Derivation Of Novel Estimation Formulas For Distribution Feeder Reconfiguration

Derivation of Novel Estimation Formulas for Distribution Feeder Reconfiguration Jen-Hao Teng* Yi-Hwa Liu+ Chia-Yen Chen**

*Department of Electrical Engineering I-Shou University Kaohsiung, Taiwan

+Department of Electrical Engineering

Chi-Fa Chen*

** Department

of Computer Science, The University of Auckland, New Zealand

National Taiwan University of Science and Technology, Taipei, Taiwan

Abstract: Some efficient and accurate approximation formulas for the variations of line loss, bus voltages and branch currents caused by a switch operation are derived in this paper and then integrated into feeder reconfiguration problems. Two matrices developed from the topological structures of distribution systems are used to analyze the relationship between bus voltages, bus current injections and branch currents. Therefore, the line loss, bus voltages and branch current caused by a switch operation can be approximated accurately and efficiently. The proposed formulas are then integrated into branch exchange algorithm and used to solve minimum loss reconfiguration. Test results compare the efficiency and accuracy of the proposed formulas.

feeder configuration thus becomes a complex decision-making process for dispatchers. Extensive numerical computation is often required if the conventional load flow techniques have to be used. Therefore, approximated formulas have great value to improve solution efficiency. This paper tries to derive some novel approximation formulas for the variations of line loss, bus voltages and branch currents, caused by a switching operation, and then integrate those derived formulas into feeder reconfiguration problems. Test results demonstrate the performance of the proposed algorithm.

Keywords: Distribution Automation, Optimum Switching, Feeder Reconfiguration, Branch Exchange

II. PROBLEM FORMULATION

I. INTRODUCTION

D istribution Automation (DA) is one of the most important tools to improve the reliability and efficiency in distribution system planning and operation. Many applications, such as network optimization, var planning, optimum switching/feeder reconfiguration, state estimation, short circuit analysis and so forth, are necessary to effectively construct DA [1]. Among those applications, an accurate and efficient feeder reconfiguration program is very important for off-line planning and real-time operation of DA. Many algorithms dealing with feeder reconfigurations have been proposed [2-11]. Aoki et al. [2] used a quasi-quadratic nonlinear programming technique to minimize power loss. Civanlar et al. [3] and Baran et al. [4] proposed some approximated power flow method for loss reduction resulting from a switch operation on distribution networks. Unbalanced three-phase distribution system was considered in the solution process proposed by [5]. Service restoration was integrated into the feeder reconfiguration problem as presented in [6]. A two-stage solution based on a modified simulated annealing technique for general multi-objective optimization was presented in [7]. The solutions based on expert system, fuzzy set technique and combinatorial optimization solution techniques such as genetic algorithm and simulated annealing were also developed and integrated into the optimal switching

problems [8-11]. Feeder reconfiguration can be used for off-line planning and real-time operation. For real-time operation, the primary concern for algorithm development should be the computation efficiency. By contrast, solution accuracy should be the primary concern for off-line planning. Therefore, a good algorithm should achieve both efficiency and accuracy. Besides, there are numerous of switches in distribution systems and the number of possible switching operations is tremendous; 1-4244-0549-1/06/$20.00 (2006 IEEE.

Feeder reconfiguration can be used to minimize line loss or to relieve overloads for distribution networks. Branch currents and bus voltage magnitudes are the operation constraints that commonly need to be taken into account. Meanwhile, the network must remain radial with all loads connected. Therefore, the formulations of minimum loss reconfiguration for loss reduction can be expressed as NF

PIoss =Z Ploss

min

(1)

i=l

where NF is the number of feeders in this network. P1s is the loss of feeder i and can be further rewritten as

Ss

(2)

R'

B

/11

where NlJ is the number of branch of feeder i. BJ and RJ are the current and line resistance for branch I of feeder i, respectively. The operation constraints for branch currents and bus voltages can be written as BJ

<

Birmax

i

=

VZ Vi,rmax <

NF, i

I

=

Nl NF,

=

J

(3) NB

B

where NB is the number of bus of feeder i.

voltage for bus j of feeder i.

Bi,max

VJ

is the

is the maximum

alowblcrrntfob h Ior

i

Vjiimax

are the minimum and maximum allowable voltages for busj of feeder i, respectively. Relationship Matrices for Distribution Networks

The proposed method is developed based on the BIBC (Bus-Injection to Branch-Current Matrix, BIBC) and BCBV (Branch-Current to Bus-Voltage Matrix, BCBV) matrices. These two matrices provide novel viewpoints in observing the relationship between bus voltages, branch currents and bus current injections. The detailed derivation of these two matrices can be found in [12]. In this paper, only the building algorithms of these two matrices are shown. The relationship of bus current injections and branch currents can be expressed as [B]= [BIBC] [I]

(5)

where V and VO are the vectors of bus voltages and no-load bus voltages, respectively. Equation (5) can also be rewritten as

[AV]= [BCBV] [B]

-

Procedure 5) If a line section (Bk) is located between bus i and bus j, copy the row of the i-th bus of BCBV matrix to the row of the j-th bus and fill the line impedance (Zi) to the position of the j-th bus row and the k-th column. Procedure 6) Repeat Procedure 5) until all line sections are included in the BCBV matrix. The BIBC matrix represents the relationship between bus current injections and branch currents. The corresponding variations of branch currents, caused by a switch operation, can be calculated directly by using the -

-

Fig. 1: Switching operation of a two-feeder network

Fig. 1 is an illustration for a switching operation of a two-feeder network. In Fig. 1, the load point n can be transferred from feeder j to feeder i by the operation of closing switch si and opening switch sj. If the load at the point n is Sn the line loss and bus voltage variations can be derived. First, based on the load flow solution, the equivalent current injection variation at load point m can be expressed as

(7)

-( Vi )

Ai m

m

Substituting (7) into (4), the variations of the branch currents caused by the switch operation can be expressed as 0

(6)

where [AV]= [VO]-[V]. The constant BCBV matrix has non-zero entries consisted of line impedance values. The building algorithm for BCBV matrix can be developed as follows: Procedure 4) For a distribution feeder with m branch sections and n buses, the dimension of BCBV matrix is

(n-I)xm.

III. FORMULA DERIVATION

(4)

where B and I are the vectors of branch currents and bus current injections, respectively. The constant BIBC matrix is an upper triangular matrix and has non-zero entries of +1 only. The building algorithm for BIBC matrix can be developed as follows: Procedure 1) - For a distribution feeder with m branch sections and n buses, the dimension of BIBC matrix is mx(n-1). Procedure 2) - If a line section (Bk) is located between bus i and bus j, copy the column of the i-th bus of BIBC matrix to the column of the j-th bus and fill a +1 to the position of the k-th row and the j-th bus column. Procedure 3) - Repeat Procedure 2) until all line sections are included in the BIBC matrix. The relationship between branch currents and bus voltages can be written as

[Vo ]- [V] = [BCBV] [B]

BIBC matrix. The BCBV matrix represents the relationship between branch currents and bus voltages. The corresponding variations of bus voltages, caused by a switch operation, can be calculated directly by using the BCB V matrix. These two matrices are very useful for the derivation of the proposed estimation formulas.

[BIBC']

(8)

Al~ 0

where LBIBCi] is the BIBC matrix of feeder i. Eq. (8) can be rewritten as Bi] = BIBC'4

(9)

where LBIBC'm] is the column vector of LBIBCi] corresponding to bus m. Therefore, the line loss of feeder i after the switch operation can be estimated by N; Pioss

/11

2

B+ AB

AI'

R±+

2

R2i

(10)

where Ri is the resistance of branch si. The bus voltage variations caused by the switch operation can be expressed as

LAV'] = LBCBV] LAB ]

(1 1)

The voltage at bus n after the switch operation can be estimated by n-

m

(12)

si

where Zsi is the impedance of line si. Similarly, the current injection variation for bus n of feederj can be estimated by (13) n

Thus, the line loss of feeder j after the switch operation can be estimated by Nl s

=

3

2

Bi A +/

(14)

RJ

1=1

The line loss before and after the switch operation can be expressed as N'

Z

1 los

B,'±BRI'

2

(15)

solution procedure implemented can be summarized as: a) Using (4) to (6) to calculate the bus voltages, bus current injections and line loss for each feeder before feeder reconfiguration. b) Executing branch exchange by closing a normal open switch and opening a closed switch, and then using (7)-(17) to estimate the line loss and bus voltage variations. c) If AP,,,, is less than 0, then exchanges these two switches and update BIBC and BCBV accordingly. Return to b). d) If no switch exchange can further reduce the line loss then stop the procedure and report the results. IV. TEST RESULTS

Fig. 2 is a two-feeder test system which has two radial feeders served from two substations. A sectionalizing switch is assumed on every section of the feeder shown in Fig. 2 [3]. The accuracy of the proposed formulas is compared to the load flow solution by successively opening of each located switch in Fig. 2 and then calculated the line loss and bus voltage variations by the proposed estimation formulas and the load flow solution. Fig. 3 shows the power loss versus open switch location. In Fig. 3, the LF(kW) and Estimated(kW) mean the solutions obtained by load flow and the proposed method, respectively. The error(%) is the percentage error between the load flow solution and the proposed method, it can be calculated by error(%)=

|LF - Estimated *100

± AI, Rj F__er A

(16)

(18) F~eeder-B

wXA

A.j

/11

The line loss variation caused by the switch operation is AP loss

-_

pafter loss

pbefore loss

(17)

Obviously, the proposed formulations can be used to estimate the variations of line loss, bus voltages and branch currents caused by switch operations effectively. For minimum loss problem, if APl%ss is less than 0, then the switch should be operated and BIBC and BCBV matrix can be updated accordingly. The feeder reconfiguration problem as formulated in Section 11 is a combinatorial and nonlinear optimization problem. This kind of optimization can be solved by artificial intelligent methods such as genetic algorithm, ant colony system or particle swarm etc. However, a straightforward and effective solution method such as branch exchange algorithm [4] is more

suitable to be used to demonstrate the accuracy and efficiency of the proposed formulas. Therefore, the branch exchange algorithm is used in this paper. The

I

4

'0~~ Fig. 2: Two-feeder test system [3]

I I

It can be seen that the error(%) for loss is negligible when the open switches are close to the tie switch and increases as the amount of transferred load increases. The maximum error(%) for loss is almost 10% in this test system; however, the amount of load transferred has exceeded 9 MW in this situation. Fig. 4 is the bus voltage profile while all load points of feeder B transferred to feeder A. From Fig. 4, it can be seen that the maximum error(%) for voltage magnitude is less than 10%. Figs. 3 and 4 mean that the proposed method has high accuracy. Meanwhile, the system status caused by switch operation can be efficiently calculated and a systematic procedure can be used to update the network configuration.

The proposed method is also used to solve the minimum loss reconfiguration for the network shown in Fig. 5. Table 1 show the line loss reduction after feeder

reconfiguration. The results obtained by the proposed estimated formulas and LF program are also compared. From Table 1, it can be seen that the proposed formulas are accurate. The line loss is 9.06% less than the original network. The switch operations are the same as those obtained in [3, 7]. 1400

12

-&- LF(kW)

1200

10

1000

8,

W 800 V

0

600 400 200 0

6

;.

4

()

2 4

3

2 1 1' 2' Open Section

3'

4'

1.02

1.5 1.3 1.1 0.9 0.7 0 0.5 ) 0.3 0.1 -0.1

0.98

0.96
0.92 0.9 0.88 A 5 4 3

2

1 1' 2' 3' 4' 5' Bus

Fig. 4: Bus voltage profile while all load points of feeder B transferred to feeder A

Feed mI

s16

15.

Jer4l

Feeder- Il

l

~~~~S22l

r 20t.

@e~~~~~~ I

i,.1 s26

s23

VI. ACKNOWLEDGEMENTS

This work was sponsored by National Science Council, Taiwan, under research grant NSC 92-2213-E-2 14-050. VII. REFERENCES [1] [2]

[3] [4]

[5]

[6]

[8]

s2I [9]

Fig. 5: Three-Feeder Sample System

V. CONCLUSIONS AND DISCUSSIONS

This paper proposed some efficient and accurate approximation formulas for the variations of line loss, bus voltages and branch currents caused by a switching operation and then integrated those formulas into a feeder reconfiguration problem. Test results demonstrated the performance of the proposed formulas. For large-scale distribution systems, the branch exchange method can only find the local optimum; therefore, artificial intelligent based methods are good choices to

Loss Reduction

Estimated 91.77kW 360.20kW 20.21kW LF 91.78kW 360.03kW Error (%) 0.0111% 0.047% -Estimated 318.52kW 56.55kW LF (S21, S17) 317.08kW 56.89kW 27.03kW Error (%) 0.454% 0.598%

[7]

s17

Table 1: Loss Reduction Comparison I III II Feeder

(S15, S19)

5'

F'ig. 3: Power loss versus open-switch location

wDsl

Switch Pair

0 5

>

improve the results. Besides, after the network topology changes caused by a switch operation, the BIBC and BCBV matrices could be updated more efficiently and technically. The application of the proposed formulas to artificial intelligent based methods and the modification of BIBC and BCBV matrices caused by switch operation will be derived in the future research.

[10]

[11]

[12]

"IEEE tutorial course on distribution automation," IEEE Power Engineering Society, 88 EH0280-8 PWR. Akoi, K.; and Ichimori, M; and Kanezashi, M.; "Normal state optimal load allocation in distribution system," IEEE Transactions on Power Delivery, No. 1, 1987, pp. 147-155 Civanlar, S.; Grainger, J.J.; Yin, H.; Lee, S.S.H., "Distribution feeder reconfiguration for loss reduction," IEEE Transactions on Power Delivery, Vol. 3 (3), pp. 1217 -1223, July 1988. Baran, M.E.; Wu, F.F., "Network reconfiguration in distribution systems for loss reduction and load balancing," IEEE Transactions on Power Delivery, Vol. 4 (2), pp. 1401 -1407, April 1989. Chiang, H.-D.; Jean-Jumeau, R.; "Optimal network reconfigurations in distribution systems. I. A new formulation and a solution methodology," IEEE Transactions on Power Delivery, Vol.5 (4), Jul 1990, pp. 1902 -1909 Shirmohammadi, D.; "Service restoration in distribution networks via network reconfiguration," IEEE Transactions on Power Delivery, Vol. 7 (2), Apr 1992, pp. 952 -958 Whei-Min Lin; Hong-Chan Chin, "A new approach for distribution feeder reconfiguration for loss reduction and service restoration," IEEE Transactions on Power Delivery, Vol. 13 (3), pp. 870 -875July 1998. Ganapathy, V.; Jasmon, G.B.; Kashem, M.A.; "Network reconfiguration for load balancing in distribution networks," IEE Proceedings-Generation, Transmission and Distribution, Vol. 146 (6), Nov 1999, pp. 563 -567 Liu, C.C.; Lee, S.J.; and Venkata, S.S., "Loss minimization for distribution feeders: optimality and algorithms," IEEE Transaction on Power Delivery, Vol. PWRD-3, 1989, pp. 1281-1289 Darling, G.R.; Hsiao-Dong Chiang; Jin-Cheng Wang; "An efficient algorithm for real-time network reconfiguration in large scale unbalanced distribution systems," IEEE Transactions on Power Systems, Vol. 11 (1), Feb 1996, pp. 511 -517 Jeon, Y.-J.; Kim, J.-C.; Kim, J.-O.; Lee, K.Y.; Shin, J.-R.; "An efficient simulated annealing algorithm for network reconfiguration in large-scale distribution systems," IEEE Transactions on Power Delivery, Vol. 17 (4), Oct 2002, pp. 1070 -1078 Teng, J.H., "A direct approach for distribution system load flow solutions," IEEE Trans. on Power Delivery, Vol. 18, Issue: 3, July 2003, pp. 882 -887