Loss allocation in radial distribution networks with various distributed generation and load models

Electrical Power and Energy Systems 75 (2016) 173–186

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Loss allocation in radial distribution networks with various distributed generation and load models Kushal Manoharrao Jagtap ⇑, Dheeraj Kumar Khatod Alternate Hydro Energy Center, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India

a r t i c l e

i n f o

Article history: Received 18 December 2014 Received in revised form 9 July 2015 Accepted 25 July 2015

Keywords: Distributed generation Load models Power flow Radial distribution network Types of DG

a b s t r a c t This paper proposes a new method for loss allocation in radial distribution networks (DNs) considering different models of distributed generation (DG) and load in context of a deregulated environment. In the proposed method, a direct relation between real/reactive power flow in a branch and its losses has been developed without taking any assumption and approximation. Suitable expressions/relations for network power flow have been developed employing power summation algorithm. The developed expressions do not contain any cross-terms. For allocating the losses among network participants, the proposed method uses a circuit based branch oriented approach. Using only power flow results, this method employs a backward sweep network reduction technique to allocate the network losses to load/DG at various nodes. This method does not require additional step of normalization to collect the exact amount of total network losses. In the present study, different types of DG, e.g. DG injecting only real power, DG injecting only reactive power, DG injecting real power and absorbing reactive power, and DG injecting both real and reactive power are considered to allocate losses. In addition to this, various load models based on impact of voltage variation on real/reactive power consumption are also considered. To test the proposed method, modified 9-node and 33-node radial DNs have been considered. In order to show the effectiveness of the proposed method, its numerical results have been compared with those by other methods available in the literature. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction Nowadays, electrical power system is experiencing major changes and is adopting deregulated operation of electricity market. The vertically integrated systems are being restructured and unbundled into generation, transmission, and distribution segments which has introduced the competition among the network participants (consumers and generators). Unlike the sale of electrical energy by generation companies, activities of transmission and distribution are generally considered as natural monopoly. Therefore, electricity market does not have any control over the cost of services provided by transmission and distribution networks (DNs). Like transmission network, power losses in the DN have large share of service charges. Thus, distribution power losses are to be allocated among network participants, fairly and justifiably. Distributed generation (DG), when introduced in DN, changes the losses depending on its location and rating [1,2]. Hence, DG should be rewarded/penalized according to its impact on losses ⇑ Corresponding author. E-mail addresses: [email protected] (K.M. Jagtap), [email protected] (D.K. Khatod). http://dx.doi.org/10.1016/j.ijepes.2015.07.042 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

of DN. Further, power loss in a branch of DN is a quadratic function of power flowing through it due to loads and DGs [3]. Hence, in bundled power flow [3], it is difficult to trace the exact share of load and DG in the network. The interdependency among network participants is expressed by the cross-terms which also have significant impact on allocated losses to loads and DGs. Hence, the allocation of total network losses cannot be carried out among consumers and DGs in the straightforward way. The critical nature of the loss allocation problem is made evident by the fact that early formulated loss allocation mechanisms, even adopted at the regulatory level, have been found to be inconsistent [4]. Various methods in the literature dealing with the problem of loss allocation are mentioned below: Based on the proportional principle, Pro rata (PR) method [5,6] allocates the network losses to consumers/DGs based on their real power consumption/injection. While allocating the losses, this method does not consider the location of consumer/DG with respect to (w.r.t.) root node and hence produces unfair result of loss allocation. MW-mile method [7,8] overcomes the drawback associated with PR method by considering the power rating as well as location of a load/DG w.r.t. root node. PR and MW-mile methods are simple and easy to implement. However, these methods do not

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Nomenclature Pt and Q t

real and reactive power, respectively, available at receiving end of branch t PD;t and Q D;t real and reactive power, respectively, of load at receiving end of branch t P0;t and Q 0;t real and reactive power, respectively, of load at receiving end of branch t under rated condition PG;t and Q G;t real and reactive power, respectively, of DG at receiving end of branch t V s;t and V r;t phasor voltages of sending and receiving nodes, respectively, of branch t a exponent for different load model Nt set of branches incident to node t Kt set of branches ahead of branch t Bt susceptance of branch t PSt and QSt real and reactive power loss, respectively, in branch t

take the power flow into account for loss allocation. Thus, to overcome these limitations of PR and MW-mile methods, marginal loss coefficient (MLC) method [9,10] came into existence for loss allocation. MLC method allocates the losses to a load/DG using the MLCs and power rating of load/DG. This method does not allocate the losses to the root node and therefore, results in over-recovery of total network losses, which is compensated by using suitable normalization procedure. Direct loss coefficient (DLC) method [9] allocates total losses based on the direct relationship between the node power injection and network losses. Z-bus method [11] considers network parameters for loss allocation. It can yield negative allocation to those loads and DGs, which contribute to reduce network losses due to their strategically well positioned in the system. Both MLC and DLC methods are based on the results of Newton–Raphson (NR) power flow, while Z-bus method depends on formation of Z-bus matrix in order to allocate losses. Since a distribution lines have higher R/X ratio in comparison with transmission lines, many times NR method fails to converge for load flow analysis of radial DNs. Also distribution lines have negligible shunt admittance which offers difficulty in formulation of Z-bus. Due to these facts, MLC and DLC methods cannot be applied to radial DN [4]. In the absence of shunt admittance of lines, succinct method [12] is able to calculate allocated losses. However, this method is not able to provide equitable loss allocation in terms of reactive power loads, when the ratio of reactance to resistance of a line is greater than that of reactive to real power available at its receiving node. Substitution method [9] calculates the allocated loss to a consumer/DG by taking the difference of network losses before and after connecting it to the network. In this method, the sum of allocated losses to consumers/DGs is not equal to the total network losses, and therefore additional step of normalization is required. Proportional sharing method [13,14] uses the results of power flow and linear proportional sharing principle which states that the power flow reaching a bus from the incoming lines is distributed among the outgoing lines proportionally to their corresponding power flows. However, this method does not Table 1 Different values of exponent. Load models

Values of a

CP CC CI

a=0 a=1 a=2

DPSD;u and DQSD;u allocated real and reactive power losses, t t respectively, of branch t to load connected at receiving end of branch u DPSG;u and DQStG;u allocated real and reactive power losses, t respectively, of branch t to DG connected at receiving end of branch u 0u P0u D;t and Q D;t updated value of PD,t and QD,t, respectively, when connected at receiving end of branch u 0u P0u G;t and Q G;t updated value of PG,t and QG,t, respectively, when connected at receiving end of branch u DPSD;t and DQSD;t total allocated real and reactive power losses, respectively, to load at receiving end of branch t DPSG;t and DQSG;t total allocated real and reactive power losses, respectively, to DG at receiving end of branch t

consider the interdependency of consumers and DGs, and allocates entire network losses to consumers or DGs. The issues related to loss allocation in radial DN with DG are addressed in [15]. It covers the issues such as characteristic of loads and DGs, formulation of the loss allocation problem for radial DNs with respect to transmission networks, and treatment of the root node in radial DNs. A comparison of different practical algorithms is presented in [5] for loss allocation in transmission networks. In context of deregulated environment, Savier and Das [16] presented an exact method of loss allocation based on the relation between node voltages and branch current in radial DN. They implemented their method for traditional passive DN. Later, Savier and Das [17] extended their method as in [16] for energy loss allocation. Carpaneto et al. [18] presented a branch current decomposition based loss allocation method by representing the power loss in a branch as a function of branch current and load/DG current at various nodes ahead of it in radial DN. Atanasovski and Taleski [4] proposed a power summation method for loss allocation (PSMLA) by establishing a direct relation between loss in a branch and injected real and reactive power at various nodes connected ahead of it. Further, they employed quadratic loss allocation scheme in order to deal with cross-terms. Atanasovski and Taleski [19] presented energy summation algorithm for allocation of energy loss in DN with DG. It is a statistical approach which uses daily load and generation curve. Using quadratic loss allocation scheme for cross-terms, Costa and Matos [20] presented a current based approach, which allocates entire variation of losses to DGs by using upstream looking algorithm. Brief literature review on various loss allocation techniques presented above shows that these techniques deal with DG having constant real and reactive power injection, and load having constant power model for loss allocation in DNs. Practically, loads normally encountered in low and medium voltage DNs are dependent on the node voltage. Further, real and reactive power injections by DG into network depend on technology employed and resources available at the site. However, to the best of authors’ knowledge, the issue of loss allocation in radial DNs with various DG types and voltage dependent load models has not been addressed so far. The present work proposes a new solution for the problem of loss allocation considering the effects of various DG types and voltage dependent load models. Based on power summation algorithm, the proposed method adopts a branch oriented approach for loss allocation in radial DNs. This method does not make any assumptions and approximations, and hence it is an accurate, simple, efficient, and useful methodology for loss allocation in

K.M. Jagtap, D.K. Khatod / Electrical Power and Energy Systems 75 (2016) 173–186

175

(a)

(b)

(c) Fig. 1. Various stages involved in backward sweep network reduction technique for n-node radial DN.

radial DNs. The developed method establishes a direct relation between node voltages and injected power at a node to calculate the losses in a branch and then employs a backward sweep network reduction technique to allocate the network losses to load/DG at various nodes. The network reduction technique starts from terminal branches and ends at the root node. The proposed method employs only outcomes of the power flow. This method is tested on modified 9-node and 33-node radial DNs and the obtained results are compared with those by power based PR method and PSMLA for validation of the proposed method. The rest of the paper consists of six sections. Section ‘Types of DGs’ describes different types of DG. Section ‘Load modeling’ explains various voltage dependent load models. Section ‘Proposed formulation for loss allocation’ deals with the mathematical formulation of the proposed method of loss allocation.

Section ‘Results and discussion’ presents the results obtained by proposed method on two radial DNs and comparison of obtained results with those by other methods available in the literature. Finally, Section ‘Conclusion’ draws the conclusions. Types of DGs There are several bases to classify DG which is available in [21]. In this paper, DGs are classified in the following four groups depending on their real and reactive power delivering capability: 1. Type-I DG: DG injecting only real power; 2. Type-II DG: DG injecting only reactive power;

K.M. Jagtap, D.K. Khatod / Electrical Power and Energy Systems 75 (2016) 173–186

Manitude of voltage (in p.u.)

176

CP load model

0.995

CC load model

Constant power (CP) load model: A CP load model is a static load model in which the real and reactive power of a load does not vary with the voltage magnitude. Constant current (CC) load model: A CC load model is a static load model in which the real and reactive power of a load varies linearly with the voltage magnitude. Constant impedance (CI) load model: A CI load model is a static load model in which the real and reactive power of a load varies with the square of voltage magnitude. In general, for above mentioned static load models, the relation between the real and reactive power of the load and the voltage magnitude can be expressed by the following equations:

CI load model

0.985 0.975 0.965 0.955 1

2

3

4

5

6

7

8

9

Bus number Fig. 2. Voltage profile of 9-node radial distribution network with various load models and without DG.

PD;x þ j Q D;x ¼ ðP0;x þ j Q 0:x ÞjV r;x ja

ð2Þ

3. Type-III DG: DG injecting real power and absorbing reactive power; and 4. Type-IV DG: DG injecting both real and reactive power.

The different value of exponent depends on the load models [23–26] and can be summarized in Table 1.

Type-I DG belongs to small power generation such as photovoltaic, battery, and fuel cell, which are operated at unity power factor (pf). This type of DG is connected to the network through suitable power electronics based interface. Type-II DG belongs to synchronous condenser which is operated at zero pf to provide reactive power support to the system. Type-III DG represents DG utilizing induction generators. This type of DG injects real power to the network but takes reactive power from network. Type-IV DG is a generation employing synchronous generator which is capable of delivering both real and reactive power to the network. In radial DNs, the apparent power supplied/absorbed by a DG connected at receiving end of branch x can be expressed as follows:

Proposed formulation for loss allocation

SG;x

8 PG;x > > > < jQ G;x ¼ > P G;x  j Q G;x > > : PG;x þ j Q G;x

Type I DG Type II DG

Operating at unity pf Operating at zero pf

Type III DG

Operating at leading pf

The following considerations have been made to develop the proposed formulation: 1. Starting from the root node which is always numbered as 1, and different nodes are assigned a unique integer number sequentially; 2. A branch is assigned a number equal to the one less than its receiving end node; 3. Branch from substation to root node is marked as 0 and assumed as loss-less; and 4. DGs are treated as a negative load, whereas consumers are treated as a positive load. The apparent power available at the receiving end of a branch x is the algebraic sum of all apparent power due to load and DG connected at receiving end of branches ahead of it, power losses in different branches supplied by it and reactive power injection due to half of shunt charging susceptances of different branches incident to its receiving end as:

ð1Þ

Type IV DG Operating at lagging pf

X 1 Px þ j Q x ¼ ðPD;x  PG;x Þ þ j Q D;x  Q G;x  jV r;x j2 Bm 2 m2N x ( X ðPD;n  PG;n þ PSn Þ þ

Load modeling The power consumption by majority of loads such as domestic, industrial, and commercial, encountered in radial DNs depends on voltage magnitude and frequency [22,23]. However for static analysis, the frequency deviation is insignificant, and thus only the effects of the voltage variation on the real and reactive load powers may be considered [24]. Depending upon the voltage dependency, following load models are suggested [24–26]:

n2K x

X 1 þ j Q D;n  Q G;n þ QSn  jV r;n j2 Bn 2 m2N n

!

!) ð3Þ

Substituting real and reactive power of load models from Eq. (2) in Eq. (3), the following equation can be given:

Table 2 Performance of 9-node radial distribution network with various load models and DG types. Without DG

Real load (kW) Reactive load (kVAr) Real power loss (kW) Reactive power loss (kVAr) Real power supplied by root node (kW) Reactive power supplied by root node (kVAr)

With Type-1 DG

With Type-2 DG

With Type-3 DG

With Type-4 DG

CP

CC

CI

CP

CC

CI

CP

CC

CI

CP

CC

CI

CP

CC

CI

747.6 559.2 24.04 13.92

727.06 543.55 22.5 13.06

708.44 529.33 21.11 12.28

747.6 559.2 15.05 9.09

732.76 547.88 14.28 8.63

719.05 537.47 13.55 8.2

747.6 559.2 18.82 11.13

729.25 545.21 17.69 10.47

712.52 532.45 16.64 9.87

747.6 559.2 23.77 13.32

730.49 546.17 22.67 12.69

714.8 514.22 21.66 12.11

747.6 559.2 10.01 6.39

734.87 549.52 9.48 6.06

723.13 546.59 8.99 5.73

536.68

517.66

500.43

545.66

531.6

518.61

541.87

524.67

508.99

536.93

520.93

506.24

550.71

538.5

527.25

405.11

390.32

376.88

409.94

399.08

389.1

407.9

394.57

382.41

405.71

393.31

361.94

412.64

403.29

400.69

177

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Allocated losses (in kW)

10 8 6

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 -2

1

2

3

4

5

6

7

8

9

Bus number

-4 -6

(a) Allocated losses (in kW)

10 8 6

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0

1

2

3

4

-2

5

6

7

8

9

Bus number

-4

(b) Allocated losses (in kW)

10 8 6

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 -2

1

2

3

4

5

6

7

8

9

Bus number

-4 -6

(c) Allocated losses (in kW)

10 8 6

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 -2

1

2

3

4

-4

5

6

7

8

9

Bus number

-6 -8

(d) Fig. 3. Loss allocation in 9-node radial distribution network for CP load model and with (a) Type-1 DG, (b) Type-2 DG, (c) Type-3 DG, and (d) Type-4 DG.

X   1 Px þ jQ x ¼ P0;x jV r;x ja  PG;x þ j Q 0;x jV r;x ja  Q G;x  jV r;x j2 Bm 2 m2N x X n  P 0;n jV r;n ja  PG;n þ PSn þ n2K x

X 1 Bn þ j Q 0;n jV r;n ja  Q G;n þ QSn  jV r;n j2 2 m2N n

!

In later step, reactive power generated by half of shunt charging susceptance is considered to be a part of reactive load at respective ends of the branch. Now, the real and reactive power losses in a branch x can be expressed by the following equation:

!) ð4Þ

PSx þ j QSx ¼

V s;x  V r;x ðPx þ j Q x Þ V r;x

ð5Þ

178

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10

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

Allocated losses (in kW)

8 6

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 1

-2

2

3

4

5

6

7

8

9

Bus number

-4 -6

(a)

Allocated losses (in kW)

10

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

8 6

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 1

2

3

4

-2

5

6

7

8

9

Bus number

-4

(b) 10

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

Allocated losses (in kW)

8 6

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 1

-2

2

3

4

5

6

7

8

9

Bus number

-4 -6

(c) 10

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

Allocated losses (in kW)

8 6

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 1

-2

2

3

4

-4

5

6

7

8

9

Bus number

-6 -8

(d) Fig. 4. Loss allocation in 9-node radial distribution network for CC load model and with (a) Type-1 DG, (b) Type-2 DG, (c) Type-3 DG, and (d) Type-4 DG.

Substituting V s;x ¼ cx þ j dx ; and V r;x ¼ ex þ j f x in Eq. (5) and simplifying it results the following equation:

PSx þ j QSx ¼ ðcx Px  wx Q x Þ þ j ðwx Px þ cx Q x Þ

ð6Þ

cx f x xf x where cx ¼ cx ee2x þd  1; and wx ¼ dxee2xþf 2 . þf 2 x

x

x

x

The real and reactive power losses in the branch x are decomposed and assigned to loads and DGs connected at receiving end of branches ahead of it as:

    G;x G;x PSx þ j QSx ¼ DPSD;x þ j DQSD;x x þ DPSx x þ DQSx   i X h G;n G;n þ DPSD;n þ j DQSD;n x þ DPSx x þ DQSx

ð7Þ

n2K x

Objective of any loss allocation problem states that summation of allocated losses to loads and DGs in the network must be equal to total network losses. To collect exact amount of total network losses without using normalization technique, proposed method

179

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Allocated losses (in kW)

10 8 6

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 -2

1

2

3

4

5

6

7

8

9

Bus number

-4 -6

(a)

Allocated losses (in kW)

10 8 6

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 1

2

3

4

-2

5

6

7

8

9

Bus number

-4

(b)

Allocated losses (in kW)

10 8 6

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 1

2

3

4

5

6

7

8

9

-2

Bus number -4

(c)

Allocated losses (in kW)

10 8 6

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

4 2 0 -2

1

2

3

4

-4

5

6

7

8

9

Bus number

-6 -8

(d) Fig. 5. Loss allocation in 9-node radial distribution network for CI load model and with (a) Type-1 DG, (b) Type-2 DG, (c) Type-3 DG, and (d) Type-4 DG.

employs backward sweep network reduction technique. This technique begins with terminal branch of the network. In a n-node radial DN as shown in Fig. 1(a), the load and DG connected at node n (end node) cause the power flow and thereby losses in the branch

n  1 (terminal branch). Using Eq. (3) for computation of power flow at the receiving end of branch n  1 and then substituting the obtained expression in Eq. (6), the real and reactive power losses in branch n  1 can be given as follows:

180

K.M. Jagtap, D.K. Khatod / Electrical Power and Energy Systems 75 (2016) 173–186

CP load model CC load model CI load model

Manitude of voltage (in p.u.)

1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91

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 30 31 32 33

Bus number Fig. 6. Voltage profile of 33-node radial distribution network with various load models and without DG.

Table 3 Summary of loss allocation (kW) for 9-node radial distribution network. Proposed method

Power based PR method

PSMLA

Without DG

With DG

Without DG

With DG

Without DG

With DG

Load

Load

DG

Total

Load

Load

DG

Total

Load

Load

DG

Total

CP

Type-1 Type-2 Type-3 Type-4

24.04

20.02 18.25 24.43 14.35

4.97 0.57 0.66 4.34

15.05 18.82 23.77 10.01

24.04

19.07 18.74 27.44 13.34

4.02 0.08 3.67 3.33

15.05 18.82 23.77 10.01

24.04

18.8 20.9 22.12 15.75

3.75 2.08 1.65 5.74

15.05 18.82 23.77 10.01

CC

Type-1 Type-2 Type-3 Type-4

22.5

19.04 17.08 23.4 13.65

4.76 0.61 0.73 4.17

14.28 17.69 22.67 9.48

22.5

18.19 17.59 26.21 12.71

3.91 0.1 3.54 3.23

14.28 17.69 22.67 9.48

22.5

17.99 19.74 21 15.18

3.71 2.05 1.67 5.7

14.28 17.69 22.67 9.48

CI

Type-1 Type-2 Type-3 Type-4

21.11

18.15 15.99 22.78 13.02

4.6 0.65 1.12 4.03

13.55 16.64 21.66 8.99

21.11

17.34 16.53 25.09 12.12

3.79 0.11 3.43 3.13

13.55 16.64 21.66 8.99

21.11

17.23 18.66 19.97 14.65

3.68 2.02 1.69 5.66

13.55 16.64 21.66 8.99

   PSn1 þ jQSn1 ¼ cn1 ðP D;n1  PG;n1 Þ  wn1 Q D;n1  Q G;n1    þ j wn1 ðPD;n1  PG;n1 Þ þ cn1 Q D;n1  Q G;n1 ð8Þ The losses in terminal branch n  1 should be allocated between the load and DG connected at its receiving end (i.e. node n). Hence, using Eq. (7) for branch n  1 results in the following relation:

    G;n1 G;n1 PSn1 þ j QSn1 ¼ DPSD;n1 þ j DQSD;n1 n1 þ DPSn1 n1 þ DQSn1

ð9Þ Now, rearranging different terms of RHS of Eq. (8) and equating it to RHS of Eq. (9), the allocation of losses of terminal branch n  1 to the load and DG connected at its receiving end can be expressed by the following equations:





D;n1 DPSD;n1 n1 þ j DQSn1 ¼ cn1 P D;n1  wn1 Q D;n1   þ j wn1 PD;n1 þ cn1 Q D;n1

DPSG;n1 n1

þ

j DQSG;n1 n1



ð10Þ



¼  cn1 PG;n1  wn1 Q G;n1    j wn1 PG;n1 þ cn1 Q G;n1

ð11Þ

After allocating the losses of terminal branch n  1 to load and DG at its receiving node, these allocated losses are added to respective power ratings of load and DG so as to set their updated power ratings. Then, load and DG with their updated ratings are connected at receiving end of branch n  2 and branch n  1 is eliminated as shown in Fig. 1(b). The following equations are used to compute the updated ratings of load and DG:

    0n2 D;n1 þ j Q D;n1 þ DQSD;n1 P0n2 n1 D;n1 þ jQ D;n1 ¼ P D;n1 þ DPSn1

ð12Þ

    0n2 G;n1 þ j Q G;n1 þ DQSG;n1 P0n2 G;n1 þ j Q G;n1 ¼ P G;n1 þ DPSn1 n1

ð13Þ

Now, branch n  2 becomes the terminal branch of reduced network (containing n  2 branches) as shown in Fig. 1(b) and its power losses are allocated among load PD;n2 þ j Q D;n2 connected at its receiving node and new ratings of load and DG that are 0n2 0n2 0n2 P0n2 D;n1 þ j Q D;n1 and P G;n1 þ j Q G;n1 , respectively, using a similar method as in case of branch n  1 of original network. After this, updated ratings of loads and DGs connected at receiving end of branch n  2 are computed using equations similar to as

Table 4 Difference of allocated losses (kW) to nodes 4 and 7 of 9-node radial distribution network with CP load model. Without DG Proposed method

Power based PR method

PSMLA

0.03

0

0

Type of DG

With DG Proposed method

Power based PR method

PSMLA

Type-1 Type-2 Type-3 Type-4

0.03 0.02 0.05 0.02

0 0 0 0

0 0 0 0

181

  o DPSD;n þ DPSG;n þ j DQSD;n þ DQSG;n

ð17Þ

n2K 0

Results and discussion The effectiveness of the proposed method has been tested on two test networks, i.e., 9-node and 33-node radial DNs. Both of these test networks have been considered for two conditions, i.e., without DG and with DG. In order to calculate the losses in network with DG, both test networks have been modified by adding different types of DGs at suitable locations. The real power injection by Type-1, Type-3 and Type-4 DG is taken as 25% of total network real load (obtained with CP load model) [27]. The reactive power injection/absorption by Type-2, Type-3 and Type-4 DG is taken as 25% of total network reactive load (with CP load model). Type-3 and Type-4 DGs are operated at leading and lagging pf, respectively. The size and location of DG are kept fixed in the network for different load models. To analyze the impact of DG type on allocated losses, one type of DG is connected to the network at a time. To analyze the effect of various load models on the allocated losses, both networks have also been tested with CP, CC, and CI load models. The results obtained by the proposed method have been compared with those by other methods such as power based PR method and PSMLA with different load models.

CI CC

3167.79 2241.21 82.11 60.27 2156.94 1373.41 3715 2300 89.65 65.37 2696.62 1427.1 3427.47 2104.29 207.68 148.29 2291.05 1148.47 3558.54 2193.17 224.99 160.04 2404.8 1225.6 3715 2300 247.54 175.36 2538.7 1317.11 3451.33 2124.25 125.68 86 2396.88 1230.72 3571.49 2204.02 139.37 95.12 2503.39 1301.37

CC

With Type-2 DG

CP

3715 2300 155.87 106.21 2630.41 1386.26 3484.56 2150.57 110.05 76.87 2445.75 1266.17 3590.06 2218.68 120.82 84.06 2540.51 1327.09 3715 2300 133.58 92.66 2652.66 1399.81

X n

3401.51 2083.53 155.69 103.38 2317.06 1172.62

¼

ðPSn þ j QSn Þ

3543.26 2181.02 176.61 117.51 2437.88 1255.98

n2K 0

CC

X

Table 5 Performance of 33-node radial distribution network with various load models and DG types.

PS þ j QS ¼

CI

The sum of allocated losses to all the loads and DGs in the network should be equal to total network losses in order to get fair and accurate loss allocation. Hence, the following relation must be satisfied:

With Type-1 DG

ð16Þ

CP

    00 DPSG;x þ j DQ G;x ¼ P 00 G;x  P G;x þ j Q G;x  Q G;x

CI

ð15Þ

CC

    00 DPSD;x þ j DQSD;x ¼ P00 D;x  P D;x þ j Q D;x  Q D;x

CI

Now, the total real power losses allocated to load and DG connected at the receiving end of a branch can be determined by taking the difference between corresponding updated ratings connected at root node and their actual ratings. The allocated losses to load and DG connected at receiving end of branch x can be calculated from the following equations:

3715 2300 202.67 135.14 2583.57 1357.33

With Type-4 DG

CP CI

ð14Þ

n2K 0

CP

X   ðPD;n  P G;n þ PSn Þ þ j Q D;n  Q G;n þ QSn

CC

With Type-3 DG

n2K 0

¼

CP

  o X n 00 00 00 P D;n  P00 G;n þ j Q D;n  Q G;n

Without DG

P0 þ j Q 0 ¼

Real load (kW) Reactive load (kVAr) Real power loss (kW) Reactive power loss (kVAr) Real power supplied by root node (kW) Reactive power supplied by root node (kVAr)

given in Eqs. (12) and (13). Now, loads and DGs at receiving end of branch n  2 are connected to receiving end of branch n  3 with their updated ratings and branch n  2 is eliminated. The process of allocation of losses; updating of loads and DGs power with their allocated losses; and elimination of branches are continued till root node occurred as shown in Fig 1(c). At the end of this procedure, there is no branch exists in the network, and all the loads and DGs with their updated ratings are connected at the root node as shown in Fig. 1(c). The power at the receiving end of branch 0 becomes equal to the updated ratings of all the loads and DGs connected at root node. It is also equal to the sum of all loads, DGs and branch losses in the network and can be expressed as follows:

3534.53 2191.39 75.62 55.91 2530.16 1327.95

K.M. Jagtap, D.K. Khatod / Electrical Power and Energy Systems 75 (2016) 173–186

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Allocated losses (in kW)

60

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

40

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20 0 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 30 31 32 33

-20

Bus number

-40 -60

(a)

Allocated losses (in kW)

60

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

40

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20 0 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 30 31 32 33

Bus number

-20 -40

(b) Allocated losses (in kW)

80

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

60 40

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20 0 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 30 31 32 33

-20

Bus number

-40 -60

(c)

Allocated losses (in kW)

60

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

40 20

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

0 -20

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 30 31 32 33

-40

Bus number

-60 -80 -100

(d) Fig. 7. Loss allocation in 33-node radial distribution network for CP load model and with (a) Type-1 DG, (b) Type-2 DG, (c) Type-3 DG, and (d) Type-4 DG.

9-node test radial distribution network A 11 kV, 9-node radial DN is used to observe the effects of various load models and DG types on the results of allocated losses. The line and load data of this test network are taken from [16]. Fig. 2 shows the voltage profile of 9-node network with CP, CC and CI load models without DG. It can be observed from this figure that the lowest voltage is observed at node 9 for all the load models. Hence, DG is connected at node 9. For different load models, Table 2 shows the real and reactive power of network load, network losses and power supplied by the root node without and with DG. It can be noticed from this table that the network load is

maximum with CP load model, while it is minimum with CI load model. That is why among the three load models, CP load model gives worst voltage profile, while CI load model provides best voltage profile as seen from Fig. 2. The voltage profile with CC load model lies in between those with CP and CI load models. Because of the lower load and better voltage profile, the network losses with CI load model are also less as compared to those with CP load model. Similar observation is made for supplied power from root node for different load models from Table 2 irrespective of DG type employed. With different types of DG, Figs. 3–5 show the node-wise allocated losses with CP, CC and CI load models, respectively.

K.M. Jagtap, D.K. Khatod / Electrical Power and Energy Systems 75 (2016) 173–186

Allocated losses (in kW)

60 40

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

183

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20 0 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 30 31 32 33

-20 -40

Bus number

-60

(a)

Allocated losses (in kW)

60 40

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20 0 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 30 31 32 33

-20

Bus number

-40

(b)

Allocated losses (in kW)

80 60 40

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20 0 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 30 31 32 33

-20 -40

Bus number

-60

(c)

Allocated losses (in kW)

60 40 20

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

0 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 30 31 32 33

-20 -40

Bus number

-60 -80

(d) Fig. 8. Loss allocation in 33-node radial distribution network for CC load model and with (a) Type-1 DG, (b) Type-2 DG, (c) Type-3 DG, and (d) Type-4 DG.

In these figures, the loss allocation obtained by power based PR and PSMLA methods is also plotted for the sake of comparison. Table 3 summarizes the total allocated losses of consumer and DG. Without DG integration in the network, total losses, supplied by root node, are caused by various loads and hence positive losses are allocated to all the loads by all the methods as seen from Table 3 and Figs. 3–5. This positive value of allocated loss indicates that consumers should pay for losses caused by them. With DG integration in the network, total power demand of the network remains unchanged, but network losses are reduced as in Table 2. For this case, the proposed method allocates positive losses to all the

consumers and negative losses to Type-1, Type-3 and Type-4 DGs as seen from Table 3 and Figs. 3–5. This negative value of allocated loss to DG may be viewed as a reward to DG for its contribution toward loss reduction in the network. From Table 3 and Figs. 3–5 (a), it is observed that the proposed method assigns more reward to Type-1 DG due to only real power injection by it and provides less cross subsidies to consumers as compared to power based PR and PSMLA methods. As observed from Table 3 and Figs. 3–5 (b), proposed and power based PR methods do not assign any reward to Type-2 DG as it injects only reactive power in the network. However, in this case, PSMLA method assigns reward to

184

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Allocated losses (in kW)

60 40

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20 0 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 30 31 32 33

-20 Bus number -40 -60

(a)

Allocated losses (in kW)

60 40

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20 0 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 30 31 32 33

-20

Bus number

-40

(b)

Allocated losses (in kW)

60 40

To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

20 0 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 30 31 32 33

-20 Bus number -40 -60

60 40

Allocated losses (in kW)

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

20

(c) To load without DG by proposed method To DG by proposed method To load with DG by PR method To load without DG by PSMLA

To load with DG by proposed method To load witout DG by PR method To DG by PR method To load with DG by PSMLA

0 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 30 31 32 33

-20 Bus number -40 -60 -80

(d)

Fig. 9. Loss allocation in 33-node radial distribution network for CI load model and with (a) Type-1 DG, (b) Type-2 DG, (c) Type-3 DG, and (d) Type-4 DG.

Type-2 DG and provides less cross subsidies to consumers. With Type-3 DG (injecting real power, but absorbing reactive power), power based PR method assigns more reward to DG as compared to proposed method, and PSMLA method does not provide any reward to DG as seen from Table 3 and Figs. 3–5(c). In this case, proposed method provides no cross subsidies but power based PR method provides over cross subsidies to consumers which is not acceptable in the electrical market. Now, from Table 3 and Figs. 3–5(d), it can be observed that PSMLA method provides more reward to Type-4 DG as compared to proposed and power based PR

methods. The reward assigned to DG by proposed method is moderate and lies in between those provided by power based PR and PSMLA methods. Also, with Type-4 DG, cross subsidies provided by proposed method are less and nearly equal to the PSMLA method. Now, the performance of different methods on the basis of discrimination between the consumers of same ratings at different electrical locations is also evaluated. The real power consumptions at nodes 4 and 7 are same (there is a small mismatch on reactive power consumption at these nodes), but electrical locations of

185

K.M. Jagtap, D.K. Khatod / Electrical Power and Energy Systems 75 (2016) 173–186 Table 6 Summary of loss allocation (kW) for 33-node radial distribution network. Proposed method

Power based PR method

PSMLA

Without DG

With DG

Without DG

With DG

Without DG

With DG

Load

Load

DG

Total

Load

Load

DG

Total

Load

Load

DG

Total

CP

Type-1 Type-2 Type-3 Type-4

202.67

179.15 132.74 258.8 170.38

45.57 23.13 11.26 80.73

133.58 155.87 247.54 89.65

202.67

175.23 147.71 300.36 118.02

41.65 8.16 52.82 28.37

133.58 155.87 247.54 89.65

202.67

190.49 184.75 222.85 173.03

56.91 28.88 24.69 83.38

133.58 155.87 247.54 89.65

CC

Type-1 Type-2 Type-3 Type-4

176.61

161.35 118.5 231.69 157.71

40.53 20.87 6.7 75.6

120.82 139.37 224.99 82.11

176.61

160.12 131.4 274.63 108.97

39.3 7.97 49.64 26.86

120.82 139.37 224.99 82.11

176.61

173.8 165.94 197.43 161.83

52.98 26.57 27.56 79.72

120.82 139.37 224.99 82.11

CI

Type-1 Type-2 Type-3 Type-4

155.69

146.54 102.01 211.65 146.15

36.49 23.67 3.97 70.53

110.05 125.68 207.68 75.62

155.69

147.23 117.91 254.81 101.09

37.18 7.77 47.13 25.47

110.05 125.68 207.68 75.62

155.69

159.78 150.32 177.82 152.31

49.73 24.64 29.86 76.69

110.05 125.68 207.68 75.62

Table 7 Difference of allocated losses (kW) at selected nodes of 33-node radial distribution network with CP load model. Difference of allocated losses of nodes 6 and 28 Without DG

Difference of allocated losses of nodes 9 and 10

With DG

Without DG

With DG

Proposed method

Power based PR method

PSMLA

Type of DG

Proposed method

Power based PR method

PSMLA

Proposed method

Power based PR method

PSMLA

Type of DG

Proposed method

Power based PR method

PSMLA

1.44

0

0.22

Type-1 Type-2 Type-3 Type-4

0.7 0.51 1.31 0.46

0 0 0 0

0.21 0.22 0.23 0.2

0.37

0

0.27

Type-1 Type-2 Type-3 Type-4

0.41 0.37 0.38 0.25

0 0 0 0

0.27 0.26 0.29 0.23

these nodes are different. The load connected at node 4 is electrically closer to the root node as compared to that at node 7. Table 4 shows the differences of allocated losses to consumers at nodes 4 and 7 with CP load model. From this table, it is seen that the difference of allocated losses to consumers at nodes 4 and 7 without DG by proposed method is 0.03 kW, while the same by power based PR and PSMLA methods is 0 kW. Further, with different types of DG also, the difference of allocated losses to consumers at nodes 4 and 7 by proposed method is more as compared to power based PR and PSMLA methods. This indicates that the proposed method is able to identify the electrical locations and contribution of consumers in total network losses. 33-node test radial distribution network The proposed method is also tested on a 12.66 kV, 33-node radial DN. The line and load data of this test network are taken from [28]. Fig. 6 shows the voltage profile of 33-node network with CP, CC and CI load models without DG. From this figure, it can be observed that among three load models, CI load model has better voltage profile than CP and CC load models. The voltage profile with CC load model lies in between those with CP and CI load models. DG is connected to node 33 which has poor voltage and is far away from the root node. For different load models, Table 5 shows the real and reactive power of network load, network losses and power supplied by root node without and with DG. From this table, it can be noticed that the network load is maximum with CP load model, while it is minimum with CI load model. Because of the lower load and better voltage profile, the network losses with CI load model are also less as compared to those with CP load model. Similar observation is made for supplied power from root node with different load models from Table 5 irrespective of DG type employed.

With different types of DG, Figs. 7–9 show the node-wise allocated losses with CP, CC and CI load models, respectively. In these figures, the loss allocation obtained by power based PR and PSMLA methods is also plotted for the sake of comparison. Table 6 summarizes the total allocated losses of consumer and DG. Without DG integration in the network, positive losses are allocated to all the loads by all the methods as seen from Table 6 and Figs. 7–9. With DG integration in the network, the proposed method allocates positive losses to all the consumers and negative losses to Type-1, Type-3 and Type-4 DGs as seen from Table 6 and Figs. 7–9. PSMLA method assigns more reward to Type-1 DG and provides less cross subsidies to consumers as compared to power based PR and proposed methods. With Type-1 DG, the proposed method assigns moderate reward and cross subsidies to DG and consumers, respectively. As observed from Table 6 and Figs. 7–9(b), proposed and power based PR methods do not assign any reward to Type2 DG as it injects only reactive power in the network. However, in this case, PSMLA method assigns reward to Type-2 DG and provides less cross subsidies to consumers. With Type-3 DG, the network losses are increased as seen from Table 5, still power based PR method assigns reward to DG and provides over cross subsidies to consumers. PSMLA method does not provide any reward to DG as seen from Table 6 and Figs. 7–9(c). In this case, the performance of the proposed method is moderate. Now, from Table 6 and Figs. 7–9(d), it can be observed that PSMLA method provides more reward to Type-4 DG as compared to proposed and power based PR methods. The reward assigned to DG by proposed method lies in between those provided by power based PR and PSMLA methods. Also, with Type-4 DG, cross subsidies provided by proposed method are nearly equal to the PSMLA method. Now, the real and reactive power consumptions at nodes 6 and 28 are same, but electrical locations of these nodes are different. The load connected at node 6 is electrically closer to the root node

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as compared to that at node 28. Further, the power consumptions at nodes 9 and 10 are same and these nodes are situated close to each other. Table 7 shows the differences of allocated losses to consumers at nodes 6 and 28; and at nodes 9 and 10 with CP load model. From this table, it is seen that the differences of allocated losses to consumer at nodes 6 and 28 without DG are 1.44 kW, 0 kW and 0.22 kW by proposed, power based PR and PSMLA methods, respectively. Similarly, with different types of DG, proposed method creates more difference in allocated losses at nodes 6 and 28 as compared to power based PR and PSMLA methods. Further, from Table 7, it is seen that the differences of allocated losses to consumer at nodes 9 and 10 without DG are 0.37 kW, 0 kW and 0.27 kW by proposed, power based PR and PSMLA methods, respectively. With different types of DG, proposed method is able to differentiate between consumers of same rating and located close to each other in a better way as compared to power based PR and PSMLA methods. From above mentioned results and discussion, it can be concluded that the proposed method has potential to identify the exact electrical location of consumers without DG. It is also able to differentiate between the same rating of consumers located anywhere in the network and allocates the losses based on their power consumption and contribution in the total network losses. It always provides reward to DG which injects real power to the network. Conclusion In the paper, a new method for loss allocation in radial DN with DG has been presented. Based on power summation method, the proposed method is a branch oriented approach which considers the network power flow and establishes a direct relation between real and reactive power at the receiving end of branch and its losses. In order to eliminate the additional steps of normalization, the proposed method employs backward sweep network reduction technique. In order to get fair allocation of losses, this method does not consider any assumptions and approximations; therefore, it is a simple, efficient and easy method for loss allocation in radial DNs. Further, this method does not require any additional information other than the result of power flow. While allocating losses to a participant, proposed method considers not only power injection/absorption by it but also its location, characteristic, and contribution in the total network losses and thereby sends correct economic signal. The proposed method has been tested on two test networks and the obtained results are compared with those by other existing methods to show its effectiveness. References [1] Jagtap KM, Khatod DK. Impact of different types of distributed generation on radial distribution network. In: IEEE conf. ICROIT Faridabad India; 2014. p. 473–6.

[2] Khatod DK, Pant V, Sharma J. Evolutionary programming based optimal placement of renewable distributed generators. IEEE Trans Power Syst 2013;28:683–95. [3] Gómez A, Riquelme JM, González T, Ruiz E. Fair allocation of transmission power losses. IEEE Trans Power Syst 2000;15:184–8. [4] Atanasovski M, Taleski R. Power summation method for loss allocation in radial distribution networks with DG. IEEE Trans Power Syst 2011;26:2491–9. [5] Conejo AJ, Arroyo JM, Alguacil N, Guijarro AL. Transmission loss allocation: a comparison of different practical algorithms. IEEE Trans Power Syst 2002;17:571–6. [6] Gonzalez JJ, Basagoiti P. Spanish power exchange market and information system. Design concepts, and operating experience, IEEE conf. PICA Santa Clara USA; 1999. p. 245–52. [7] Shirmohammadi D, Gribik PR. Evaluation of network capacity use for wheeling transection. IEEE Trans Power Syst 1989;4:1405–13. [8] Happ HH. Cost of wheeling methodologies. IEEE Trans Power Syst 1994;9:147–55. [9] Mutale J, Strbac G, Curcic S, Jenkins N. Allocation of losses in distribution systems with embedded generation. IEE Proc Gener Trans Distrib 2000;147:7–14. [10] Galiana FD, Conejo AJ, Kockar I. Incremental transmission loss allocation under pool dispatch. IEEE Trans Power Syst 2002;17:26–33. [11] Conejo AJ, Galiana FD, Kochar I. Z-bus loss allocation. IEEE Trans Power Syst 2001;16:105–10. [12] Fang WL, Ngan HW. Succinct method for allocation of network losses. IEE Proc Gener Trans Distrib 2002;149:171–4. [13] Bialek JW. Tracing the flow of electricity. IEE Proc Gener Trans Distrib 1996;143:313–20. [14] Bialek JW, Kattuman PA. Proportional sharing assumption in tracing methodology. IEE Proc Gener Trans Distrib 2004;151:523–6. [15] Carpaneto E, Chicco G, Akilimali JS. Characterization of the loss allocation techniques for radial systems with distributed generation. Elect Power Syst Res 2008;78:1396–406. [16] Savier JS, Das D. An exact method for loss allocation in radial distribution systems. Int J Elect Power Energy Syst 2012;36:100–6. [17] Savier JS, Das D. Energy loss allocation in radial distribution systems: a comparison of practical algorithms. IEEE Trans Power Del 2007;22:2473–80. [18] Carpaneto E, Chicco G, Akilimali JS. Branch current decomposition method for loss allocation in radial distribution systems with distributed generation. IEEE Trans Power Syst 2006;21:1170–9. [19] Atanasovski M, Taleski R. Energy summation method for loss allocation in radial distribution networks with DG. IEEE Trans Power Syst 2012;27:1433–40. [20] Costa PM, Matos MA. Loss allocation in distribution networks with embedded generation. IEEE Trans Power Syst 2004;19:384–9. [21] Hung DQ, Mithulananthan N, Bansal RC. Analytical expression for DG allocation in primary distribution network. IEEE Trans Energy Convers 2010;25:814–20. [22] Liu J, Salama MMA, Mansour RR. An efficient power flow algorithm for distribution systems with polynomial load. Int J Electr Eng Edu 2002;39:372–86. [23] El-Hawary ME, Dias LG. Correspondence incorporation of load models in load flow studies: form of model effects. IEEE Proc 1987;134:27–30. [24] Haque MH. Load flow solution of distribution systems with voltage dependent load models. Electr Power Syst Res 1996;36:151–6. [25] Eminogle U, Hocaoglu MK. A new power flow method for radial distribution system including voltage dependent load models. Electr Power Syst Res 2005;76:106–14. [26] Ribeiro JR, Lang FJ. A new aggregation method for determining composite load characteristics. IEEE Trans Power App Syst 1982;101:2869–75. [27] Ackermann T, Andersson G, Soder L. Distributed generation: a definition. Electr Power Syst Res 2001;57:195–204. [28] Baran M, Wu FF. Network reconfiguration in distribution systems for loss reduction and load balancing. IEEE Trans Power Del 1989;4:1401–7.