The optimal loss reduction of distribution feeder based on transformer rearrangement

Electrical Power and Energy Systems 23 (2001) 343±348

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The optimal loss reduction of distribution feeder based on transformer rearrangement Ming-Tong Tsay*, Shun-Yu Chan Department of Electrical Engineering, Cheng-Shiu Institute of Technology, 840 Cherng-Ching Road, Neau-Song Country, Kaohsiung 833, Taiwan, ROC Received 7 April 1998; revised 11 September 2000; accepted 10 October 2000

Abstract This paper presents an ef®cient strategy for transformer planning to reduce the system losses by means of transformer rearrangement. The load patterns of residential, commercial and industrial customers are derived by load survey and then used to calculate the transformer hourly loading. The three-phase load ¯ow is used to solve the system losses and unbalance factor according to the load patterns and transformer hourly loading. With regard to transformer rearrangement, the Dynamic Programming (DP) technique is employed to ®nd out the phase assignment of transformers. In the DP process, a backward procedure and a forward procedure are used to search for the best phase arrangement of transformers. The objective function including the cost of loss reduction and switching is formulated in this paper. To verify the ef®ciency of the proposed method, a practical feeder is selected for computer simulation. By comparison of the results between without transformer rearrangement and with transformer rearrangement, the proposed method is veri®ed to be valid and available. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic programming; Three-phase load ¯ow; Unbalance factor

1. Introduction A power system comprises many components with various compositions and the behavior of the load can usually be look upon as a function of time [1±3]. When a distribution system is built up to supply energy to the customers, the power quality and operation costs are the primary considerations. Problems arising from improper design during the planning stage would bring about unnecessary cost and system losses. For example, many utility companies used Opne-Wye/Open-Delta transformers to serve various customers. It is possible to serve both the threephase and single-phase loads by using two single-phase transformers, and it is more economical for transformer management. However, Opne-Wye/Open-Delta connections could result in severe phase unbalance problems. It also caused extra line losses and degraded system security. A few papers have pointed out this problem and developed some models for such transformers [4,5]. Thus, we would like to lay special emphasis on the reduction of system losses and unbalance factor by transformer regulation. In the Taiwan Power Company (Taipower) distribution * Corresponding author. Tel.: 1886-7-7310606, ext. 311; fax: 1886-77315367. E-mail address: [email protected] (M.-T. Tsay).

system, there exist a number of Opne-Wye/Open-Delta transformers for the sake of economic and future expansion considerations [6,7]. The frequent phase unbalance of systems is one of the main reasons causing extra line losses and unexpected ground relay tripping in the distribution system. It is found that the phase unbalance introduced could be one of the most serious problems for a practical three-phase distribution system. Since the distribution transformers may be single-phase, two-phase or three-phase, the assignment of phase types for each transformer must follow the goal of phase balance and minimization of system losses. Many researchers [8±12] employed feeder recon®guration and capacitor placement to minimize power loss and system unbalance. Nevertheless, surprisingly few studies have so far been made about the transformer arrangement [13,14]. This paper will focus on the optimal transformer rearrangement to solve the problems. Optimization by changing transformer phases for balancing load and minimizing loss is a complicated mathematical formulation. It is dif®cult to de®ne a direct correlation between the variables, such as transformer phase types, cost of loss reduction and operation constraints. In this issue, the Dynamic Programming (DP) technique with backward and forward procedures is employed to search for the best arrangement of phase types for all transformers in a feeder. With the feeder divided into several stages, the backward

0142-0615/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0142- 061 5( 00) 00082-X

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M.-T. Tsay, S.-Y. Chan / Electrical Power and Energy Systems 23 (2001) 343±348

customers served by each distribution transformer are identi®ed by ®eld investigation according to the feeder circuit diagram and load distribution diagram. The load percentage of customer i is calculated by Eq. (1) and the load of each load customer is solved by Eq. (2). Therefore, the transformer loading can be easily obtained by summing the hourly loading of each customer served by the transformer as expressed in Eq. (3) Fig. 1. Determination of customer hourly loading by the typical daily load curve.

searching procedure will ®nd out and record the paths, which meet the empirical current constraints. Once the backward search arrives at the substation, the forward searching is successively practiced. The voltage of each stage can be calculated step by step, and any path violating an empirical constraint is canceled. In this way, the system losses can be evaluated at the same time. Only ten optimal cases are stored as starting conditions for the ongoing iteration. Based on the proposed transformer arrangement data, a three-phase load ¯ow program [15] is performed on a practical Taipower distribution feeder. To execute the analysis more precisely, much data such as the length of line segments, the customers served and the location and connection of distribution transformers, is prepared by the ®eld survey. Results will suf®ce to show the impact of the proposed methodology on the reduction of system losses and unbalance. By envisaging the demonstrated tables and ®gure, etc. there is enough evidence to show that the issued methods have positive and valid effects on the reduction of distribution system losses and balance of the system. 2. Transformer hourly loading In order to get more realistic customer data with the load patterns, some linkage must be maintained between Customer Information System (CIS) [16] and the Distribution Database. The record of each customer in the CIS must be connected with a distribution transformer in the Distribution Database. In this paper, CIS combined with the load patterns of various customers and the Distribution Database has been utilized to obtain the transformer hourly loading. A systematic method was presented to calculate the hourly power loading of each distribution transformer by the ®eld survey and typical load patterns as illustrated in Fig. 1. All

Fig. 2. A three-line diagram.

PARTit ˆ

PARit 24 X PARit

…1†

iˆ1

Pit ˆ PARTit £ Pttr ˆ

n X iˆ1

PKWHi NDAY

Pit

…2† …3†

where PARit is per unit of customer i at the t-th hour, PARTit the load percentage of customer i at the t-th hour, PKWHi the monthly energy consumption of customer i in CIS, NDAY the number of sample days, Pit the load of customer i at the t-th hour and Pttr the transformer loading at the t-th hour. 3. Transformer rearrangement methodology In the Taipower distribution feeder, Open-Wye/OpenDelta and single-phase transformers are utilized to meet the requirements of either single-phase or three-phase loads. It can be clearly seen that unbalanced phase has led to the extra energy losses. Accordingly, appropriate arrangements of transformers and swapping of phase type are studied to accomplish a more rigid and ef®cient distribution system. 3.1. Backward procedure The Dynamic Programming of the backward procedure is ®rst used to compute the maximal loss reduction in interval K with state I, i.e. Fcost …K; I† ˆ Max ‰Pcost …K; I† 1 Fcost …K 1 1; J†Š {J}

…4†

Fcost …K; I† ˆ Lp p Loss 2 SWcost Loss ˆ DPloss ˆ Ploss old 2 Ploss new where Pcost …K; I† is the maximal cost in the search node during interval K with given state I, Fcost …K; I† the maximal total cost from state I in interval K to the ending interval M, {J} the set of feasible states in interval K 1 1; SWcost the cost of switching (NT$3450/per) and Lp the cost of loss (NT$4620/kw-year). Ploss old is the initial system losses including transformer core loss, transformer copper loss, and line loss. Ploss new is the system losses after transformer

M.-T. Tsay, S.-Y. Chan / Electrical Power and Energy Systems 23 (2001) 343±348

rearrangement. If the Fcost …K; I† is positive, the phase type of transformer must be swapped to reduce the total cost. 3.1.1. Current variation with transformer rearrangement Fig. 2 shows an example of a three-line diagram of a distribution feeder. It can be seen that phase c in the bus i is absent in the three-phase system. It is not a good arrangement and may lead to system unbalance. With the entire system taken into account, the effective operation can be reached by performing phase rearrangement. At bus i, the node current calculation will be described as follows: 2

Iai

3

2

…PaLi 1 jQaLi †p =Vaip

3

6 7 6 b 7 6 Ibi 7 ˆ 6 …PLi 1 jQbLi †p =Vbip 7 4 5 4 5

…5†

…PcLi 1 jQcLi †p =Vcip

Ici

where Iai ; Ibi ; Ici are the current injections at node i, Pabc Li ; Qabc Li are the power injections at node i and Vai ; Vbi ; Vci are the voltages at node i.The line current between bus i and bus j can be calculated with 2

2 3 Iak X 6 7 6 7 6 7 6 Ibi2j 7 ˆ 26 Ibj 7 1 6 Ibk 7 4 5 4 5 4 5 k[K Ici2j Icj Ick Iai2j

3

2

Iaj

3

…6†

where Iai2j ; Ibi2j ; Ici2j are the line section currents between bus i and bus j and Iak ; Ibk ; Ick are the set of line section connections to node j. If the phase of an Open-Wye/Open-Delta transformer is changed, three cases must be considered to re-calculate the line current. In the following description, the variables x, y and z are used to denote the phase a, b or c and, therefore, cover all the possible cases of phase rearrangement. They are:1. Phase xy ! xz. !p Py 1 jQy new old old Iy ˆ Iy 2 DIy ˆ Iy 2 …7† Vy Iznew ˆ Izold 1 DIz ˆ Izold 1



Py 1 jQy Vz

p

…8†

2. Phase xy ! yx. In this case, the switching cost is doubled. It will increase the total cost and is not taken into consideration in this paper. 3. Phase x ! y.   Px 1 jQx p Ixnew ˆ Ixold 2 DIx ˆ Ixold 2 …9† Vx Iynew

ˆ

Iyold

1 DIy ˆ

Iyold

1

Px 1 jQx Vy

!p …10†

The line current can be re-calculated as follows: 2 new 3 2 new 3 2 new 3 Iaj Iai2j Iak 6 new 7 6 new 7 X 6 new 7 6 Ibi2j 7 ˆ 26 Ibj 7 1 6 Ibk 7 4 5 4 5 4 5 k[K new new Ici2j Icjnew Ick

345

…11†

The new loss, inclusive of core loss, copper loss, and line loss, is calculated in each bus and used to compare with the original transformer arrangement. An empirical rule based on the branch current for path searching can be effectively used to reduce the number of selected paths. The justi®ed rules for branch currents are described as follows: Ia 1 Ib 1 Ic ˆ In # A…amp†

…12†

Imax ˆ max{uIa u; uIb u; uIc u} # B…amp† where A and B are constants dependent upon the used line types. If the currents can meet Eq. (12), the searching action is kept on from terminal bus to substation and the selected path is recorded. On the contrary, any combinations of the transformer, which violate the constraints, are kicked out without further backward searching. After the backward procedure, the corresponding branch currents based on Eq. (11) are memorized. 3.2. Forward procedure The forward procedure is performed to ®nd out the voltages of each bus. The searching procedure gets started from the substation and arrives at the feeder terminus. With reference to branch impedance data, the bus voltage of the next stage can be conducted as: 2 3 2 3 2 32 3 Vja Iai2j Via Zaa Zab Zac 6 7 6 7 6 76 7 6 Vjb 7 ˆ 6 Vib 7 2 6 Zba Zbb Zbc 76 Ibi2j 7 …13† 4 5 4 5 4 54 5 Vjc Ici2j Vic Zca Zcb Zcc where the Z-matrix is the line impedance between bus i and bus j. When voltages of each bus are evaluated, some constraints must be followed to reduce the number of transformer combinations. The voltage unbalance (VU) is de®ned as follows: 3…Vmax 2 Vmin † £ 100% Va 1 Vb 1 Vc

…14†

0:95 p:u: , Va ; Vb ; Vc , 1:05 p:u:

…15†

%VU ˆ

where Va ; Vb ; Vc are the phase voltages of a, b and c and Vmax, Vmin are the maximal and minimal phase voltages. Any combination, which violates the constraints, is no longer selected and better combination of transformer arrangements can be obtained. Once the backward and forward procedures are accomplished in one iteration, the system losses can be computed and the cost of all possible paths are summed up. For saving of computer memory, ten of the best combinations with

346

M.-T. Tsay, S.-Y. Chan / Electrical Power and Energy Systems 23 (2001) 343±348

Fig. 3. The sample distribution feeder.

minimal losses are recorded in the ®rst iteration. In the same way, the best ten combinations are updated as the calculation is going on. Of course, the enumerations must be iterated step by step until all voltage deviations converge to a speci®ed value. The voltage ¯uctuations are less than a speci®ed tolerance. 4. Practical feeder simulation Fig. 3 shows the one-line diagram of a feeder on the Table 1 Summary of the distribution transformer Transformer type

The no. of transformer

Unit capacity (kVA)

Three-phase Open-Wye/Open-Delta Phase A Phase B Phase C Total

4 52 11 9 7 83

1600.0 3919.5 1100.0 1492.0 704.5 8816

Taipower system for the urban area of Kaohsiung City. The feeder is selected for demonstrating the proposed method. The load is served including residential, commercial and small-industrial customers. There are about 2982 different customers connected with the feeder. The total capacity of the connected transformers is 8816 kVA and a summary of these transformers is listed in Table 1. As this paper is aimed at the effects of the proposed method, the two cases studied will be examined through the simulation of a three-phase load ¯ow program. One case is considered without rearrangement and another case is with rearrangement. By employing the calculated results from the proposed cases, the system losses and unbalance factors can be clearly envisaged. Fig. 4 shows the ¯owchart of the proposed method. Table 2 demonstrates the daily system unbalance factor. The unbalance factor between 8 AM and 5 PM is much smaller than that of the other periods. If the unbalance of system is very critical, the tripping actions may occur in the distribution system. In Table 2, it is clearly observed that the conditions with rearrangement display lower unbalance

M.-T. Tsay, S.-Y. Chan / Electrical Power and Energy Systems 23 (2001) 343±348

347

Fig. 5. The daily system losses pro®les.

Fig. 4. The ¯owchart of the proposed method.

Table 2 The daily system unbalance factor Time (h)

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

Unbalance factor(%) Original arrangement

DP arrangement

15.65435 14.89148 14.68865 14.56525 14.65468 13.70638 13.09517 12.37866 9.62982 9.34418 9.49249 9.61949 10.62612 9.83533 9.77247 9.93774 10.73156 13.25853 15.20534 15.70080 15.63437 15.68941 15.93553 15.85878

2.28498 2.13390 2.08357 2.04482 2.02967 1.88280 1.76758 1.49529 0.65093 0.49220 0.38112 0.40438 0.53879 0.52204 0.51330 0.49536 0.66855 1.40188 1.92539 2.05530 2.02735 2.08907 2.32545 2.35128

factor than those without rearrangement. Fig. 5 illustrates the pro®les of daily system losses of two cases. From Fig. 5, the pro®les with rearrangement still sound better performance. The loss reduction at around 3 PM is manifest, thus leading to the reduction of the system peak. Table 3 summarizes the maximal unbalance factors and daily losses of the two studied cases. As expected, it can hardly be denied that the case with rearrangement has achieved signi®cant curtailment in daily loss and kept better system balance. During the searching process, the number of phase changed is 10 and the cost reduction is equal to 145495.2NT$/year (38.96kw £ 4620NT$/kw-year 2 10 £ 3450NT$/per-change). Table 4 is a comparison of the phase load before and after transformer arrangement. The Open-Wye/Open-Delta transformers are looked upon as two single-phase transformers. From Table 4, the load of each phase is better balanced after the transformer arrangement. Table 5 shows the maximal ground current …In † and the minimal bus voltage. The current In is reduced and the voltage quality is also effectively promoted in the proposed process.

5. Conclusion and discussion This paper proposes a searching algorithm based on the DP method to ®nd out the optimal phase assignment for the overall transformers in a distribution feeder. The maximal reduction cost including the cost of loss reduction and the cost of switching is formulated as the objective function. The DP process swaps the phase type of transformers to accomplish the appropriate arrangement of transformers. According to the optimal phase assignment of the transformer and an actual Taipower distribution feeder, the system Table 3 Summary of the critical value

Without rearrangement With rearrangement Reduction

Max. unbalance factor(%)

Daily loss (kW)

15.85878 2.35128 13.5075

1202.150 1163.190 38.96

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M.-T. Tsay, S.-Y. Chan / Electrical Power and Energy Systems 23 (2001) 343±348

Table 4 Load distributions pre- and post- rearrangement

Original status Rearrangement status Deviation

Phase-A load (kW)

Phase-B load (kW)

Phase-C load (kW)

3-Phase load (kW)

707 604 2 103

689 599 2 90

397 590 1 193

1673 1673 0

Table 5 The maximal In and the minimal bus voltage

Original rearrangement DP arrangement

Max. In (A)

Min. bus voltage (p.u.)

47.8 10.5

0.951 0.984

losses and unbalance factor are solved by the three-phase load ¯ow analysis. By comparing the computer simulation, there is suf®cient evidence to show that the transformer rearrangement method is both effective and ef®cient.

[7] [8] [9] [10] [11]

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