system and it consists of 33 buses and five tie lines. The total load conditions are 3715kW and 2300kVAR. The initial operating cost of this system is 34,049.74 $/year. The buses 5, 27 and 28 are identified as candidate locations for capacitor location through sensitivity analysis. ODE tunes for the optimum capacitor size for the identified locations.
The proposed method reduces the power loss from 202.67kW to 159.89kW, and maintains the bus voltages well above minimum value. The bus voltages before and after capacitor placement have been shown in the
Fig. 4
. From the
Fig. 4
, it is observed that all the bus voltages are maintained above 0.95 pu. The details of the results received are shown in
Table 4
. It is found that through the proposed algorithm 16.61 % of cost saving has been achieved.
Bus voltages of 33-bus RDS before and after capacitor placement
Summary results for33-bus RDS
Summary results for33-bus RDS
- 5.3 34-Bus RDS
The proposed method has been tested with 34-bus balanced radial distribution system
[24]
. The initial uncompensated system annual cost is 37,241 $/year. As per the sensitivity analysis, the sensitive buses are identified. For this test system, buses 19, 22 and 20 are selected as optimal locations for the capacitor placement. The proposed method reduces the power loss from 221.67kW to 161.07kW, and maintains the bus voltages well above minimum value. The bus voltages before and after capacitor placement have been shown in the
Fig. 5
. The results are compared with previous works addressed for optimization and shown in
Table 5
.
Bus voltages of 34-bus RDS before and after capacitor placement
Comparison with other methods in literature for 34-bus RDS
Comparison with other methods in literature for 34-bus RDS
- 5.4 85-Bus RDS
The proposed method has been validated further by implementing to 85-bus balanced radial distribution system
[25]
. The sensitive buses 8, 58 and 7 were identified through sensitivity analysis. With the use of ODE, the capacitor sizes have been identified for those buses. The power loss is reduced from 315.714kW to 161.4kW.
The bus voltages before and after capacitor placement have been shown in the
Fig. 6
. The results are compared with previous works and shown in
Table 6
.
Bus voltages of 85-bus RDS before and after capacitor placement
Comparison with other methods in literature for 85-bus RDS
Comparison with other methods in literature for 85-bus RDS
6. Conclusion
In this paper, distribution system optimization through capacitor placement has been handled through ODE along with sensitivity analysis. The optimal locations of the capacitors were identified through conventional sensitivity analysis. For finding the optimum capacitor sizes on the optimal locations, ODE has been proposed.
This combined effort of sensitivity analysis with ODE ensured global optimum and bus voltages has been maintained above the minimum limit. The proposed algorithm has been validated with four distribution systems of different sizes such as 10, 33, 34 and 85-bus radial distribution system. The results were compared with other papers referred in the literature.
7. Scope for Extension
This research work provides multiple directions for future work. This proposed method used conventional sensitivity analysis for finding the optimal location for the capacitors. Instead it can be carried out with suitable soft computing technique to improve the efficiency of searching or it can be combined with the proposed optimization technique used for finding optimum sizes of capacitor. The proposed algorithm handled only the balanced distribution systems; it can be tested with unbalanced distribution system. The proposed method applied only for capacitor placement, it can also be combined with reconfiguration and/or phase balancing.
Acknowledgements
The authors would like to thank the reviewers for their constructive suggestions which have helped to enhance the quality of this manuscript.
BIO
R. MuthuKumar received his B.E Degree in Electricaland Electronics Engg. from Bharathiyar University, Coimbatore, TamilNadu in the year 2002, M.E., in Power Systems Engg. From GCT, Coimbatore TamilNadu in the year 2006 and pursuing Ph.D in Power System Planning at Anna University, Chennai, TamilNadu.He has published three international journals and has four International/National conference publications. His research interest includes power system planning, voltage stability analysis and application of evolutionary algorithms to power system optimization.
K. Thanushkodi, born in Theni District, Tamilnadu State, Indiain 1948, received BE in Electrical and Electronics Engineering from Madras University, Chennai. MSc(Engg) from Madras University, Chennai and Ph.D in Electrical and Electronics Engineering from Bharathiar University, Coimbatore in 1972, 1976 and 1991 respectively. His research interests lie in the area of Computer Modeling and Simulation, Computer Networking and Power System. He has published 26 technical papers in National and International journals. Presently he is the Director of Akshaya College of Engineering and Technology.
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