A New Fuzzy Reasoning Approach For Load Balancing In Distribution System

1426

IEEE TI-tions

on Power Systems, Vol. 10. No. 3, August 1995

A NEW FUZZY REASONING APPROACH FOR LOAD BALANCING IN DISTRIBUTION SYSTEM B. Naga Raj

K.S.Prakasa Rao, Senior Member, IEEE. Department of Electrical Engineering Indian Institute of Technology, Delhi Hauz Khas, NEW DELHI - 110 016. INDIA

-

Abstract Transformer and feedef load balancing in a distribution system reduces the risk of overloads due to load changes. The possible out of service area fobwing the occurrenceof a fault is also reduced. However, keeping in view the l i e expectancy of the switches, it is desirable that the number of switching operations is kept to a minimum. A new algorithm for load balancing based on fuzzy set decision theory is presented in this paper. The decision regarding switching operations is amved at by considering transformer and feeder load balancing together. The proposed method is illustrated through an example.

1. INTRODUCTION Primary distribution systems are usually operated in a radial configuration, with each load-point being supplied by one end only one transformer. It is always desirable to operate the system with the loads on the transformers and feeders kept balanced in order to reduce overloading and the possible out-of-service area. Load balancing also helps in optimal utilization of transformers and feeders so that investments for capacity enhancement could be deferred.

In normal operation, load balancing of a distribution system is achieved by reconfiguring the feeders thereby redistributing the loads among feeders and transformers. Some loads can be transferred from heavily loaded feedersltransformers to relatively lightly loaded neighbouring feedersltransformers. In this way, by changing the status of openlclosed switches on distribution feeders,the loads on the system get evenly distributed among the various feedersltransformers. Essentially, load balancing is a combinatorial optimization problem involving a decision making regarding the position of all the sectionalizing switches in a distribution system. Usually there are a large number of sectionalizing switches in a typical distribution

95 WM 126-3 PWRS A paper recommended and approved by t h e IEEE Power Syatem Engineering Committee o f t h e IEEE Power Engineering S o c i e t y f o r p r e s e n t a t i o n a t t h e 1995 IEEE/PES Winter Meeting, January 29, t o February 2, 1995, New York, NY. Manuscript submitted J u l y 28, 1994; made a v a i l a b l e f o r p r i n t i n g January 18, 1995.

system and hence the possible switching options are also extremely large. Hence it is very difficult to obtain an optimal solution to this problem in a short time. A number of heuristic approaches have been proposed in the past [1-71 to obtain a near optimal solution to the above problem in a short time. A simple search technique for service restoration and load balancing was proposed by Castro et al. (11 considering the data base and implementation requirements given by the operators for on-line distribution automation application. In the method proposed by Aoki et al. [2], load transfer is camed out initially for a pair of transformers which have the highest and the least load indices (load to capacity ratio). By appropriate switching operations, the load indices of other transformers are equalised as closely as possible. Load balancing for feeders is also performed in a similar way, through open loop switches. Baran and Wu [3] proposed a method, by which a gradual reduction of system load index is achieved through a search process. A heuristic method for load balancing was proposed by Hsu et a1.[4]. The method is applicable to bwh constant as well aswhanging load conditions. Chen and Cho [S] evaluated the optimal switching operations based on the hourly load patterns. The critical switches are identified by investigating the optimal switching patterns. Expert systems have also been used for load balancing [6-71. Most of the above approaches consider transformer and feeder load balancing independent of each other. Loads on any transformer are non-homogenous in nature. That is, a lightly loaded transformer might have some lightly loaded feedersllaterals and some heavily loaded feedersllaterals connected to it. Similarly, a heavily loaded transformer might have some heavily loaded feedersllaterals and some lightly loaded feedersllaterals connected to it. In any load transfer from a heavily loaded transformer (source transformer) to a lightly ,loaded transformer (sink transformer), preference should be given to load transfer from a heavily loaded feederllateral on the source transformer. to lightly loaded feederllateral on the sink transformer. System operators do not usually consider the loading level of transformers, main feeders and lateral feeders as equally significant. Overloading of a transformer necessitates a switching operation more urgently than the overloading of a main feeder. Likewise, alleviation of overload on a main feeder is more critical than that on a lateral feeder. Load balancing is thus a multiple objective decision making problem and a compromise is required between the number of switching operations and the degree of balancing achieved in the process.

0885-8950195/$04.00 0 1995 IEEE

1427 An operator's preference in finding a compromise solution is required in such an environment where different options satisfy the various requirements to varying degrees. Classical approaches do not have a mechanism to incorporate the vague or "fuzzy" preference of the operator in obtaining an optimal solution in the presence of such multiple objectives.

In the proposed method, the degree of satisfaction of various components (such as transformers, feeders and laterals) is considered along with the desirability of minimum switching options in a fuzzy set theoretic framework during the load balancing. The effectiveness of the proposed method for load balancing is demonstrated on a typical distribution system which consists of four transformers, six main feeders, seventy eight sections and a hundred and six sectionahzing switches.

11. FORMULATION OF THE PROBLEM

11. T,

Sink transformer.

12. ns

Number of switching operations.

Ideal Balanced Distribution Svstem: An ideal balanced distribution system is that system in which every transformer and feeder is loaded to the same extent so that the load indices of all components are identically equal to the system load index. It widely acknowledged that due to the discrete nature of system loads, an ideally balanced system state can seldom be attained in practice. Load Balancing: The objective of reconfiguration for load balancing is to identify a proper set of switches that should b e closed or opened such that appropriate load transfer among transformers and feeders results in a practical balanced system which is as-close to the ideal balanced system as possible.

The following assumptions [4] have been made in this work: 111. SYSTEM MODELLING IN FUZZY FRAMEWORK 1. All feeder section loads are known. 2. The loads are 3-phase and are balanced. 3. Loads are modelled as constant current sinks.

Applicability of fuzzy set theory to different powel: system problems is being investigated widely over the past several years. A representative list of published work can be found in [8-131.

NOMENCLATURE: 1.

-

2'. p

3. CAPTG), CAPF(i) and CAPLF (k) :

Difference operator. Membership function. Rated capacities (in A) of transformer j , feeder i and lateral feeder k respectively.

4. LOADTG), LOADF(i)

and LOADLF(k): 5. SLI

6. TLlG), FLI(i) and LFLI(k)

Loads incident (in A) on transformer j, feeder i and lateral feeder k. System Load Index which is obtained as the ratio of the total system load and the sum of capacities of the system transformers.

Optimization in Fuzzy Environment 1141: In fuzzy domain, each objective is associated with a membership function. The membership function indicates the degree of satisfaction of the objective. In the crisp domain, either the objective is satisfied or it is violated, implying membership values of unity and zero respectively. On the contrary, Fuzzy sets entertain varying degrees of membership function values from zero to unity. Thus fuzzy set theory is an extension of standard set theory[8]. When there are multiple objectives to be satisfied simultaneously, a compromise has to be made to get the best solution. One solution methodology for multi-objective optimization in fuzzy framework is based on Max-min principle which is described as follow:

- For each option considered, the degrees of satisfaction of all the different objectives are evaluated.

- The degree of overall satisfaction for this option is the minimum of all the above degrees of satisfaction.

Load indices of transformer j , feeder i and lateral feeder k, given as load on the component divided by the rating of the component.

- The optimal solution for the system is the maximum of all such degrees of satisfaction. This has been proved by Zimmermann [14].

7. ILTTQ)

Ideal load transfer (in A) that would make the transformer load index equal to the system load index, and i s given by CAPTG) * SLI LOADTG)

-

8. ILTF(i)

Ideal load transfer (in A) that would make the feeder load index equal to the system load index, and is given by CAPF(i) * SLI LOADF(i)

-

9. 1

Amount of load transferred by a switching operation.

10. To

Source transformer.

I

In the proposed method for load balancing, the terms DOSF), D O S O , DOS(LF) and DOS@) indicate the degree of satisfaction for load balancing of Transformer, Feeder, Lateral Feeder and the number of switching operations respectively. A brief explanation of the above terms will be in order:

i) Degree of Satisfaction for Transformers, DOSQ: Consider Fig.1 which is a part of a large distribution system with SLI = 0.5.

1428

sw

TO

If--,

;

TS

n

f3

5

l-

5 Rg.1 Example to Illustrate

f2

DOS (1)

Fig.3 Example to Illustrate DOS(F)

1

1

L o a d Transfer91 ( A )

Fig.2 Membership Function IJ DOS(T) for Transformers

L

Load Transfer, I ( A )

Fig.4 Membership Function ~ D O S ( FFor ) Feeders

Assume CAPT(To) = 1 W A LOADT(T,J = 700A CAPT(TJ = lo00 A LOADT(TJ = 400 A Then ILT(T,) = 700 ILT(TJ = 400

- (lo00 * 0.5) = 200 A - (lo00 * 0.5) = 100 A

Define a ‘Target Load Transfer for Transformers’, TLTT as

Assume SLI = 0.5 CAPT(T0) = 1000 A LOADT(To) = 675 A CAPT(TJ = lo00 A LOADT(T,) = 400 A CAPF(fl) = 500 A CAPF(f2) = 500 A CAPF(f3) = lo00 A

LOADF(fl) = 400 A LOADF(f2) = 275 A LOADF(f3) = 400 A

TLTT = Min ( ILT(TiJ, ILT(T,) ) From the above system, a balanced feeder loading would be obtained if the ideal loading of transformer To is equitably distributed among feeders fl and f2, even while performing a load transfer from Toto T..

For this case, TLTT = Min (200,100) = 100 A This implies that one should attempt to transfer the minimum load that would take one of the two transformers to a TLI = SLI. Here, the target for load transfer is 100 A. However, a load transfer of exactly 100 A may not be available for switching. In order to quantify the degree of satisfaction for various switching options, a fuzzy membership function for DOS(T) is formulated as shown in Fig.2. A load transfer of 1 = TLTT is given a membership value of unity. The larger the deviation from the target, the lesser is the degree of satisfaction. For all load transfers greater than twice the TLTT, the membership function is zero, which is completely undesirable. This can be represented as PDOql.)

= 1/TLTT

= 2-UTLTT

for I
* TLTT

For example, a load transfer of 80 A would have a membership value of 0.8 while a transfer of 110 A would have a value of 0.9. Thus, a transfer of 110 A is more desirable than a transfer of 80 A when the TLTT is 100 A.

ii) Degree of Satisfaction for Feeder, DOS@) : Consider Fig.3 which is a part of a large distribution.

Define the Target Load Transfer for a Feeder ‘i’ as Rating of the feeder TLTF(i)=LOADF(i) -

(Ideal Transformer (Sum of the ratings of Load.) the feeders connected to the same transformer)

*

To quantify the degree of satisfaction for switching operations on feeders, a membership function pDWF)is defined as shown in Fig.4 Here also, as in the case of transformers, a load transfer of 1 = TLTF(i) is given a degree of satisfaction of unity. The greater the divergence from this target, the lesser is the degree of satisfaction. For 1 > 2 * TLTF(i), the value is zero. This can be expressed as

PrnB

= UTLTF(i) = 2-UTLTF(i)

for 1 < TLTF(i) for TLTF(i) < 1 < TLTF(i)

Considertwo points in the search process for the system shown in Fig.3 the details of which are given in Table 1.

1429 Table I shows that a load transfer of 125 A on f l is more though 100 A preferable than a load transfer of 100 A on feeder is the target load transfer between the two transformers. This is due to the fact that the first case offers more compensation with regard to feeder load balancing. The final loads on the transformers and feeders, for the above options, are indicated in Table U, for the sake of comparison.

a,

number of switching operations is given a lower memebership value

vol. TABLE I. Details of the switching options ( for system in Fig.3 ) to illustrate DOS(F).

From Table 11, it can be seen that the first option allows greater feeder load balancing compared to the second option and hence is preferred.

iii) Degree of Satisfaction for Later Feeders, DOS(LF): The balancing of lateral feeders is also considered for the following reason: In typical distribution system, each transformer is supported by another source, not only via an open sectionalizing switch on the main feeders, but also by a similar arrangement on the lateral feeders. Usually, the lateral feeders have lesser capacity compared to the main feeders. For load balancing, the case of lateral feeders is somewhat different from the transformers and main feeders. Since lateral feeders are spread out along each main feeder and their capacities are diverse, it is not appropriate to prescribe a definite 'target for load transfer' for lateral feeders. Hence, a membership function for degree of satisfaction for lateral feeders is defined as in Fig.5.

TABLE 11. Final load level of transformers and feeders for the svstem in Fig.3 Feeder h d ( A )

I

I

550 575

1. 2.

525 MO

n

I

I

275 400

R

r3

275 I75

525 MO

n

LL

0

0

2,

The degree of satisfaction for lateral feeders is equal to unity as long as the lateral load remains less than 50% of its rated capacity ( LFLI(LF) < 0.5 ). From 50% rated capacity to 150 5% rated capacity, the membership function value keeps decreasing. More than 150% loading ( LFLI(LF) > 1.5 ) is absolutely undesirable. This is due to the fact that in most utilities, a higher loading is allowed on lateral feeders as compared to the transformers and main feeders. This can be described as ILDOs(LF3

= 1 = 1.EUCAP(LF)

for UCAP(LF) < 0.5 for 0.5< UCAP(LF) < 1.5

iv) Degree of Satisfaction of Number of Switching Operation, PDOS,,:

As stated above, one of the objectives of load balancing is to have a minimal number of switching operations. Hence a membership function for DOS(ns) is also defined as in Fig.6 which decreases with an increase in the number of switching operations. Each switching option involves the operation of two switches: closing an open sectionalizing switch and opening another switch in the loop that is formed. This can be written as Pm-)

= 1 = 1.25 - (ns/8)

for ns 5 2 for 2<ns<10

Switching options in which only a single pair of switches are operated are known as single-switching options. Cases where two or more pairs of switches are operated are known as double and multiple switching options respectively. It is noted that, in fuzzy set rotation, a high membership valve indicates a desirable situation. For example, the degree of satisfaction for a single switching operation which is a highly desirable situation, is assigned a value equal to unity. A larger

.-a ' r

!!! a, n E

a,

z

Lateral Feeder Load Index LFLI (If)

Fig. 5 Membership Function ~ D O S ( L F .For ) Lateral n Feeders v1 O1 0

.a,

n

a,

2 No of Switchings

ns

10

c

Fig.6 Me mbership Function DOS (ns) For No. of Swit chings

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1V. PROPOSED FUZZY-REASONING APPROACH FOR LOAD BALANCING

a) b)

The proposed fuzzy reasoning algorithm for load balancing is described as follows:

c) d)

STEP 1 : Compute the system load index (SLI) and the

STEP 5 : If the position of any loop switch is altered in STEP 4, then proceed to STEP 1. Else STOP.

transformer load indices (TLI) of all the transformers in the system.

STEP 2 : Select the two transformers which have the highest and the least load indices. Evaluate the Target Load Transfer TLTT and TLTF (wherever applicable).

Select a transformer. Evaluate the loads and the target load transfer for all feeders provided with open loop switches. Execute load balancing as in STEP 3. Repeat for all transformers.

It may be noted that the transformer load balancing is not exhaustive since the loop switches remain fmed during the calculation. Hence, if any loop switch is altered during STEP 4, one has to search for the possibility of a greater load balancing.

STEP 3 : Execute load balancing as explaining below: For all single switching options, calculate D O S O , DOS(F) and DOS(LF) wherever applicable. Extract the best possible switching option through MaxMin principle of fuzzy numbers. If the best possible option obtained as above has a for ns = 2 then degree of satisfaction less then p,, search for double - switching options and so on. Else, go to step e). Under multiple switching options for load balancing between a pair of transformers, evaluate the best compromise amongst the best options obtained for the various switching combinations. Go to STEP 1, till all transformers are considered and no further balancing is possible.

STEP 4 : After balancing of all transformers is camed out, balancing of main feeders via. loop switches is performed, in a similar fashion to that of the transformer balancing, as follow:

FIG07

In the search process of transferring the load between a pair of transformers, the load is gradually shifted from the source transformer to the sink transformer by sequentially opening and closing the appropriate switches until1 the desired load balancing has been achieved. If at any juncture, the load transferred '1' exceeds the target load transfer TLT, then further options along this path need not be considered. This is justified due to the fact that when one moves further along the path the amount of load transferred would be increasing, there by decreasing the Degree of Satisfaction of load balancing (refer Fig.:! and Fig.4).

V. APPLICATION OF FUZZY REASONING APPROACH The effectiveness of the proposed fussy reasoning approach is illustrated through an example system shown in Fig.7 [15]. For this system, each transformer has a rating of 1000 A. Feeders f3 and f4 also have a rating of 1000 A while f l , f2, M and f6 have a rating of 500 A each. The system load index SLI is 0.408. Load balancing has been performed for this system using the proposed method. The following switching operations are suggested to achieve load balancing:

EXAMPLE SYSTEM

I

143 1

Switches to be closed : 19,29,37 and 69 Switches to be opened : 14,28,35 and 67

Table 111. Comparision of loads and transformers and feeders

VI DISCUSSION

S.NO.

F d a

BaQ Losd

(4 Trl fl

a 2.

Tn

0.646 0.516

4.

413 220 I93

0.323

388

B 3.

of

Nta

1.

The solution for the example system was obtained in 50 secs. of CPU time on a PC 286. Aoki et a1 [2] have used a similar technique on the Hiroshima City System for load balancing. In crisp domain i.e., without incorporating the fuzzy operator's preference. They reported a reasonable CPU time (3.1 - 3.6 secs. on a HITAC M - 200H Computer) for obtaining solution in real time. Since the fuzzy-logic based technique does not impose any extra significant computational burden, this technique can also be used in real time Distribution Management Systems OMS).

TrPNrama

indices

I

I

I

A comparison of the loads on the transformers and feeders before and after load balancing is provided in Table 111.

load

i4

0.298 0.298

f5

0.536

Tr3

421 421 410

TA f6

0.421 0.410 0.504 0.316

VII. CONCLUSIONS An approach based on fuzzy reasoning has been developed for determining a proper set of switching operations to balance the loads on a distribution system. A characteristic of the fuzzy reasoning approach is that it considers the desirability of a switching option vis-a-vis the components for which load balancing is contemplated. In most of the existing works, the loads of various components are viewed as being equally critical. Whereas, an experienced operator would give a higher preference to balancing of transformer loads, then to feeder loads and lastly to lateral loads. Such an intelligence is incorporated in the proposed fuzzy reasoning approach and is expected to give more realistic options in practice.

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[7] G.Chang, J.Zrida and J.D.Birdwell,

"KnowledgeBased Distribution System Analysis and Reconfiguration", IEEE Tans. on Power Delivery, Vol. PWRS-5, Aug. 1990, pp 744749.

[SI K.Tomsovic, "A Fuzzy Linear Programming Approach to the Reactive PowerNoltage Control Problem", IEEE Trans. on Power System, Vol. PWRS-5, Feb. 1992, pp 287-293. [9] H.C.Kuo and Y.Y.Hsu, "Distribution System Load Estimation and Service Restoration Using a Fuzzy Set Approach", IEEE Trans. on Power Delivery, Vol. PWRD-8, Oct. 1993, pp 1950-1957. [lo] Y.Y.Hsu and H.C.Kuo, "A Heuristic Based Fuzzy Reasoning Approach for Distribution System Service Restoration", IEEE Trans. on Power Delivery, Vol. PWRD-9, April 1994, pp 948-953. [ l l ] C.C.Su and Y.Y.Hsu, "Fuzzy Dynamic Programming: An Application to Unit Commitment", IEEE Trans. on Power Systems, Vol. PWRS-6, Aug. 1991, pp 1231-1237. [12] Y.Y.Hsu and H.C.Kuo, "Fuzzy-set Based Contingency Ranking", IEEE Trans. on Power System, Vol. PWRS-7, Aug. 1992, pp 1189-1196. [131 D. Srinivasan, C.S. Chang and A. C .Liew,

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[14] H.J.Zimmermann, "Fuzzy Programming and Linear Programming with Several Objective Functions", TIMSlStudies in the Management Sciences, Vol. 20, 1984, North-Holland, pp 109-121. [15] K.Aoki, K.Nara and T.Satoh, "New Reconfiguration Algorithm for Distribution System-Priority Contrained Emergency Service Restoration", Proc. IFAC Conference on Power Systems and Power Plant Control, Seoul, 1989, pp 443448.

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.,a I

I

I

“I.\

B.Naea Rai was born in Tenali, A.P, India, on Feb 9, 1971. He received his B.E.(Electrical & Electronics) from Andhra University, Visakhapatnam, A.P, in 1992. From 1992 to 1993 he was with the Coromandel Fertilizers Ltd., Visakhapatnam, A.P. He is currently working for his M.Tech (Power Systems) degree in the Dept. of Electrical Engg., IIT Delhi, India. His research areas include applications of AI techniques to Power System.

K.S.Prakasa Rao (S’70 - M’74 - SM’81) was born in Prakkilanka, A.P, India, on July 15, 1942. He received his B.E.(Electrical) and M.E.(Power Systems) from Osmania University, Hyderabad, in 1964 and 1966 respectively. He obtained his Ph.D degree from the Indian Institute of Technology, Kanpur in 1974. He is presently a professor in the Electrical Engineering Department at IIT Delhi. His fields of interest are Power Systems Planning, Operation and Reliability.