Optimal Conductor Selection Using Fuzzy Logic Engineering Essay

Published: 2021-07-01 18:10:05
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4.1 INTRODUCTION
Distribution system is that part of the electric power system which connects the high voltage transmission network to the low voltage consumer service point. In any distribution system the power is distributed to various users through feeders, distributors and service mains. Feeders are conductors of large current carrying capacity, which carry the current in bulk to the feeding points. Distributors are conductors from which the current is tapped off for the supply to the consumer premises. The size of the feeder is determined based upon the current carrying capacity and the size of the distributor is determined based upon the permissible voltage drop. Also the size and cross – section of feeders is affected by the increase in supply voltage.
In normal practice, the conductor used for radial distribution feeder is of uniform cross section. However, the load at the substation end is high and it reduces gradually as we proceed on to the tail end of the feeder. This indicates that the use of a higher size conductor, which is capable of supplying load from the source point, is not necessary at the tail end point. Similarly, use of different conductor cross section for intermediate sections will lead to a reduction in respect of both capital investment and cost of line losses.
The use of larger number of conductors of different cross section will result in increased cost of inventory. A judicious choice can however be made in the selection of number and size of cross section for considering the optimal design. It has been established that 70% of the total losses occur in the primary and secondary distribution system, while transmission and sub transmission lines account for only 30% of the total losses. Distribution losses are approximately 15.5% of the generation capacity and the target level is to reduce it to about 7.5%. Therefore the primary and secondary distribution network must be properly planned to ensure losses are within the acceptable limits.
Hence, Power losses in the lines account for the bulk of the distribution system losses. The capital investment in laying distribution network lines accounts for a considerable fraction of total capital investment. Therefore, considerable attention has been shown on distribution system planning over last few years. Many mathematical models have been reported to determine the best locations, size and interconnection of substation and feeder to meet the present and predicted future load demands [1, 4, 5, 7, 43]. In Funkhouser and Huber [1] have proposed a method based on the uniform load distribution for the feeders, however this method cannot be used in general because load on the feeders can't be uniformly distributed. In most of the distribution systems planning methods [9, 23] the distribution feeders have been assumed to be of uniform cross section. In [9] dynamic programming approach is used to obtain the solution to the optimization problem.
Anders et al. [29] have analyzed the parameters that affect the economic selection of cable sizes. The authors also did a sensitivity analysis of the different parameters as to how they affect the overall economics of the system. Das et al. [32] have proposed an analytical method for grading of conductors based on current carrying capacity of the conductors. Many authors [76, 103, 116, 128] have proposed different methods using genetic, evolutionary programming, Particle swarm optimization, plant growth techniques for selecting optimal branch conductor for radial distribution system. Srinivasa Rao [133] has proposed Differential Evolution (DE) algorithm for optimal selection of conductors in each branch of the distribution system by considering constraints of voltage and maximum current carrying capacity of each conductor in the optimization problem. The sum of capital investment and capitalized energy loss cost has been considered as objective function.
In this chapter, a method is proposed based on Fuzzy expert system for selecting the optimal type of conductor for radial distribution systems. For selecting an optimal conductor, a current deviation and voltage deviation parameters have been calculated and given to fuzzy expert system as inputs. The conductor, which is determined by this method, will satisfy the maximum current carrying capacity and maintain acceptable voltage levels in the radial distribution system. In addition, it gives the maximum saving in capital cost of conducting material and cost of energy loss.
The objective function and its constraints of the proposed method are described in Section 4.2. In Section 4.3, the modifications to be carried out in load flow calculations to select an optimal conductor of distribution system are explained. The implementation aspects of Fuzzy Expert System to select optimal conductor of distribution system is described in Section 4.4. In Section 4.5, the procedure to select optimal conductor taking future load growth into consideration is explained. The algorithm of the proposed method is presented in Section 4.6. The effectiveness of the proposed method is tested with different examples of radial distribution system is given in Section 4.7 and conclusions are presented in Section 4.8.
4.2 OBJECTIVE FUNCTION
The objective is to select the optimal size of the conductor in each branch of the system, and suitable type of conductor which minimizes the sum of depreciation on capital investment and cost of energy losses.
The objective function for optimal selection of conductor for branch k with ‘ff’ type of conductor is formulated as follows:
… (4.1)
where
= real power loss of branch k with ‘ff’ type of conductor in kW
Kp = annual demand cost of power loss in `./kW
Ke = annual cost of energy loss in `./kWh
Lsf = Loss factor = 0.8 × (LF)2+ 0.2×LF
LF = load factor
λ = Interest and depreciation factor
A(ff) = Cross sectional area of ‘ff’ type of conductor in mm2
cost(ff) = Cost of ‘ff’ type conductor in `./ mm2 /km
len(k) = Length of branch k in km
A radial distribution system has several branches. When these branches are reconductored, it alters the flow of current and it changes the resulting power losses and voltage profile. The objective of the method is to select the best conductor type for each branch of the RDS, such that the resulting RDS requires the least cost for conductor grading, which yields the minimum real power losses and better voltage profile.
4.2.1 Constraint equations
i) The bus voltages at all buses of the feeder must be above the acceptable voltage level. i.e., |Vi,ff| > |Vmin|, for i=1,2,..nbus, ff=1,2,…ntype
where
nbus = total number of buses
ntype= number of types of conductor
ii) Maximum current carrying capacity: Current flowing through branch k with ‘ff’ type conductors should be less than maximum current carrying capacity of ‘ff’ type conductor, Imax(ff)
|Ik,ff| < |Imax (ff)|, for k=1,2,....nbus-1, ff=1,2,…ntype
4.3 LOAD FLOW METHOD FOR OPTIMAL BRANCH CONDUCTOR
SELECTION
A simple load flow algorithm developed in Chapter 2 with a little modification is used for the optimal branch conductor size selection of radial distribution system.
Fig. 2.3 shows a single line diagram of equivalent distribution system. Consider branch 1, the receiving end voltage of branch -1 for ‘ff’ type conductor can be written from Eqn. (2.7) as
… (4.2)
where
|V2,ff| = Voltage magnitude of bus 2 with ‘ff’ type conductor of branch 1, ff = 1,
2,…ntype
V1 = substation voltage (constant for all type of conductors)
R1,ff = resistance of branch 1 with ff type of conductor, ff = 1, 2,....ntype
X1,ff = reactance of branch 1 with ff type of conductor, ff = 1, 2,....ntype
As the substation voltage V1 is known, calculate V2,ff for all type of conductors.
The generalized equation of receiving end voltage with ‘ff’ type conductor can be written as
… (4.3)
where
i =1, 2……nbus.
k =1, 2,3…..nbus-1.
nbus = total number of buses.
Real and reactive power losses of branch, k with ‘ff’ type conductor are given by
… (4.4)
… (4.5)
The total real and reactive power losses are given by
… (4.6)
… (4.7)
4.4 IMPLEMENTATION ASPECTS OF FUZZY EXPERT SYSTEM TO
IDENTIFY OPTIMAL BRANCH CONDUCTOR
The fuzzy logic is used to identify the optimal conductor size of a branch in a radial distribution system under normal or varying load conditions so as to minimize the losses while keeping the voltage at buses within the limit and also by taking the cost of the conductors into account.
4.4.1 Procedure to generate optimal set of conductor combinations using fuzzy
logic
Let the vector B= [B1, B2, B3,….., BNC ] refer NC different combinations of conductors initially chosen for the branches in RDS. In the proposed algorithm, these initial combinations are considered as a starting guess to generate NC more combinations BNC+1 to B2NC through a random process. The best combination BNC+k having lowest value of objective function in the set of NC combinations BNC+1 to B2NC is generated as follows [58]:
… (4.8)
where
NC refers to number of combinations
γ is a random factor appropriately chosen
x is a random number between 0 to xM
is satisfaction parameter value of combination Bk
BMIN and BMAX are the combinations of B that yields minimum and maximum satisfaction parameter values respectively
is maximum satisfaction parameter value
4.4.2 Procedure to determine satisfaction parameter value
In the proposed algorithm, a satisfaction parameter is used which takes voltages and an objective function into account to obtain the optimal branch conductor selection of radial distribution system. The satisfaction parameter value μB (B) is computed as follows
… (4.9)
where
is the membership function of an objective function
… (4.10)
where
FMAX and FMIN are the maximum and minimum values among the set of objective functions F(B).
is the membership function of voltage index
… (4.11)
where
vMAX and vMIN are the maximum and minimum values among the set of permissible values of v.
v(B) is the voltage-deviation index value for the combination B is given by
… (4.12)
where
NL = Number of time intervals
= voltage deviation index for the iith interval.
The voltage-deviation index in the interval ii, is calculated by
… (4.13)
where
BNV is the number of buses that violate the prescribed voltage limits
Vi is the voltage at ith bus.
ViLIM is the upper limit of the ith bus voltage if there is an upper-limit violation
or lower-limit if there is a lower limit violation.
nbus is the number of buses in the system
Considering all the combinations [B1, B2, B3,….., B2NC ], evaluate them using Eqn. (4.1) and obtain the best combination having minimum value of objective function and finally the voltage deviation index is to be calculated using Eqn. (4.13).
4.5 OPTIMAL CONDUCTOR SELECTION FOR LOAD GROWTH
Load growth in a distribution system with time is a natural phenomenon. The growth in feeder load may be due to the addition of new loads or due to the incremental additions to the existing loads. A feeder is well designed and constructed on a long term planning basis can accept additional loads to the extent it can accommodate satisfying the voltage constraint and current carrying capacity. Once, the load exceeds the feeder capacity, limited by voltage or thermal constraints, new facilities such as substations or additional feeders need to be created. Till such time, the substation feed area and the configuration of the feeders may be assumed to remain unchanged. It is further assumed that the feeder load grows at a predetermined annual rate, in proportion to the connected loads.
The real and reactive power loads at any year 'N' is given by
... (4.14) ... (4.15)
where
PL, QL = Real and reactive power load at Nth year
PL0 , QL0 = Real and reactive power load at base year (0th year)
N = number of years
g = Annual load growth rate (assumed as 7.0%)
The Eqns. (4.14) and (4.15) can be used to determine the maximum allowable load growth in a period of ‘N’ years.
4.6 ALGORITHM FOR OPTIMAL TYPE OF CONDUCTOR SELECTION
Step 1 : Read line and load data
Step 2 : Perform load flow study
Step 3 : Generate randomly NC combination of solution vectors [B1, B2,…., BNC.]
Step 4 : Set the iteration count '1'.
Step 5 : Evaluate the objective function using Eqn. (4.1)
Step 6 : Generate NC more combinations using Eqn. (4.8)
Step 7: Choose the best NC combinations among the set of 2NC combinations with
lower values of objective function.
Step 8 : Increment the iteration count. If iteration count < Maximum go to Step 4.
Else replace existing NC combinations with best NC combinations and go to
Step 9.
Step 9 : Perform load flow with the best NC combinations. Print the total real power
loss, reactive power loss and voltages.
Step10: Stop
4.7 FLOW CHART FOR OPTIMAL TYPE OF CONDUCTOR SELECTION
Read Distribution System line and load data, maximum number of iterations ‘Max.’
Start
Perform load flows and calculate voltages, real and reactive power losses and total cost
Generate randomly NC combination of solution vectors [B1, B2,…., BNC] and set iteration count (IC)=1
Evaluate the objective function using Eqn. (4.1)
Perform load flow with the best NC combinations
Choose the best NC combinations among the set of 2NC combinations with lower values of objective function
Generate NC more combinations using Eqn. (4.8)
Check for iteration count Stop
Compute voltages, angles, power flows, real and reactive power losses and Print the results
Yes
No
IC=IC+1
Fig. 4.1 Flow chart for optimal conductor selection
4.8 ILLUSTRATIVE EXAMPLES
The effectiveness of the proposed method is demonstrated with two examples, consisting of practical 26 bus and 32 bus radial distribution systems. Before analyzing the results, it is worth mentioning that presently in India, utilities are using three or four different types of conductors for distribution feeders. The electrical properties of these conductors are given in Appendix C (Table C.1).
4.8.1 Example – 1
The single line diagram for practical 26 bus, 11kV feeder from 33/11kV substation at Vidyuth Nagar in Anantapur town, Andhra Pradesh is shown in Fig. 4.2. The line and load data [65] of this system are given in Appendix C (Table C.2).
Fig. 4.2 Single line diagram of practical 26 bus radial distribution system
The voltage profile of the system before and after conductor grading is given in Table 4.1. The summary of results is given in Table 4.2. The modifications of the conductor type before and after conductor grading is given in Table 4.3. From Table 4.3, it can be seen that reconductering is necessary for most of the branches. The minimum voltage is improved from 0.9311 p.u. to 0.9646 p.u. The improvement in voltage regulation is 3.35%. The total real power loss reduces from 154.6852 kW to 58.8836 kW and reactive power loss reduces from 64.4861 kVAr to 57.3115 kVAr after conductor selection. The total cost reduction after conductor selection is `. 3, 16,989/-.
Table 4.1 Voltage profile of 26 bus radial distribution system before and after
conductor grading
Bus No.
Before conductor selection
After conductor selection
Voltage Magnitude (p.u.)
Angle (deg.)
Voltage Magnitude (p.u.)
Angle (deg.)
1
1.0000
0.0000
1.0000
0.0000
2
0.9827
0.2423
0.9911
-0.0663
3
0.9782
0.3059
0.9888
-0.0835
4
0.9740
0.3663
0.9866
-0.0998
5
0.9679
0.4540
0.9835
-0.1233
6
0.9583
0.5965
0.9785
-0.1612
7
0.9564
0.6241
0.9776
-0.1685
8
0.9546
0.6505
0.9767
-0.1755
9
0.9512
0.7012
0.9749
-0.1888
10
0.9480
0.7496
0.9733
-0.2015
11
0.9450
0.7958
0.9717
-0.2136
12
0.9402
0.8682
0.9693
-0.2325
13
0.9369
0.9191
0.9676
-0.2457
14
0.9339
0.9664
0.9660
-0.2579
15
0.9332
0.9773
0.9657
-0.2566
16
0.9323
0.9916
0.9652
-0.2548
17
0.9317
1.0011
0.9649
-0.2362
18
0.9311
1.0095
0.9646
-0.2194
19
0.9824
0.2459
0.9909
-0.0628
20
0.9824
0.2471
0.9909
-0.0616
21
0.9448
0.7996
0.9716
-0.2146
22
0.9447
0.8009
0.9715
-0.2145
23
0.9443
0.8009
0.9711
-0.2151
24
0.9441
0.8035
0.9708
-0.2156
25
0.9336
0.9713
0.9656
-0.2592
26
0.9335
0.9730
0.9655
-0.2560
Table 4.2 Summary of results after conductor grading of 26 bus RDS
Description
Before Conductor selection
After Conductor selection
Min. Voltage (p.u.)
0.9311
0.9646
Total real power loss (kW)
154.6852
58.8836
Reactive power loss (kVAr)
64.4861
57.3115
Total Cost (`.)
5,27,829/-
2,10,840/-
Table 4.3 Modifications in the feeder conductor type after conductor grading
Branch no
Existing conductor (from)
Modified conductor(to)
1 to 17
Weasel
Raccon
18 to 20
Weasel
Rabbit
22 to 25
Weasel
Squirrel
The voltage profile of the system and variation of real power loss in each branch before and after conductor selection is shown in Figs. 4.3 and 4.4 respectively.
Fig. 4.3 Voltage (p.u.) of 26 bus RDS before and after conductor selection
Fig. 4.4 Real power loss (kW) of 26 bus RDS before and after conductor selection
4.8.1.1 The effect of annual load growth on optimal conductor selection
The load on the distribution system is increasing year after year. Under these circumstances, the existing system may not able to meet the load demand and maintain the voltages within the specified limits which requires expansion of the existing system or a new system to be planned, which involves additional investment.
It is observed that the optimal conductor selected by the proposed method will be able to maintain the voltage within the specified limits taking annual load growth into consideration. The total loads, losses and minimum voltage for 26 bus system considering the load growth before and after conductor selection are given in Tables 4.4 and 4.5 respectively.
Table 4.4 Total loads, losses and minimum voltage for 26 bus system before
conductor selection when considering the load growth
Year
Total Real Power Load (kW)
Total Reactive Power Load (kVAr)
Total Real Power loss (kW)
Total Reactive Power loss(kVAr)
Min. Voltage at bus 18 (p.u.)
Base year (0th year)
2368.0
1776.0
154.69
64.49
0.9311
Table 4.5 Total loads, losses and minimum voltage for 26 bus system after
conductor selection when considering the load growth
Year
Total Real Power Load (kW)
Total Reactive Power Load (kVAr)
Total Real Power loss (kW)
Total Reactive Power loss (kVAr)
Min. Voltage at bus 18 (p.u.)
Base year (0th year)
2368.0
1776.0
58.88
57.31
0.9646
1st year
2533.8
1900.3
67.7
65.9
0.9620
2nd year
2711.1
2033.3
77.9
75.8
0.9593
3rd year
2900.9
2175.7
89.652
87.272
0.9563
4th year
3104
2328
103.22
100.48
0.9531
5th year
3321.2
2490.9
118.9
115.74
0.9507
It is observed that, the same optimal conductor selected by the proposed method is able to maintain voltage profile and reduction in power loss up to next 5 years. But due to optimal conductor selection this maximum real and reactive load can be increased from 2368 kW to 3321.2 kW and 1776 kVAr to 2490.9 kVAr without violating minimum voltage constraint over a period of 5 years. From Table 4.5, it can be seen that till 5th year the same set of conductors can be used even taking annual load growth into consideration.
4.8.2 Example – 2
The single line diagram of practical 32 bus, 11kV feeder is shown in Fig. 4.5. The line and load data of this system are given in Appendix C (Table C.3).
Fig. 4.5 Single line diagram of practical 32 bus radial distribution system
Based on the proposed algorithm, the results of conductor type selection are presented in Table 4.6. From Table 4.6, it can be seen that reconductering is necessary for most of the branches. Summary of results are given in Table 4.7. The minimum voltage is improved from 0.9312p.u. to 0.9598p.u. The improvement in voltage regulation is 2.86%. The total real power loss reduces from 421.3789kW to 168.8492kW after conductor selection. The total cost reduction after conductor selection is `. 24, 33, 400/-.
Table 4.6 Modifications in the feeder conductor type after conductor grading
Branch no
Existing feeder (from)
Modification (to)
1 to 14
Weasel
Raccon
15 - 20
Weasel
Rabbit
23 - 31
Weasel
Squirrel
Table 4.7 Summary of results after conductor grading of 32 bus RDS
Description
Before conductor selection
After Conductor selection
Min. Voltage (p.u.)
0.9312
0.9598
Total real power loss (kW)
421.3789
168.8492
Total Cost (`.)
33, 24, 400/-
8, 91, 000/-
The voltage profile of the system and variation of real power loss in each branch before and after conductor selection is shown in Figs. 4.6 and 4.7 respectively.
Fig. 4.6 Voltage (p.u.) of 32 bus RDS before and after conductor selection
Fig. 4.7 Real power loss (kW) of 32 bus RDS before and after conductor selection
4.8.2.1 The effect of annual load growth on optimal conductor selection
Loads, losses and minimum voltage for 32 bus system before and after conductor selection when load growth is considered are given in Tables 4.8 and 4.9 respectively. From Table 4.8 it is observed that, the same optimal conductor selected by the proposed method is able to maintain voltage profile and reduction in power loss up to next 3 years. But due to optimal conductor selection this maximum real and reactive load can be increased from 6480 kW to 7938.3 kW and 4580kVAr to 5610.7 kVAr without violating minimum voltage constraint over a period of 3 years. From Table 4.9, it can be seen that till 3rd year the same set of conductors can be used even taking annual load growth into consideration.
Table 4.8 Total loads, losses and minimum voltage for 32 bus system before conductor selection when considering the load growth
Year
Total Real Power Load (kW)
Total Reactive Power Load (kVAr)
Total Real Power loss (kW)
Total Reactive Power loss (kVAr)
Min. Voltage at bus 29 (p.u.)
Base year (0th year)
6480
4580
421.3789
175.6671
0.9312
Table 4.9 Total loads, losses and minimum voltage for 32 bus system after conductor selection when considering the load growth
Year
Total Real Power Load (kW)
Total Reactive Power Load (kVAr)
Total Real Power loss (kW)
Total Reactive Power loss (kVAr)
Min. Voltage at bus 29 (p.u.)
Base year (0th year)
6480
4580
168.8492
156.7761
0.9598
1st year
6933.6
4900.6
194.2066
180.3112
0.9568
2nd year
7419.0
5243.6
223.4512
207.452
0.9537
3rd year
7938.3
5610.7
257.2003
238.7705
0.9503
4.9 CONCLUSIONS
It is very important to select an optimal set of conductors for designing a distribution system. In this chapter, an algorithm has been proposed for selecting the optimal branch conductor using fuzzy expert system. The proposed method selects the optimal branch conductor by minimizing the sum of cost of energy losses, demand cost of power losses and depreciation cost of feeder conductor while maintaining the minimum voltage within prescribed limit and current flowing through branches below the current capacity of the conductors. It is also investigated to determine the period for which the same set of optimal conductors selected will be able to maintain the voltage profile even taking the annual load growth into consideration. The proposed algorithm has been implemented on practical 26 and 32 bus radial distribution systems and results are presented.
CHAPTER - 5
OPTIMAL CAPACITOR PLACEMENT USING FUZZY LOGIC
5.1 INTRODUCTION
The power supplied from electrical distribution system is composed of both active and reactive components. Overhead lines, transformers and loads consume the reactive power. So voltage/VAr control is an essential measure to reduce the power losses through the switching operations of capacitors and load tap changing transformers. Reactive power compensation plays an important role in the planning of an electrical system. A proper control of the reactive power will improve the voltage profile, reduces the system losses and improves the system efficiency. Proper generation and control of reactive power is important for maintaining the network voltages under normal and abnormal conditions and to reduce system losses. The system voltage collapse due to lack of global control of reactive power flow during crucial contingencies is emerging as a serious problem. The aim is principally to provide an appropriate placement and sizing of the compensation devices to ensure a satisfactory voltage profile while minimizing the cost of compensation.
Mainly capacitors are used to develop reactive power near the point of consumption. Series and shunt capacitors in a power system generate reactive power to improve power factor and voltage, thereby enhancing the system capacity and reducing the losses. Due to various limitations in the use of series capacitors, shunt capacitors are widely used in distribution systems. The general capacitor placement problem is formulated as an optimization problem to determine the location of capacitors, the types and size of capacitors to be installed and the control scheme for the capacitors at the buses of radial distribution networks.
Several methods of loss reduction by placing capacitors in distribution systems have been reported over the years. The early approaches to this problem include those using analytical methods, heuristic methods, artificial intelligence methods and those using dynamic programming technique to include the discrete nature of the capacitor size. Baran and Wu [18] have proposed an analytical method based on mixed integer programming to find the optimal size of capacitor to reduce the losses of a radial distribution system. Haque [51] has proposed a method of minimizing the loss associated with the reactive component of branch currents by placing capacitors at proper locations. The method first finds the location of the capacitor in a sequential manner. Once the capacitor locations are determined, the optimal capacitor size at each selected location is determined by optimizing the loss saving equation.
More recently, the use of various non-deterministic methods like tabu search, genetic algorithms, fuzzy expert system and simulated annealing to determine the location and size of capacitor to improve the voltage profile of the system have been reported. Mekhamer et al. [73] have proposed a method to select an optimal location of capacitors using fuzzy logic and its allocation by analytical method. Ng et al. [40] presents a methodology to convert the analytical method stated in Salama and Chikani [28] from crisp solution into fuzzy solution by modeling the parameters using possibility distribution function, thus accounting for the uncertainties in these parameters.
Prasad et al. [102] have presented a genetic approach to determine optimal size of capacitor. The optimal location to install capacitor is determined by taking values of Power Loss Indices randomly. Power loss indices are calculated by compensating the self-reactive power at each bus and run the load flows to determine the total active power losses in each case. Most of the previous studies [30, 31, 45, 57, 64, 124] have presented a method to find the location and size of capacitors using heuristic, genetic, simulated annealing techniques.
In this chapter, a method is proposed to determine the optimal location of capacitors using fuzzy expert system by considering power loss indices and voltage at each bus simultaneously and size of the capacitor by an index based method to obtain good results without violating the voltage constraints. This method has the versatility of being applied to the large distribution systems and having any uncertain data. The proposed method is tested with different radial distribution systems.
The mathematical formulation of the proposed method is explained in Section 5.2. In this Section, the objective function and its constraints are defined. The identification of sensitive bus for capacitor placement using fuzzy logic is described in Section 5.3. Also this Section explains the calculation of Power Loss Index and implementation aspects of Fuzzy Expert System (FES) to identify sensitive bus to place capacitor. The effectiveness of the proposed FES is tested with one example and the results are presented in Section 5.4. The size of the capacitor using Index based method is explained in Section 5.5. In Section 5.6, algorithm to be followed to obtain optimal location and size of capacitor are presented. The effectiveness of the proposed method is tested with different examples of distribution system and the results obtained are compared with the results of existing methods. In Section 5.7, conclusions of the proposed method are presented.
5.2 MATHEMATICAL FORMULATION
The objective function is to maximize the net savings function (F) by placing the proper size of capacitors at suitable locations is formulated as:
… (5.1)
where
F = net savings (`.)
Plr = Reduction in power losses due to installation of capacitor
= (Power loss before installation of capacitor - Power loss after
installation of capacitor)
Ke = Cost of energy in `./kWh
= Installation cost in `.
= Total number of capacitor buses
QC = total size of capacitor
KC = Capital cost of each capacitor
λ = rate of annual depreciation and interest charges of capacitor
5.2.1 Constraints
The objective function is subjected to the following constraints
The voltage at each bus should lie within the voltage limits.
Vmin.≤Vi≤Vmax. i=1,2, …..no. of buses
The size of the capacitor to be installed at suitable bus is less than the total reactive load of the system.
where nbus= total number of buses
5.3 IDENTIFICATION OF SENSITIVE BUS FOR CAPACITOR PLACEMENT USING FUZZY LOGIC
The fuzzy logic is used to identify the optimal location to place the capacitor in a radial distribution system so as to minimize the losses while keeping the voltage at buses within the limit and also by taking the cost of the capacitors in to account.
The Fuzzy Expert System (FES) contains a set of rules, which are developed from qualitative descriptions. In a FES, rules may be fired with some degree using fuzzy inference, where as in a conventional Expert System, a rule is either fired or not fired. For the capacitor placement problem, rules are defined to determine the suitability of a bus for capacitor placement. Such rules are expressed in the following general form:
If premise (antecedent), THEN conclusion (Consequent)
For determining the suitability of a particular bus for capacitor placement at a particular bus, sets of multiple-antecedent fuzzy rules have been established. The inputs to the rules are the bus voltages in p.u., power loss indices, and the output consequent is the suitability of a bus for capacitor placement.
5.3.1 Procedure to calculate power loss index
The Power Loss Index at ith bus, PLI (i) is the variable which is given to fuzzy expert system to identify suitable location for the capacitor.
Step 1 : Read radial distribution system data
Step 2 : Perform the load flows and calculate the base case active power loss
Step 3 : By compensating the reactive power injections (Qc) at each bus (except
source bus)and run the load flows, and calculate the active power loss in
each case.
Step 4 : Calculate the power loss reduction and power loss indices using the following
equation
… (5.2)
where
X(i) = loss reduction at ith bus
Y = minimum loss reduction
Z = maximum loss reduction
nbus = number of buses
Step 5 : Stop
5.3.2 Implementation aspects of Fuzzy expert system to identify the sensitive bus
The power loss indices and bus voltages are used as the inputs to the fuzzy expert system, which determines the buses which are more suitable for capacitor installation. The power loss indices range varies from 0 to 1, the voltage range varies from 0.9 to 1.1 and the output [Capacitor Suitability Index (CSI)] range varies from 0 to 1. These variables are described by five membership functions of high, high-medium/normal, medium/normal, low-medium/normal and low. The membership functions of power loss indices and CSI are triangular in shape, the voltage is combination of triangular and trapezoidal membership functions. These are graphically shown in Figs. 5.1 to 5.3.
0 0.2 0.4 0.6 0.8 1.0
Power Loss Index
1.0
0.8
0.6
0.4
0.2
0
Degree of Membership
Low-Med
Med
Hi-Med
High
Low
1.0
0.8
0.4
0.2
0.0
Degree of membership
Lo-Norm
Low
Norm
Hi-Norm
High
0.0 0.92 0.94 1.0 1.04 1.06 1.1
Voltage (p.u.)
Fig. 5.1 Power loss index membership function
Fig. 5.2 Voltage membership function
Low-Med
Med
Hi-Med
High
Low
0 0.2 0.4 0.6 0.8 1.0
Capacitor Suitability Index
1.0
0.8
0.6
0.4
0.2
0
Degree of Membership
Fig. 5.3 Capacitor suitability index membership function
For the capacitor placement problem, rules are defined to determine the suitability of a bus for capacitor installation. For determining the suitability for capacitor placement at a particular bus, a set of multiple antecedent fuzzy rules have been established. The rules are summarized in the fuzzy decision matrix in Table 5.1. The consequent of the rules are in the shaded part of the matrix.
And
Voltage
Low
Low-
Normal
Normal
High-Normal
High
Power Loss Index
(PLI)
Low
Low-Med.
Low-Med.
Low
Low
Low
Low-
Med.
Med.
Low-Med.
Low-Med.
Low
Low
Med.
High- Med.
Med.
Low-Med.
Low
Low
High-Med.
High-Med.
High-Med.
Med.
Low-Med.
Low
High
High
High-Med.
Med.
Low-Med.
Low-Med.
Table 5.1 Decision matrix for determining suitable capacitor locations
After the FES receives inputs from the load flow program, several rules may fire with some degree of membership. The fuzzy inference methods such as Mamdani max-min and max-prod implication methods [34] are used to determine the aggregated output from a set of triggered rules.
A final aggregated membership function is achieved by taking the union of all the truncated consequent membership functions of the fired rules. For the capacitor placement problem, resulting capacitor suitability index membership function, s, of bus i for ‘m’ fired rules is
… (5.3)
Where PLI and v are the membership functions of the power loss index and p.u. voltage level respectively.
Once the suitability membership function of a bus is calculated, it must be defuzzified in order to determine the buses suitability ranking. The centroid method of defuzzification is used; this finds the center of area of the membership function. Thus, the capacitor suitability index is determined by:
… (5.4)
5.3.3 Illustration of FES for a sample system
The proposed method is explained with a sample system. Consider a 15 bus system whose single line diagram is shown in Fig. 2.3. The line and load data of this system is given in Appendix – A (Table A.1). After performing the load flows for base case, the total active power loss and minimum voltage is given as 61.7993 kW and 0.9445 p.u.
Considering one bus at a time, every bus is compensated with reactive power injection equivalent to that of self reactive load. Now perform the load flows to determine the active power loss, power loss index and the voltage in each case. These are given in Table 5.2.
Table 5.2 Power Loss Index and voltage
Bus No.
Voltage (p.u.)
PLI
Bus No.
Voltage (p.u.)
PLI
1
1.0000
0
9
0.9697
0.3231
2
0.9730
0.1874
10
0.9686
0.2128
3
0.9599
0.4478
11
0.9532
0.9891
4
0.9551
0.9676
12
0.9491
0.5621
5
0.9542
0.3289
13
0.9478
0.3706
6
0.9600
0.8375
14
0.9529
0.5221
7
0.9578
0.8661
15
0.9537
1.0000
8
0.9587
0.4528
The Capacitor Suitability Indices (CSI) of 15 bus system from FES is given in Table 5.3. The most suitable buses for capacitor placement are selected based on the maximum value of CSI of the system and they are 3, 4, 6, 11 and 15.
Table 5.3 Capacitor suitability indices of 15 bus system
Bus No.
CSI
Bus No.
CSI
1
0.0800
9
0.2574
2
0.2407
10
0.2451
3
0.7500
11
0.7500
4
0.7500
12
0.5719
5
0.3371
13
0.3715
6
0.7500
14
0.5301
7
0.4246
15
0.7500
8
0.4336
5.4 PROCEDURE TO CALCULATE CAPACITOR SIZE USING INDEX BASED METHOD
After knowing the optimal locations to place the capacitor, the size of the capacitor can be calculated by using index based method.
… (5.5)
Where
Vi = Voltage at ith bus.
Ip[k], Iq[k] = real and reactive component of current in kth branch.
Qeffectiveload,I = total reactive load beyond ith bus (including Qload at ith bus)
Qtotal = total reactive load of the given distribution system
… (5.6)
where
Qload[i] = local reactive load at ith bus
5.5 ALGORITHM FOR CAPACITOR PLACEMENT AND SIZING USING FES AND INDEX BASED METHOD
Step 1: Read the system input data
Number of buses, number of branches, resistance and reactance of each branch, from bus and to bus of each branch, active and reactive power of each bus.
Base kV, base kVA, tolerance, etc.
Step 2: Run load flow program and calculate the voltage at each bus and
calculate the active power loss before compensation.
Step 3: Run the load flow program by compensating the reactive load at each
bus, considering one bus at a time, and calculate the loss reduction at
each bus.
Step 4: The power-loss reduction indices and the bus voltages are the inputs to
the fuzzy expert system.
Step 5: The outputs of FES, the capacitor suitability index, CSI are obtained
from which the optimal location for the capacitor placement is selected
by considering the maximum value of it.
Step 6: The index vector is determined at selected buses using Eqn.(5.5).
Step 7: Calculate the size of capacitor at selected buses by multiplying the
reactive load at that bus with index vector at that bus (Eqn. (5.6)).
Step 8: Then placing the calculated size of capacitors at best locations conduct
a load flow study.
Step 9: Print the results.
Step 10: Stop
5.6 FLOW CHART FOR OPTIMAL CAPACITOR PLACEMENT USING FES
Read Distribution System line and load data, base kV and kVA, iteration count (IC) =1and tolerance (ε) = 0.0001
Start
Perform load flows and calculate voltage at each bus, real and reactive power losses
Calculate the loss reduction by running load flow by compensating the reactive load at each bus, considering one bus at a time
Calculate power loss reduction indices, PLI using Eqn. (5.2)
Calculate index vector and size of capacitor using Eqns. (5.5) and (5.6)
Select the optimal locations for the capacitor placement by considering the maximum value of CSI
Obtain Capacitor Suitability Index (CSI) from the FES by providing PLI and bus voltages as inputs to the FES
Stop
Compute voltages, angles, power flows, real and reactive power losses and Print the results
Perform load flow by placing the calculated size of capacitors at best locations
Check for convergence
Yes
No
IC=IC+1
Compute bus voltages, real and reactive power losses
Fig. 5.4 Flow chart for optimal capacitor placement using FES
5.7 ILLUSTRATIVE EXAMPLES
The proposed method is tested with four different radial distribution systems having of 15, 33, 34 and 69 buses.
5.7.1 Example – 1
Consider a 15 bus system whose single line diagram is shown in Fig. 2.3. The line and load data of this system is given in Appendix A (Table A.1). The total real power loss and minimum bus voltage before compensation are 61.7993 kW and 0.9445 p.u.
The optimal locations and the size of the capacitors obtained by the proposed method are given in Table 5.4. In addition, voltage at these buses before and after compensation, loss reduction and net savings due to compensation are also given in the same table. The effect of using the nearest standard size capacitors instead of actual size of the capacitors is presented in Table 5.5 and it is observed that the changes in loss reduction and net savings are marginal. The active power loss reduction due to compensation is from 61.7933 kW to 32.1437 kW i.e., a reduction of 47.98% of the original active power loss.
The voltage profile of the system before and after compensation is given in Table 5.6. The minimum voltage is improved from 0.9445 p.u. to 0.9667 p.u. The voltage regulation is improved from 5.55% to 3. 33%. The line flows of the system is given in Table 5.7.
Table 5.4 Capacitor allocation and loss reduction of 15 bus RDS for calculated
size of capacitor
Bus No.
Without capacitor
With capacitor
Voltage (p.u.)
Voltage (p.u.)
Q-Cap (kVAr)
3
0.9567
0.9734
189.7
4
0.9509
0.9739
349.54
6
0.9582
0.9747
292. 65
11
0.9500
0.9761
284.19
15
0.9484
0.9752
278.95
Total size of capacitor
1,395.03
Without capacitor
With capacitor
Improvement
Ploss (kW)
Qloss
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
61.7933
57.2967
31.8981
24.5325
29.8952
32.7642
Net Saving (`.)
Without Capacitor
With Capacitor
-----
6, 79, 844/-
Table 5.5 Capacitor allocation and loss reduction of 15 bus system for standard
size of capacitor
Bus No.
Without capacitor
With capacitor
Voltage (p.u.)
Voltage (p.u.)
Q-Cap (kVAr)
3
0.9567
0.9765
200
4
0.9509
0.9740
350
6
0.9582
0.9751
300
11
0.9500
0.9742
275
15
0.9484
0.9748
275
Total size of capacitor
1,400
Without capacitor
With capacitor
Improvement
Ploss
(kW)
Qloss
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
61.7933
57.2967
32.1437
24.9865
29.6496
32.3102
Net Saving (`.)
Without Capacitor
With Capacitor
-----
6,74, 695/-
Table 5.6 Voltage profile before and after compensation of 15 bus RDS
Bus No.
Before compensation
After compensation
Voltage magnitude (p.u.)
Angle (deg.)
Voltage magnitude (p.u.)
Angle (deg.)
1
1.0000
0.0000
1.0000
0.0000
2
0.9713
0.0320
0.9835
-0.6516
3
0.9567
0.0493
0.9765
-1.0673
4
0.9509
0.0565
0.9740
-1.2488
5
0.9499
0.0687
0.9730
-1.2372
6
0.9582
0.1894
0.9751
-0.8776
7
0.9560
0.2166
0.9738
-0.9327
8
0.9570
0.2050
0.9738
-0.8625
9
0.9680
0.0720
0.9802
-0.6126
10
0.9669
0.0850
0.9792
-0.5999
11
0.9500
0.1315
0.9742
-1.2571
12
0.9458
0.1824
0.9690
-1.2086
13
0.9445
0.1987
0.9677
-1.1931
14
0.9486
0.0848
0.9717
-1.2218
15
0.9484
0.0869
0.9748
-1.3096
Table 5.7 Line flows of 15 bus system
Bus No.
Before compensation
After compensation
Active power loss (kW)
Reactive power loss (kVAr)
Active power loss (kW)
Reactive power loss (kVAr)
1
37.7019
36.8772
18.2521
17.8528
2
11.2895
11.0426
5.2822
5.1667
3
2.4439
2.3904
1.1573
1.1320
4
0.0554
0.0374
0.0528
0.0356
5
0.4722
0.3185
0.4604
0.3106
6
0.0592
0.0399
0.0577
0.0389
7
5.7680
3.8906
2.8006
1.8890
8
0.3936
0.2655
0.1864
0.1257
9
0.1129
0.0762
0.1091
0.0736
10
2.1763
1.4679
1.0427
0.7033
11
0.6016
0.4058
0.5732
0.3866
12
0.0740
0.0499
0.0705
0.0476
13
0.2049
0.1382
0.1952
0.1317
14
0.4399
0.2967
0.2055
0.1386
The variations of real power loss at each branch and voltage magnitude at each bus with and without compensation are shown in Figs. 5.5 and 5.6 respectively.
Fig. 5.5 Real power loss at each branch of 15 bus RDS with and without
capacitor
Fig. 5.6 Voltages at each bus of 15 bus RDS with and without capacitor
5.7.2 Example – 2
Consider a 34 bus system whose single line diagram is shown in Fig. 5.7. The line and load data of this system is given in Appendix A (Table A.4).The total real power loss and minimum bus voltage before compensation are 221.7210 kW and 0.9417p.u. The capacitor Suitability Index and capacitor sizes (nearest standard size of capacitors to the actual value) at the best suitable buses are given in Table 5.8.
Fig. 5.7 Single line diagram of 34 bus radial distribution system
Table 5.8 CSI and size of capacitor of 34 bus RDS
Bus No.
CSI
Capacitor size (kVAr)
20
0.7500
450
21
0.7500
150
23
0.7500
300
24
0.7500
300
25
0.7500
300
Total size of capacitor
1,500
The summary of results before and after compensation is given in Table 5.9. The comparison of results with existing methods is given in Table 5.10.
Table 5.9 Capacitor allocation and loss reduction for 34 bus system
Bus No.
Without capacitor
With capacitor
Voltage (p.u.)
Voltage (p.u.)
Q-Cap (kVAr)
20
0.9549
0.9684
450
21
0.9520
0.9659
150
23
0.9460
0.9587
300
24
0.9435
0.9555
300
25
0.9423
0.9539
300
Total size of capacitor required
1,500
Without capacitor
With capacitor
Improvement
Ploss
(kW)
Qloss
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
Ploss
(kW)
Qloss
(kVAr)
221.7210
65.1093
156.4270
39.8758
65.293
25.2335
Net Saving (`.)
Without Capacitor
With Capacitor
----
16, 05, 926/-
Min. Voltage (p.u.)
0.9417
0.9509
Table 5.10 Comparison of results of 34 bus system with existing methods
Description
Existing method [45]
Existing method [36]
Proposed method
Before compensation
After compensation
Before compensation
After compensation
Before compensation
After compensation
Real power losses (kW)
221.72
168.35
221.72
181.72
221.7210
156.4270
Net saving (`.)
----
12,40,563/-
---
9,65,200/-
----
16,05,926/-
Total size of capacitor required (kVAr)
1550
---
1650
----
1500
---
From Table 5.9 it is observed that, the minimum voltage is improved from 0.9417 p.u. to 0.9509 p.u., total real power loss reduced from 221.7210 kW to 156.4270 kW (i.e., 29.45%) and total reactive power loss reduced from 65.1093 kVAr to 39.8758 kVAr (i.e., 38.76%) due to reactive power compensation. Thus, voltage regulation is improved from 5.83% to 4.91%. From Table 5.10, the size of the capacitor required is 1500 kVAr and the net saving is `.16, 05, 926/- which is comparable with the existing methods.
The variations of real power loss at each branch and voltages at each bus for with and without compensation are shown in Figs. 5.8 and 5.9 respectively.
Fig. 5.8 Real power loss at each branch of 34 bus RDS with and without
capacitor
Fig. 5.9 Voltages at each bus of 34 bus RDS with and without capacitor
5.7.3 Example – 3
Consider a 33 bus system whose single line diagram is shown in Fig. 2.5. The line and load data of this system is given in Appendix - A (Table A.2). The CSI and size of capacitor is given in Table 5.11. The summary of results before and after compensation is given in Table 5.12. From results it is observed that, the minimum voltage is improved from 0.9131 p.u. to 0.9237 p.u. The improvement in voltage regulation is 1.06%. Also, the total real power loss reduces from 202.5022 kW to 145.0658 kW (i.e., 28.36%) and reactive power loss reduces from 135.1286 kVAr to 96.956 kVAr (i.e., 28.25%) after capacitor placement. The net saving is `.14,57,428/-. The variation of real power loss at each branch and voltages at each bus with and without capacitor are shown in Figs. 5.10 and 5.11 respectively.
Table 5.11 CSI and size of capacitor for 33 bus system
Bus No.
CSI
Capacitor size (kVAr)
30
0.9180
1050
Table 5.12 Capacitor allocation and loss reduction for 33 bus system
Description
Without capacitor
With capacitor
Min. Voltage
0.9131
0.9237
Voltage regulation (%)
8.69
7.63
Total real power loss(kW)
202.5022
145.0658
Total reactive power loss(kVAr)
135.1286
96.956
Improvement in real power loss (kW)
157.4364
Improvement in reactive power loss (kVAr)
38.1726
Total capacitor size at bus 30
1050
Net saving (`.)
-----
14,57,428/-
Fig. 5.10 Real power loss at each branch of 33 bus RDS with and without
capacitor
Fig. 5.11 Voltages at each bus of 33 bus RDS with and without capacitor
5.7.4 Example – 4
Consider a 69 bus system whose single line diagram is shown in Fig. 2.6. The line and load data of this system is given in Appendix - A (Table A.3). The CSI and size of capacitor (nearest standard size of capacitor to the actual value) is given in Table 5.13. The summary of results before and after compensation is given in Table 5.14. From results it is observed that, the minimum voltage is improved from 0.9123 p.u. to 0.9341 p.u. The improvement in voltage regulation is 2.18%. Also, the total real power loss reduces from 224.9457 kW to 152.0469 kW (i.e., 32.40%) and reactive power loss reduces from 102.1397 kVAr to 70.485 kVAr (i.e., 30.99%) after capacitor placement. The net saving is `. 20, 65, 298/-.The variation of real power loss at each branch and voltages at each bus with and without capacitor are shown in Figs. 5.12 and 5.13 respectively.
Table 5.13 CSI and size of capacitor for 69 bus system
Bus No.
CSI
Capacitor size (kVAr)
61
0.9200
1350
Table 5.14 Capacitor allocation and loss reduction for 69 bus system
Description
Without capacitor
With capacitor
Min. Voltage
0.9123
0.9341
Voltage regulation (%)
8.77
6.59
Total real power loss(kW)
224.9457
152.0469
Total reactive power loss(kVAr)
102.1397
70.485
Improvement in real power loss (kW)
72.8988
Improvement in reactive power loss (kVAr)
31.6547
Total capacitor size at bus 61
1350
Net saving (`.)
-----
20,65,298/-
Fig. 5.12 Real power loss at each branch of 69 bus RDS with and without
capacitor
Fig. 5.13 Voltages at each bus of 69 bus RDS with and without capacitor
5.8 CONCLUSIONS
A method has been proposed to determine most sensitive buses to place capacitors using fuzzy logic and its size is calculated using index based method in radial distribution systems. The FES considers loss reduction and voltage profile improvement simultaneously while deciding which buses are the most ideal for placement of capacitor. Hence, a good compromise of loss reduction, voltage profile improvement and net saving is achieved when compared to existing methods. The proposed method has been tested on four distribution systems consisting of 15, 33, 34 and 69 buses. It has been noticed that losses are reduced and voltage profile is improved.

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