Optimal Conductor Selection Using Fuzzy Logic

Published Date: 02 Nov 2017

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 <Max.

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.