Maximum loading and cost of energy loss of radial distribution feeders

Electrical Power and Energy Systems 26 (2004) 307–314 www.elsevier.com/locate/ijepes

Maximum loading and cost of energy loss of radial distribution feeders D. Das* Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, West Bengal, India Accepted 30 September 2003

Abstract The paper presents a method for obtaining the maximum allowable loading of radial distribution feeders for different types of loads without violating the maximum current carrying capacity of branch conductors. Minimum voltage of the feeders can also be maintained by allowing the feeders to take load growth up to a specific period of time. A simple mathematical formula for calculating power loss in kth year is proposed. Another mathematical expression for the present worth of the cost of feeder energy loss over the entire life time is also proposed for different types of load modeling considering the effect of load growth, load factor and cost of energy. Results are also presented for a radial distribution feeder. q 2004 Elsevier Ltd. All rights reserved. Keywords: Radial feeders; Loading capability; Energy loss calculation

1. Introduction The steady increase in the demand for electrical energy is forcing utilities into higher investments; a larger part of which have to be made in distribution systems. Distribution systems must continually be adaptable to meet the maximum possible power demand without violating the voltage limit and maximum current carrying capacity of branch conductors. Service quality is the main constraint of distribution feeders. The quality of service, ensuring declared voltage to the consumers, is a function of several variables such as voltage drops in the lines and transformers, tap settings in the transformers, voltage receive at the source stations in the grid, etc. All these variables, except the voltage drop in the feeders are dependent either on the standards of manufacturing or on the behaviour of the transmission systems. The voltage drop in the distribution feeders is solely dependent on the type of branch conductors, level of loading, types of load distribution and circuit voltage. Miu and Chiang [1] are probably the first to propose a solution algorithm for distribution system load capability. They have computed the load capability under different * Tel.: þ91-3222-79507; fax: þ91-3222-55303. E-mail address: [email protected] (D. Das). 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2003.09.002

loading conditions for a given load variation pattern. However, they have not proposed any mathematical model for computing the present worth of the cost of feeder energy loss. Ponnavaikko and Rao [2,3] have proposed a mathematical model for computing the present worth of the cost of feeder energy loss. However, their model [2,3] is suitable only for constant current load and for radial main feeder only. In the present work, a simple algorithm is proposed for determining the maximum loading of the feeders without violating the maximum current carrying capacity of branch conductors. A predetermined annual load growth is also considered to determine the allowable load growth period without violating the minimum voltage limit of the feeders. For determine the power loss in the kth year, a simple loss formula which is function of power loss in the base year (0th year) and load growth is also proposed for different types of loads. A mathematical expression for present worth of the cost of feeder energy loss over the entire life period of the feeder is also proposed for different types of loads considering load growth, growth in load factor and cost of energy. Results are also presented through an example. This work can be useful for distribution system planning and also to help the study of optimum shunt capacitors placement and network reconfiguration to increase the system capacity.

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D. Das / Electrical Power and Energy Systems 26 (2004) 307–314

2. Assumptions A balanced three phase radial distribution feeder is assumed and can be represented by its equivalent single-line diagram. Line charging capacitance is negligible at the distribution voltage levels as found in most practical cases.

3. Methodology Fig. 1 shows a sample radial distribution feeder. The branch number, sending end and receiving end nodes are given in Table 1. Branch numbers of this feeder (Fig. 1) are also shown in (·). The load current at any node i; is given as ILi ¼

PLi 2 jQLi ; i ¼ 2; 3; 4; …; NB Vip

ð1Þ

For the purpose of explanation, consider branch 2 5 of Fig. 1. Current through branch 2 5 is given by I5 ¼ IL5

Table 1 Branch number, sending end node, receiving end node, and nodes Beyond branches 1,2,3,…,11 of Fig. 1 Branch number ðjjÞ

Sending end node IS ðjjÞ

Receiving end node IR ðjjÞ

Total number of nodes beyond branch 2 jj, ðNðjjÞÞ

Nodes beyond branch 2 jjðIEðjj; iÞ, i ¼ 1; 2; …; NðjjÞ)

1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 2 7 8 4 10 8

2 3 4 5 6 7 8 9 10 11 12

11 6 5 2 1 4 3 1 2 1 1

2,3,7,8,4,5,10,9,12,6,11 3,4,5,10,6,11 4,5,10,6,11 5,6 6 7,8,9,12 8,9,12 9 10,11 11 12

General expression of branch current through branch 2jj is given by [4]

ð2Þ

Now consider branch 2 4. Total number of nodes beyond branch 2 4 is two and these two nodes are 5 and 6, respectively. Therefore, current through branch 2 4 is I4 ¼ IL5 þ IL6

ð3Þ

Similarly, consider branch 2 3. Total number of nodes beyond branch 2 3 is five and these nodes are 4, 5, 6, 10 and 11, respectively. Hence current through branch 2 3 is I3 ¼ IL4 þ IL5 þ IL6 þ IL10 þ IL11

ð4Þ

From Eqs. (2) –(4), it is clear that, if we identify the nodes beyond all the branches and if the load currents are known, then it is extremely easy to compute the branch currents. A load flow algorithm based on the identification of the nodes beyond all the branches has been proposed in Ref. [4]. In the present work, this load flow algorithm [4] is used to compute the load currents and branch currents.

IðjjÞ ¼

NðjjÞ X

ð5Þ

IL{IEðjj; iÞ}

i¼1

where NðjjÞ ¼ total number of nodes beyond branch 2jj IEðjj; iÞ ¼ stores the number of nodes beyond branch 2jj for i ¼ 1; 2; …; NðjjÞ: Eq. (5) can be rewritten as IðjjÞ ¼

NðjjÞ X i¼1

PL{IEðjj; iÞ} 2 jQL{IEðjj; iÞ} V p {IEðjj; iÞ}

ð6Þ

Now, if the real and reactive power loads of all the nodes beyond branch 2jj is increased by a factor DlðjjÞ; then current through branch 2jj is given by Il ðjjÞ¼{1þDlðjjÞ}

NðjjÞ X

PL{IEðjj;iÞ}2jQL{IEðjj;iÞ} Vlp {IEðjj;iÞ} i¼1

ð7Þ

where Il ðjjÞ ¼ current of branch 2jj after the loads of all the nodes beyond branch 2jj is increased by a factor DlðjjÞ Vl {IEðjj; iÞ} ¼ new voltage of node IEðjj; iÞ after the load is increased. Note that DlðjjÞ is a real quantity. Eq. (7) may be rewritten as Il ðjjÞ ¼ {1 þ DlðjjÞ}

NðjjÞ X i¼1

PL{IEðjj; iÞ} 2 jQL{IEðjj; iÞ} V p {IEðjj; iÞ}

p

 Fig. 1. Sample radial distribution network.

V {IEðjj; iÞ} Vlp {IEðjj; iÞ}

ð8Þ

D. Das / Electrical Power and Energy Systems 26 (2004) 307–314

In a distribution system voltage angle is very very small and can be neglected. Therefore, for the sake of simplicity we assume that V p {IEðjj; iÞ} lV{IEðjj; iÞ}l < 1:0 ¼ p Vl {IEðjj; iÞ} lVl {IEðjj; iÞ}l

ð9Þ

Then Eq. (8) may be written as NðjjÞ X PL{IEðjj; iÞ} 2 jQL{IEðjj; iÞ} Il ðjjÞ ¼ {1 þ DlðjjÞ} V p {IEðjj; iÞ} i¼1 ð10Þ From Eqs. (6) and (10), we get Il ðjjÞ ¼ {1 þ DlðjjÞ}IðjjÞ

ð11Þ

As DlðjjÞ is a real quantity, therefore magnitude of branch current Il ðjjÞ can be written as lIl ðjjÞl ¼ {1 þ DlðjjÞ}lIðjjÞl

ð12Þ

Now the constraint is that the current carrying by each branch conductor of the feeder should not violate the maximum current carrying capacity of the conductor of each branch. Therefore, maximum allowable current that each branch of the feeder can carry, can be given as lIl ðjjÞl ¼ CCðjjÞ

ð13Þ

where, CCðjjÞ is the maximum current carrying capacity of the conductor of branch 2jj From Eqs. (12) and (13) we get DlðjjÞ ¼

CCðjjÞ 2 lIðjjÞl lIðjjÞl

ð14Þ

Further explanation on DlðjjÞ is as follows. After running the load flow program [4] for base case, lIðjjÞl for jj ¼ 1; 2; …; NB 2 1; must be computed. After that, DlðjjÞ for jj ¼ 1; 2; …; NB 2 1 must be computed using Eq. (14). Now the minimum of all the values of DlðjjÞ for jj ¼ 1; 2; …; NB 2 1 must be selected such that maximum current carrying capacity of the branch conductors should not be violated. Therefore, we can write Dlmin ð‘Þ ¼ min{DlðjjÞ; jj ¼ 1; 2; …; NB 2 1}

ð15Þ

where ‘ is the branch number at which loading factor is minimum. For example, consider Fig. 1, and say Dlmin ð‘Þ ¼ Dlð4Þ; i.e. ‘ ¼ 4: Therefore, beyond branch 2 4 there are two nodes, 5 and 6, respectively, and only real and reactive power loads of these two nodes must be increased by a factor Dlmin ð4Þ and rest of the loads will remain unchanged. This is a simple iterative process and complete algorithm is given in Section 4.

Step 2 IT ¼ 1 Step 3 Run distribution load flow program using the algorithm given in Ref. [4] And compute lIðjjÞl for jj ¼ 1; 2; …; NB 2 1 Step 4 Compute DlðjjÞ for jj ¼ 1; 2; …; NB 2 1 using Eq. (14) Step 5 Find out Dlmin ¼ Dlmin ð‘Þ ¼ min{DlðjjÞ; for jj ¼ 1; 2; …; NB 2 1} Step 6 Dl ¼ Dl þ Dlmin Step 7 di ¼ lDl 2 sfl Step 8 If {di , 1} go to Step 13 Otherwise go to Step 9 Step 9 sf ¼ Dl Step 10 Identify all the nodes beyond branch 2‘ using the node identification algorithm given in Ref. [4] and update the loads of all the nodes beyond branch 2‘; i.e. for i ¼ 1 to Nð‘Þ; compute m ¼ IEð‘; iÞ PLðmÞ ¼ ð1 þ DlÞPL0ðmÞ QLðmÞ ¼ ð1 þ DlÞQL0ðmÞ Step 11 IT ¼ IT þ 1 Step 12 If{IT # ITMAX} go to Step 3 Step 13 Solution has converged Write down the value of Dl and load flow results. Step 14 stop

5. Load modeling All loads were represented by their active and reactive components (i.e. PL0ðiÞ and QL0ðiÞ; for i ¼ 2; 3; …; NB) at 1.0 per unit nodes voltage [9,10]. The effect of voltage magnitude variation on load of node i is represented as follows PLðiÞ ¼ PL0ðiÞlVi lb

ð16Þ

QLðiÞ ¼ QL0ðiÞlVi lb

ð17Þ

where 4. Algorithm for computation of loading factor Step 1 Dl ¼ 0:0 sf ¼ 0.0

309

lVi l ¼ voltage magnitude of node i b ¼ 0 for constant power load b ¼ 1 for constant current load b ¼ 2 for constant impedance load

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D. Das / Electrical Power and Energy Systems 26 (2004) 307–314

For composite load modeling, the real and reactive power loads of node i are given as PLðiÞ ¼ PL0ðiÞðC1 þ C2 lVi l þ C3 lVi l2 Þ

ð18Þ

QLðiÞ ¼ QL0ðiÞðD1 þ D2 lVi l þ D3 lVi l2 Þ

ð19Þ

Constants (C1 ; D1 ), (C2 ; D2 ) and (C3 ; D3 ) are the compositions of constant power, constant current and constant impedance loads, respectively. In the present work, a composition of 40% constant power ðC1 ¼ D1 ¼ 0:4Þ; 30% of constant current ðC2 ¼ D2 ¼ 0:3Þ and 30% of constant impedance ðC3 ¼ D3 ¼ 0:3Þ loads are considered. It is worth mentioning here that for load flow calculation, substation voltage is considered as 1.0 pu, i.e. Vs ¼ V1 ¼ 1:0/08: When load is of constant power type, real and reactive power loads (i.e. b ¼ 0; PLðiÞ ¼ PL0ðiÞ and QLðiÞ ¼ QL0ðiÞ) will remain constant throughout the iterative process of load flow. For constant current load (i.e. b ¼ 1; PLðiÞ ¼ PL0ðiÞlVi l and QLðiÞ ¼ QL0ðiÞlVi l), constant impedance load (i.e. b ¼ 2; PLðiÞ ¼ PL0ðiÞlVi l2 and QLðiÞ ¼ QL0ðiÞlVi l2 ) and composite loads (Eqs. (18) and (19)), PLðiÞ and QLðiÞ have to be computed after every iteration of load flow and hence load flow results will be different for different types of loads consideration although the initial loads (PL0ðiÞ and QL0ðiÞ for i ¼ 2; 3; …; NB at 1.0 pu nodes voltage) remain same for all types of loads. Therefore, loading factor ðDlÞ as mentioned in Section 3 will also be different for different types of loads. It was found that loading factor is minimum for constant power load and maximum for constant impedance load.

loads are of constant power or constant impedance type, proposed algorithm takes four iterations to converge. But for constant current type loads, it takes only one iteration to converge. For composite load, number of iterations depends on the composition of different types of loads. In the present case (for 40% constant power, 30% each for constant current and constant impedance types of loads), the proposed algorithm takes only two iterations to converge. From Table 2, it is also seen that constant power loads give a conservative estimate of the load, while constant impedance loads are the liberal. Note that real and reactive power loads at the base year (0th year) 996 kW and 747 kVAr, respectively.

7. Determination of maximum load growth period considering minimum voltage limit of the feeder Load growth in an area with time is a natural phenomenon. The growth in feeder load may be due to addition of new loads to the feeder or due to the incremental addition to the existing loads. A feeder designed and constructed on a long term basis can accept additional loads to the extent it can accommodate satisfying the voltage constraint. Once the load exceeds the feeder capacity, limited by either voltage regulation 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. Real and reactive power load at any year k is given by

6. Example

PLOADðkÞ ¼ PLOADð0Þð1 þ gÞk

ð20Þ

The problem considered here is that a new feeder is already planned and its estimated peak load in the base year (0th year) is given. Example network is a 30 node radial distribution network. Data for this system are given in Appendix A. Table 2 shows the value of loading factor, number of iterations and maximum allowable load of the feeder without violating the maximum current carrying capacity of branch conductors. From Table 2, it is seen that when

QLOADðkÞ ¼ QLOADð0Þð1 þ gÞk

ð21Þ

Table 2 Number of iterations, loading factor and maximum feeder load Types of loads

Constant power Constant current Constant impedance Composite

Number of iterations

4 1 4 2

Loading factor ðDlÞ

0.548 0.696 0.853 0.677

Maximum real and reactive power load PLOAD (kW)

QLOAD (kVAr)

1542 1689 1849 1671

1156 1267 1387 1253

where g ¼ annual load growth rate PLOAD(0) ¼ real power loads in the base year (0th year) QLOAD(0) ¼ reactive power loads in the base year (0th year) PLOADðkÞ ¼ real power load in the year k QLOADðkÞ ¼ reactive power load in the year k Eq. (20) or (21) may be used to determine the maximum allowable load growth period. It is assumed that the annual load growth rate g ¼ 7:5% ¼ 0:075: From Table 2, for constant power load, PLOADðk ¼ kmax Þ ¼ 1542 kW and as mentioned earlier, real power load at the base year, PLOADð0Þ ¼ 996 kW. Therefore, using Eq. (20), we can write 1542 ¼ 996ð1 þ 0:075Þkmax or kmax ¼ 6:04 years

D. Das / Electrical Power and Energy Systems 26 (2004) 307–314

311

Table 3 Loads, losses and minimum feeder voltage for constant power load Different variables

Base year (0th year)

1st year

2nd year

3rd year

4th year

5th year

6th year

Real power load (kW) Reactive power load (kVAr) PLOSS (kW) QLOSS (kVAr) Vmin (pu)

996 747 61.5 37.8 0.9366

1071 803 71.8 44.0 0.9315

1151 863 83.8 51.4 0.9260

1237 928 98 60.1 0.9200

1330 998 114.6 70.6 0.9135

1430 1072 134.3 82.4 0.9063

1537 1153 157.6 96.66 0.8985

Table 4 Loads, losses and minimum feeder voltage for constant current load Different variables

Base year (0th year)

1st year

2nd year

3rd year

4th year

5th year

6th year

7th year

Real power load (kW) Reactive power load (kVAr) PLOSS (kW) QLOSS (kVAr) Vmin (pu)

996 747 54.8 33.65 0.9402

1071 803 63.35 38.9 0.9357

1151 863 73.2 44.9 0.9309

1237 928 84.6 51.9 0.9257

1330 998 97.8 60 0.9202

1430 1072 113 69.4 0.9142

1537 1153 130.6 80.16 0.9077

1652 1239 150.9 92.63 0.9008

Similarly for constant current, constant impedance and composite (40% CP, 30% CI and 30% CZ) loads, kmax ¼ 7:3, 8.55 and 7.14 years, respectively. As mentioned before, feeder should not violate the voltage constraint also. l In this work, minimum voltage limit Vmin ¼ 0:90 pu is considered and load growth of the feeder is allowed as long as voltage limit is not violated. Tables 3– 6 show the load of the feeder, real and reactive power losses and minimum system voltage at the end of each year. From Table 3, it is seen that when loads are treated as constant power, feeder can take load growth up to 6 years and minimum voltage is 0.8985 pu which is very l close to the minimum voltage limit ðVmin ¼ 0:90 puÞ: Similarly, for constant current, constant impedance and composite loads (Tables 4 – 6) feeder can take load growth up to 7, 8, and 7 years, respectively, without violating the minimum voltage limit.

8. Derivation of loss formula in terms of load growth Real power loss at any year k is given by PLOSSðkÞ ¼ PLOSSð0Þð1 þ gÞak

ð22Þ

where g ¼ annual load growth rate. PLOSS(0) ¼ real power loss in the base year (0th year) PLOSSðkÞ ¼ real power loss in the kth year ðk ¼ 1; 2; …; MÞ a ¼ a constant For determining the value of a for constant power load, loss data at different years (Table 3) is used. For example, from Table 3, PLOSSðk ¼ 0Þ ¼ 61.5 kW and

Table 5 Loads, losses and minimum feeder voltage for constant impedance load Different variables

Base year (0th year)

1st year

2nd year

3rd year

4th year

5th year

6th year

7th year

8th year

Real power load (kW) Reactive power load (kVAr) PLOSS (kW) QLOSS (kVAr) Vmin (pu)

996 747 49.45 30.37 0.9433

1071 803 56.7 34.84 0.9392

1151 863 65 39.9 0.9349

1237 928 74.5 45.8 0.9304

1330 998 85.3 52.4 0.9255

1430 1072 97.6 60 0.9203

1537 1153 111.6 68.6 0.9148

1652 1239 127.6 78.36 0.9089

1776 1332 145.6 89.48 0.9027

Table 6 Loads, losses and minimum feeder voltage for composite load Different variables

Base year (0th year)

1st year

2nd year

3rd year

4th year

5th year

6th year

7th year

Real power load (kW) Reactive power load (kVAr) PLOSS (kW) QLOSS (kVAr) Vmin (pu)

996 747 55.54 34.1 0.9398

1071 803 64.26 39.44 0.9353

1151 863 74.36 45.64 0.9304

1237 928 86.06 52.28 0.9251

1330 998 99.6 61.14 0.9194

1430 1072 115.3 70.8 0.9133

1537 1153 133.54 82 0.9067

1652 1239 154.67 94.94 0.9000

40% constant power, 30% constant current and 30% constant impedance.

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D. Das / Electrical Power and Energy Systems 26 (2004) 307–314

Table 7 Comparison of real power loss using distribution load flow program and Eq. (22) for constant power load Methods

Load growth period

Real power loss (kW) calculation using load flow program Real power loss (kW) calculation using Eq. (22)

1st year

2nd year

3rd year

4th year

5th year

6th year

71.80 71.80

83.80 83.93

98.00 98.05

114.60 114.55

134.30 132.82

157.60 156.33

PLOSSðk ¼ 1Þ ¼ 71.8 kW. It is given that g ¼ 7:5% ¼ 0:075: Therefore, using Eq. (22), we get

concepts given in Refs. [2,3,5,7,8] and details are described below.

71:8 ¼ 61:5ð1 þ 0:075Þa

9.1. Cost of feeder energy loss

or a ¼ 2:14 Similarly from Table 3, PLOSSðk ¼ 2Þ ¼ 83:8 kW and using Eq. (22), we get

The energy loss in kwh in the feeder for the loading conditions in the base year (0th year) is given by

83:8 ¼ 61:5ð1 þ 0:075Þ

2a

a ¼ 2:139 Similarly other values of a for k ¼ 3; 4; 5 and 6 are 2.147, 2.151, 2.160, 2.168, respectively. Therefore it is clear that variation of a at different year is very very small and an average value may be chosen. Author has also tested five more examples and it was found that a ¼ 2:15 works extremely well for constant power load to determine the power loss in each year. In fact power loss in the base year (0th year) must be obtained using the load flow program. It was also found that value of a is insensitive to the annual load growth rate g: Similarly, for constant current load (using the loss data given in Table 4), a ¼ 2:0 and for constant impedance load (using the loss data given in Table 5), a ¼ 1:87 are found to be highly suitable. For composite load, value of a depends on the composition of different types of loads. In the present work as mentioned in Section 5, a composition of 40% constant power, 30% of constant current and 30% of constant impedance loads are considered for composite load (using loss data given in Table 6) and it was found that a ¼ 2:02 works very well. Table 7 shows the comparison of real power loss at different years using load flow program and using Eq. (22) for constant power load. This table shows that loss calculation at various years using Eq. (22) are very very close to that obtained using load flow program. Similar findings were also observed for other types of loads.

9. Derivation of formula for the present worth of the cost of feeder energy loss After deriving the loss formula in terms of load growth for different types of load, it is very simple to derive the mathematical expression for the present worth of the cost of feeder energy loss based on some fundamental mathematical

EL ¼ 3TPLOSSð0ÞðLLFÞ

ð23Þ

where the loss load factor, LLF is a function of load factor, LF and is defined [5] as LLF ¼ AðLFÞ2 þ BðLFÞ

ð24Þ

In the present study, the values for A and B are taken as 0.8 and 0.2, respectively [5], T ¼ 8760 h and PLOSS(0) is the real peak power loss per phase in the base year (0th year). The cost of energy loss is a recurring expenditure annually to the utilities throughout the life period of the feeders. The sum of the present worth of the cost of energy loss in the life period of N years at a discount rate r; can be obtained as CP0 ¼ ðELÞCð0Þ

N X k¼1

1 ð1 þ rÞk

ð25Þ

where Cð0Þ is the cost of energy per kwh in the base year. Using Eqs. (23) and (25) we get CP0 ¼ 3TPLOSSð0ÞðLLFÞCð0Þ

N X k¼1

1 ð1 þ rÞk

ð26Þ

As could be seen from Eq. (26), the energy loss cost is a function of peak power loss per phase in the base year, loss load factor and cost of energy. Variation of any of these factors will change the cost of energy loss. Therefore, the effect of load growth, growth in load factor and change in cost of energy are also considered. 9.2. Effect of load growth Load growth in an area with time is a natural phenomenon. The growth in feeder load may be due to addition of new loads to the feeder or due to the incremental additions to the existing loads. A feeder designed and constructed in a system on long term basis can accept additional loads to the extent it can accommodate satisfying the voltage constraints. Once the load exceeds the feeder capacity, limited by either voltage regulation or thermal

D. Das / Electrical Power and Energy Systems 26 (2004) 307–314

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. With the growth in load, power loss also varies in each year as already given in Eq. (22). Therefore, the effect of load growth on the cost of energy losses can be introduced by multiplying Eq. (26) by a factor ð1 þ gÞak ; k ¼ 1; 2; …; M; where M is the number of years up to which the feeder can take load growth. Therefore, Eq. (26), may be written as CP0 ¼ 3TPLOSSð0ÞðLLFÞCð0Þ " # M N X X ð1þgÞak 1 aM  þð1þgÞ k k k¼1 ð1þrÞ k¼Mþ1 ð1þrÞ

ð27Þ

9.3. Effect of growth in load factor The system experienced growth in load factor due to many reasons such as increase in load diversity with load growth, increased in energy consumption per kW connected load, measures take by the utilities to curb the growth in peak demand, etc. According to Scheer [6], the system load factor grows cutting the difference between an ultimate load factor and the present load factor into half over a period of 16 years and the load factor at any year, k; is given by LFðkÞ ¼ LFu 2 yðkÞðLFu 2 LFp Þ

ð28Þ

where yðkÞ ¼ ð0:5Þk=16

ð29Þ

LFp ¼ present load factor LFu ¼ ultimate load factor The corresponding annual loss load factor, LLFðkÞ; k ¼ 1; 2; …; M can be obtained from Eq. (24) and rewritten as 2

LLFðkÞ ¼ ALF ðkÞ þ BLFðkÞ

ð30Þ

Since the load growth on the feeder is limited to M years, the increase in load factor due to other factors beyond M years are ignored. Therefore, when the effect of increase in load factor is considered, Eq. (27) may be written as " M X LLFðkÞð1 þ gÞak CP0 ¼ 3TPLOSSð0ÞCð0Þ ð1 þ rÞk k¼1 # N X 1 aM ð31Þ þLLFðMÞð1 þ gÞ k k¼Mþ1 ð1 þ rÞ

defined as CðkÞ ¼ Cð0Þð1 þ eÞk

ð32Þ

where Cð0Þ ¼ cost of energy in the base year (0th year) e ¼ annual increase rate of the cost of energy Therefore, when the cost of energy in the year k as given in Eq. (32) is considered, the present worth of the cost of the feeder energy loss over the entire life period of the feeder as given by Eq. (31), can be written as " M X LLFðkÞð1 þ gÞak ð1 þ eÞk CP0 ¼ 3TPLOSSð0ÞCð0Þ ð1 þ rÞk k¼1 # N X ð1 þ eÞk aM ð33Þ þLLFðMÞð1 þ gÞ k k¼Mþ1 ð1 þ rÞ Eq. (33) gives the present worth of the feeder energy loss over the life period of the feeder considering the load growth factor, load factor and cost of energy. Note that in Eq. (33), values of a is different for different types of loads as mentioned in Section 8. For computing CP0, only base case load flow run is required to calculate PLOSS(0). In Refs. [2,3,5,7,8], present worth of the feeder energy loss over the life period of the feeder is also considered but expression for that given in Refs. [2,3,5,7,8] is only suitable for constant current load and main feeder case only. However, mathematical expression given in Eq. (33), can be used for any type of loads, both for main feeder and main feeder with laterals and sublaterals.

10. Results and discussions Table 8 gives the present worth of the feeder energy loss over the entire life period of the feeder with and without considering the annual increased rate of cost of energy for different types of loads. It is clearly seen that for constant Table 8 Comparison of cost of feeder energy loss with and without considering increased rate of cost of energy for different types of loads Types of loads

9.4. Effect of cost of energy The cost of equipment, materials construction, operation and maintenance in a system increase with time. This results in a continuous increase in the cost of energy in the system. The increase in the cost of energy in the year k; may be

313

Constant power Constant current Constant impedance Composite load

Cost of feeder energy loss (Rupees in lakhs) Without considering increased rate of cost of energy ðe ¼ 0:0Þ

With considering increased rate of cost of energy ðe ¼ 0:03Þ

50.01 46.18 42.71 47.97

67.62 62.90 58.60 64.31

314

D. Das / Electrical Power and Energy Systems 26 (2004) 307–314

11. Conclusions

Table A1 Line and load data of 30 node radial feeder Branch Number.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Sending end node

1 (S/S) 2 3 4 5 6 7 8 4 10 11 12 11 28 29 5 20 21 22 21 24 25 26 6 14 15 16 16 18

Receiving end node

2 3 4 5 6 7 8 9 10 11 12 13 28 29 30 20 21 22 23 24 25 26 27 14 15 16 17 18 19

R (V)

2.1632 1.0880 0.5440 0.2720 0.5440 1.3760 2.7520 4.1280 3.6432 0.9108 0.4554 0.4554 1.3760 1.3760 4.1280 0.9108 1.8216 2.7324 0.9108 2.7520 3.0272 2.7520 2.7520 0.9108 1.8216 1.8216 0.9108 1.3760 1.3760

X (V)

1.1019 0.7346 0.3673 0.1836 0.3673 0.3896 0.7792 1.1688 1.5188 0.3797 0.1898 0.1898 0.3896 0.3896 1.1688 0.3797 0.7594 1.1391 0.3797 0.7792 0.8571 0.7792 0.7792 0.3797 0.7594 0.7594 0.3797 0.3896 0.3896

Load at receiving end node PL0 (KW)

QL0 (KVAR)

35 25 10 25 38 10 15 30 40 54 30 10 40 30 22 10 40 40 27 80 45 25 20 25 40 40 100 60 30

30 15 6 15 28 5 12 30 30 30 15 5 30 20 12 5 40 30 20 70 39 20 10 20 30 30 90 30 30

Table A2 Other data Annual load growth rate ¼ 0.075 Discount rate ¼ 0.10 Present load factor ¼ 0.26 Ultimate load factor ¼ 0.55 Energy cost ¼ Rs. 1.50/kWhr Life of the feeder ¼ 25 years

power load, present worth of the cost of feeder energy loss is more and for constant impedance load it is less. For composite load, it depends on the percentage composition of the loads. It is also seen from Table 8 that when annual increased rate of cost of energy is considered ðe ¼ 3% ¼ 0:03Þ; these cost figures are much higher. Present worth of the cost of the feeder energy loss as given by Eq. (33) will be very useful for planning of distribution systems. It will also be very useful for optimum shunt capacitor placement, network reconfiguration, etc. Main advantage of this mathematical expression is that it can handle all types of loads.

A simple method has been proposed for obtaining the maximum loading of the radial feeders for different types of loads considering the maximum current carrying capacity of branch conductors. Voltage constraint has also been satisfied by allowing the feeders to take the load growth up to a specified period of time. It has been found that load capability is higher for constant impedance load and lower for constant power load. A simple power loss formula in terms of load growth has been proposed for evaluating the power loss in each year and found that power loss calculation in each year can easily be obtained using the proposed power loss formula for different types of loads without running the load flow program. Finally, a simple mathematical expression for the present worth of the cost of feeder energy loss over the entire life period of the feeder has been proposed considering load growth, growth in load factor and cost of energy. The proposed formula for the present worth of the cost of feeder energy loss will be very useful for distribution system planning, optimum placement of shunt capacitors, network reconfiguration, etc.

Appendix See Tables A1 and A2.

References [1] Miu KN, Chiang HD. Electric distribution load capability: problem formulation, solution algorithm and numerical results. IEEE Transact Power Deliv 2000;15(1):436–42. [2] Ponnavaikko M, Rao KSP. An approach to optimal distribution system planning through conductor gradation. IEEE Transact Power Apparatus Syst 1982;PAS-101(6):1735–42. [3] Ponnavaikko M, Rao KSP. Optimal choice of fixed and switched shunt capacitors on radial distributors by the method of local variations. IEEE Transact Power Apparatus Syst 1983;PAS-102(6): 1607–14. [4] Ghosh S, DAS D. Method for load flow solution of radial distribution networks. IEE Proc Gen Transm Distrib 1999;146(6):641– 8. [5] Ponnavaikko M, Rao KSP. Optimal distribution system planning. IEEE Transact Power Apparatus Syst 1981;PAS-100(6):2269–977. [6] Scheer GB. Future power prediction. Energy Int 1996;February: 14– 16. [7] Rao PSN. An extremely simple method of determining optimal conductor sections for radial distribution feeders. IEEE Transact Power Apparatus Syst 1985;PAS-104(6):1439–42. [8] Wang Z, Liu H, Yu DC, Wang X, Song H. A practical approach to the conductor size selection in planning radial distribution systems. IEEE Trans Power Deliv 2000;15(1):350–4. [9] Das D, Nagi HS, Kothari DP. Novel method for solving radial distribution networks. IEE Proc Gen Transm Distrib 1994;141(4): 291 –8. [10] Chakravorty M, Das D. Voltage stability analysis of radial distribution networks. Int J Electr Power Energ Syst 2001;23(2):129–35.