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Voltage Drop andPower Losses Fast Evaluationthrough Equivalent Models ofFeeders for OptimalOperation ofAutomated Distribution NetworksA. Augugliaro, L. Dusonchet, E. Riva Sanseverino

Abstract

In the paper: a generalised distribution-feeder equivalent model is developed and presented. The model has beenspecijicullv developed aiming at applications in thefield ofoptimal operation ofautomated distribution networks./t aIlows to eas i Iy atul rapid Iy perform calcu Iuti onsjor opti111a I reconfig utu tion and/or compensation. These kindot'difjicl/lt combinatorial problems have indeed been extensively solved in literature by means ofheuristic stra­tegies and non-deterministic algorithms. These methodologies are intrinsically iterative. requiring many eva­luations o{ the objective function. which is often cotnputationallv time-consuming. The studied eletuentarv sy­stem is II distributionfeeder with lumped loads and capucitor hunks. The proposed model is hi-directional, lie­cause [his condition is required when simulating recoufiguration. The paperfirst shows the derivation ofthe equi­valent models puratneters when considering a single branch element. then, a calculation methodologvfor radi­(II networks is presented, //1 this case, the reduction strategv requires a preliminar» visit ofall the nodesfrom theterminal ones up to the root ofthe tree-network atul in turn considering the summed up loads below, as terminalloads. The proposed networks reduction can he uppliedforfust implementation ofpower losses and voltage dropscalculcuions when using advanced dvnaniic data structure [or the networks representation. The proposed me­thodology luis then been implemented bv means ofthese information tools and tested on difjerent distribution S\'­

stems. so us to evuluutr the errors due to thefeeders and networks reduction,

1 Introduction

The basic elements. which characterise a medium­voltage distribution network. are:

- Wide extension. due to the large number of loadnodes compared to the number of supply nodes.

- Radial topology.

- Large number of intermediate load nodes (betweentwo nodes from which laterals spread up).

- Lines having a large ratio between resistance and in­ductive reactance and negligible line-to-ground ca­pacitance.

At present, distribution systems design keeps intoaccount the automation of some management opera­tions. in order to improve distribution systems efficien­cy and to offer a reliable service to customers. namely interms of supply availability and of achievable economi­cal benefit. Two of the functions that can be performedin an automated distribution system are:

- Reactive loads compensation through swirchable ca­pacitor banks. located at some of the load nodes: thisprovides power losses reduction. voltage profile flat­tening. loading capacity increase for the elements ofthe network,

- Reconfiguration of the network through tic-switchesallowing branches disconnection; this function ul­low-, minimum losses operation. in the normal state.

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while providing service restoration in outage situa­tion. It indeed may allow the connection betweenareas supplied by different substations, providingsupply to the de-energised loads.

In an automated distribution system. these functionscan be performed in order to get optimal values for someobjective functions. Some of these express the followingobjects: management cost, voltage bandwidth, load bal­ancing.

Many calculation techniques have already been de­veloped so as to optimise these functions, either singu­larly taken or in combination. The nature of this optim­isation problem requires heuristic rather than mathemat­ical solution methods. That is indeed an evident tenden­cy in the most recent papers handling the subject.

Heuristic methods need usually the execution of anumber of iterations, each of which requires the stateevaluation of the networks, in terms of power losses en­tity and voltage level at load nodes. The state evaluationof the networks can be done using the classical proce­dures yet widely used in power systems analysis, al­though these methodologies lead to a number of seriousdrawbacks mainly due to

electrical features of the system (high resistance-in­ductive reactance lines ratio). which make not sure al­gorithmic convergence of traditional methodologies,

the large number of nodes. which implies a huge cal­culation effort and memory occupation.

217

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2 Equivalent Feeder Model

Fig. 1. Active and reactive loads and reactive powerof capacitors in the elementary feeder

0AQA ... II + 12 . l- I" .... 1,,+1 .. ')sQsA B

PI P, Pi P,,-I P"QI Q; 0 Q"-l Q"-I

Branchingout bus ortie-switch

~QC."-I~QC."-~ -~

Branchingout bus ortie-switch

jgc.1-~

A distribution network, where reconfiguration andcompensation may be performed automatically, has tie­switches located on the branches, for reconfiguration, anda set of capacitor banks allowing the reactive power-flowmanagement (see Fig. I I in Section 4). The number offeeders is lower than the number of loads. The main taskof the whole system management is that of getting the op­timal value for a certain function, or a set of functions, bymeans of remote operation on network elements, such astie-switches and switchable capacitor banks.

In the feeder-equivalent model the represented net­work control parameters must explicitly appear, and inthe studied case, these can be the binary variables indi­cating either the open-closed status of capacitor switch­es or of tie-switches.

The elementary feeder, whose equivalent modelmust be determined, is the segment of real feeder start­ing and ending either with a tie-switch or a branching outbus. The elementary feeder representation, wherelumped loads, capacitors and the terminal loads areclearly indicated, is shown in Fig. 1, where:

- Pi' Qi are active and reactive loads derived at the i-thnode of the feeder;

- QC.i is the capacitor bank reactive power at the i-thnode of the feeder: this quantity is obviously equal tozero for not compensated nodes;

PA' QA (Ps, Qs) are active and reactive power flowsthrough A (B) node; they are directed towards theareas below, when the feeder is supplied at the B (A)end bus;

8A (8s) is a binary variable assuming value I, whenthe feeder is supplied from the B (A) end bus and 0when it is supplied from the A (B) end bus.

On the basis of the definition given for terminalloads, it can be said that:

- PA (P s ) is the total active power flow supplying theloads below the A (B) node;

QA (Qs) is the total reactive power flow supplying theloads below the A (B) node.

In the equivalent feeder model development, theloads have been assumed as constant current elements;this assumption turns to be convenient for the calcula­tion to be executed and does not lead to errors within anoptimisation procedure, as it was noticed [4].

The equivalent parameters calculated for voltagedrops evaluations differ from those calculated for power

the solution algorithms, for the optimisation prob­lems, are iterative processes and at any iteration thesystem state evaluation is required; in this way, thewhole calculation time increases, making an effec­tive on-line control of the system even more difficult;

the control parameters are discrete quantities andthe related objective function also assume discretevalues.

Moreover, among those functions that can be per­formed in an automated network it is necessary to keepinto account that

It is then clear that an analysis methodology allow­ing a fast calculation of a number of state evaluations,even if approximate, can be quite favourable. The exactstate evaluation of the system is not necessary; approxi­mation does not influence indeed the goodness of the so­lution found at the end of the optimisation process be­cause, in this case, the objective function should mainlybe a good indicator of the quality of the solution, ofcourse depending on the function value leading the op­timisation process (e. g. power losses).

A reduction in the calculation time and in the mem­ory occupation can be obtained through the use of a net­work reduction methodology, which allows the transfor­mation of each branch having a large number of lumpedloads along, into one having a single lumped load.

In this way, it is important to define some relationskeeping safe equivalence for the required evaluations.Two equivalent feeder models have been proposed in [I]and [2]. The first model [I] has been developed for feed­ers where supply can come only from one end; the sec­ond one [2], even if bi-directional, does not give anycomputational benefit, when applied to optimisationstrategies for automated distribution networks, where itis required to repeatedly perform the state evaluation ofthe networks, as the compensation level and the loadsentity vary. At each of these changes, the models devel­oped in [2], require the entire calculation of the equiva­lent model parameters.

In this paper a generalised equivalent feeder modelis presented. The model refers to a real feeder whichmay have lumped loads and capacitor banks both alongand at the end, it is bi-directional and it can be usedwhen dealing with automated networks optimisationproblems. The equivalent feeder model parametersmust be evaluated taking into account the functionevaluations to be afterwards executed (voltage dropsand/or power losses). Later on, a solution methodolo­gy specifically developed for radially operated net­works is presented. The solution method is based on theanalysis of the topological features of the network andit is clearly put into evidence that the proposed reduc­tion can be feasibly used together with an efficient dy­namic representation of the network, yet experimentedby the authors [3]. The proposed calculation methodhas then been implemented and its precision has beentested. Finally, the calculation time of the desired quan­tities is much lower, and this can be seen in some testruns using the proposed model within an applicationfor optimal reconfiguration and compensation in dis­tribution systems.

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purely reactive. Therefore. for voltage drops evalua­tions. the feeder in Fig. I can be represented in an equiv­alent way as it is shown in Fig. 2 and Fig. 3. respective­ly. referred to /:"Ur and /:,.U,.

The quantities I',!, eq and 1'/. eq represent the distancesat which the sum of intermediate loads, respectively. ac­tive and reactive. must be derived, so as to obtain thesame voltage drops that can be calculated in the realfeeder. These lengths can easi Iy be calculated equatingthe voltage drops expressions in the two feeders in Fig. 2and in Fig. 3a and 3b.

The case of feeder supply at the A end (6,\ = 0) isstudied first, In this case. the voltage drop. evaluatedwith regard to the systems in Fig. 2a and in Fig. 3a. in­dicating with Uthe rated voltage of the network. is givenby:

/:,.U= /:,.Ur+/:,.U,

= ~ fIh[±P; ±t., + P;3'£ Ir.i]U 1 i~1 )~I i~1

+ x!J[~ Q;~I/,.)+ Q<~ Ir.i- ~ QC;~I/I.J11· (4)

The same voltage drop. evaluated by means of thesystems in Fig. 2b and in Fig. 3b. is given by:

Equating /:"Ur and /:,.U, terms in eqs. (4) and (5). weobtain (the equivalent length for t1U, has been evaluatedonly taking into account the loads: indeed. the reactivepower val ues of the capaci tor banks are control variablesin the optimisation processes on automated networks.therefore they have to appear in explicit form in theequivalent model):

( I)

(2)

(3)

r,Ir .i = --'--Ii'

rh

I x; Ix.i=-i"

x h

Obviously. if the real feeder has constant sectionthen for each segment:

a) .-\ B

,iAP.\ 11'.1 .~ 11'.2 ,illP ll,PI P, Pi P,,-I P"

b) A B

(iA;:l ~iPi I~PBDI r.c ,!

i~ I

2.1 Equivalent Feeder Modelfor Voltage Drop Evaluation

losses evaluations. due to their non-linear behaviour.Since non-linearity is not related to resistance and reac­tance of the segments composing the real feeder. someof these can have different sections. If that is the case, thefollowing steps should be taken:

- The conductor size of one of the segments compos­ing the considered feeder is taken as a reference, suchas the corresponding resistance and reactance, whichcan be indicated with r h and .rh'

- The equivalent lengths of one single segment can thenbe calculated through the following relationships:

The approximate evaluation of voltage drops /:,.U indistribution networks can be executed by summing upthe active power contributions of loads /:"Ur for feedersconsidered ,IS purely resistive. and the reactive powercontributions of loads /:,.U, for feeders considered as

11'.; =11.;=l;

Fig. 2. Representations for !'J.Ur evaluationa) Real feeder modelb) Equivalent feeder model

(6)

a) A " ; /"I~Cll = L Q;L ll.) L Q; .;~, j~l ;~l

(7)

Fig. 3. Representations for !'!,.U, evaluationa) Real feeder modelb) Equivalent feeder model

(8)

If the feeder is supplied at the B end, the expressions,corresponding to eqs. (4) and (5), are the following:

t1U= /:,.Ur+ /:,.U,

I 1[" ,,+1 ,,; ,,+1 1= - rh L P; L 11'.; - L P; L t., + PA L t.,

U i~1 i~1 i~1 j~1 i~1

Q"

"L. o,i = I

i,.2IJ

II.CII

11.1

b) A

i,.1

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Equating the terms expressing I1U, and I1U, ineqs. (8) and (9) we again obtain eqs. (6) and (7).

Finally, the general expression for the voltage dropin the feeder, either supplied at A and B end, is givenby

LI B.eq

41r.1I ."..Ir.II+I.OBQB

P; PII - 1 PIIQ; QII-I QII

LI A.eq _

II II rIP;.2.(Qi-Qc;);= I ;= I .

From eq. (13) we obtain:

I1P = \ fi I,.i[[ i Pj)2 + (iQ; )2]U 1,=1 J=I .1=/

5 Q I I(A A i4 ,.1.. ,..2.

PI P,QI Q;

c) A B

~~r-------------<-------itb~-----------<.I-II--II----k:

2. Pi. 2. (Q;- QCi);= I i= I '

Fig. 4. Feeder representations for /:;Pevaluationa) Real feederb) Equivalent feeder model. supply node Ac) Equivalent feeder model. supply node B

(9)

( I I)

( 12)

II ["+1 k J])- I QC.k I/r.; - I/r.j .k=1 ;=1 j=l

I1U= I1U r + I1U,

I {[("+1 D) II 11+1]= - rh I t., - Ir .eq I P; + PA I I,.;U ,=1 1=1 1=1

[("+1 )II 11+1+ X b ~ ! v.i - I~eq ~ Q; + QA~ I,.i

I [ II 11+1 II 11+1 ]I1UC = U rh~ P; ~ I,.; + X h~ Q;~ t., .

I (" ")I1Uo = - It, I,~eq I P; + X h I~cq I Q; ,U i=1 ;=1

assuming:

I1U = 8AI1UC + (_1)8, [ I1Uo - .~ ~ QC.i~ I r .) J

xhII 11+1 I [("+1 )-8A - I Qc.;I/r.i+- rh I/,.iU i=1 ;=1 U ;=1

.(8A P" +8BPB)+ x{g t.,}8A QA +8BQB)].(10)

2.2 Equivalent Feeder Modelfor Power Losses Evaluation

The power losses expression, evaluated for the out­lined equivalent feeder model in Fig. 4b, is given by:

For power losses calculations in the following onlythe line losses have been considered not taking into ac­count the transformer losses. These are indeed almostconstant, due to the adopted load model, and then not in­fluencing the optimisation process. In this way uselesscalculations and time consumption are avoided.

If the feeder is supplied at theA end (8A = 0), Fig. 4a,the power losses expression is: Comparing eqs. (14) and ( 15), we obtain

(16)

n (" )"" ""A, +A, PIl+A,Q" +~ 1,-.1 ~,Qc.j -~, M(!_I~, Qc.i-2QB~ I,-.,~, Qc.i

2D, f],+ 2D, QIl + D,+ DJ - 2D,W, + QIl)

assuming:

AI = i l,.i[(ip ;)2+(iQ;)]./=1 .1=1 .1=/

(17)

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II 1/

A2= 2II,.;I Pi'i~1 i>!

( 18)Comparing eqs. (26) and (27). we obtain

/I /I

A,= 2I I,..; I c..i=l i=i

( 19)II - I I I J' - II· I I I 11 ~ I I I

Ii,+ Ii, 1', +IiJJ, +,~ t. ~,IJI., -,~ N,", ~,IJ,'" - 21J., ,~t,., ~,IJ",

2D, r,» 2D, Q, + fJ,+ fJ;- 2IJ,( IJ,+ Q., I

II

M(}.i= 2/,..; I o.. i = 1,2, .. . . 11,

;=i(20)

(28)

where:

II

DI = I t;;~\

(21 )(29)

II

D2 = IQ;,i~1

(22) 11+1 ;-1

82 = 2I/,..; I Pi';=2 j=l

(30)

which results in:

(31)

(32)

(33)

i-I

Nu.i = 2/,..; I Qi' i = 2.3 ..... 11 + I.j=i

11+1 i-I

8." = 2 I I,..; I o.i=2 j=l

As a resul t. from eqs. ( 15) to (27) we can get a gen­eral expression for power losses. valid whatever it is thesupply end:

!'!.P = ~'2 {(/~,c'18B +1~.c,l\)[(DI +P'\6" +~l6B)2

+(D2 + Q)5\ + Qu6 11 - D~)2]

+ ['~\/I.i - (/~·.eAl + I[~·.c,l\ )]

3.1 Solution Method Description

3 Radial Networks Solution Methodthrough Equivalent Models

The evaluation of the reactive power variations!'!.Qin the feeder can be performed in a similar way comingto expressions similar to those obtained for power loss­es !'!.P replacing the resistance rh with the inductive re­actance Xh and the lengths I,;; with I,,;. For this reasonthese expressions are not here quoted. also consideringthat the proposed solution method does not require theuse of this quantities.

The proposed models for a single feeder analysis canbe favourably applied in optimisation processes taking partof automatic distribution systems management. In thiscase. a possible application. in distribution systems wherereconfiguration and compensation may be performed. isthe automatic management of networks where solutionmethodologies requiring the evaluation and analysis ofseveral possible systems states are employed. Indeed. inthe calculation of the model parameters. only a few termsdepend on the network configuration and on capacitorbanks state. In voltage drops evaluations. the equivalent

(24)

(23)

II

D~= I QC.i·;::::::1

[ , 'jr J, 1 1/+1 1/+1 i-I - ;-1 -

8.P =~ l(r:+QA)I I,..; +I I,.; (I PiJ+(I a, JU I~I 1~2 .I~I .I~I

t, J [/+1 ]8.P = ~'2 1(p.~ + Qn ~ 1,;- lii.e'1 + li;.c '1

,11+1 ;-1 11+1 ;-1 11+\ (;-1 J-

+ 2P,,;~ I,,;.~ Pi+2QA i~ I,;.~ Qj+;~1,.1 .~ Qc.j

11P~ ~, jf,,(I:;+Q,;)++f,{[~,+ ~,p;r

+(Q,+ ~,(Qj-Qc;)J}+f".,

[[1\+ ~p}( QA+ ~(Qj-Q,,)Jl (25)

1/+1 ;-1 1/+1 i-I ;-1 }

- 2QA ;~ I,..; j~ Qc.j - 2;~ I,..; j~ aj~ Qc.j' (26)

The terms AI' A 2, A 1• M(}.i (i = I. .... II), D I• D2, 0.1and D~ are not depending on the control variables andthen keep a constant value wi thin the whole optimisationprocess. such as the equivalent lengths in eqs. (6) and (7).The quantity 1\.e'1 depends on the terminal load entity andon the capacitor hanks reactive power: this length mustthen be evaluated at any step of the process.

When the power supply flows from B towards A,(611 = 0). Fig. 4c. then the power losses expression is

Power losses evaluated on the basis of the equivalentmodel in Fig. 4c are given by:

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lengths ofeach feeder can be straightforward calculated bymeans ofeqs. (6) and (7), where only the supplied load val­ues appear. In the power losses evaluation instead, only theterms in eqs. (17) to (24) and eqs. (29) to (32) are constant,as the capacitor banks states and the network configura­tion vary; these terms can indeed be calculated only on thebasis of the total supplied load. As a result, keeping intoaccount the statements above, the network radiality andthe constant current hypothesis for the model, a favour­able and efficient solution procedure can be articulated inthe following stages:

I. On the basis of the radial configuration of the net­work the power-flow directions are defined in all thebranches, then the values assumed by the binaryvariables, (j", and !5a, are also known.

II. The terminal branches are then considered, beingcharacterised by having PA = QA = 0 (or PB = QB= 0) and f.U and f.P can be calculated througheqs. (10) and (33).

III. Active and reactive power inflows are then calculat­ed for any terminal branch setting a balance equa­tion at the supply node:

/I

P, = I p,. (34);~I

/I

Q,. =IQ;· (35)i~1

VI. The branches right above the terminal ones arethen considered; for each of them the terminal loadPA, QA (or PB, QB) is evaluated by summing up ac­tive and reactive power flows into the feedersbelow the node A (or B). Through eqs. (10) and(33) f.U and f.P can be calculated; active and re­active power values at the supply end of the feed­er are then given by:

of equivalent models resides in the large reduction of thenumber of nodes to be visited, which produces a de­crease in either calculation time and memory occupa­tion. This is a very important element when dealing withreal-time control problems in distribution systems.

4 Performance Evaluationin Equivalent Models

The approximations introduced in the equivalentmodel definition for voltage drops calculations, lead tosome errors in the evaluation of these quantities, com­pared to those values obtained with traditional analysismethods, where loads can be considered as constant cur­rent elements. Some trials have then been executed so asto evaluate the error entity as some meaningful parame­ters vary. The obtained results have been indeed com­pared to those obtained through the bus impedance ma­trix applied to the original network.

The following have been tested:

- Single feeder with five lumped loads located at equaldistance one from the other, having the same powerfactor, Fig. Sa.

- Single feeders made up of the combination of two orfour feeders having the characteristics of the Fig. 5atype; they are represented in Fig. Sb to Sc.

- Radial networks composed of 3, 7 and 15 feeders, ofthe Fig. 5a type, represented in Fig. 6a to 6c.

In these test cases we considered

- the load equal to the rated value and half of it (loadfactor kL = I; 0.5);

- the power factor cos tp varying between 0.6 and 1.0;

- the presence and absence of capacitor banks.

/I

t; = I p, + PA /B 'i~1

For the systems in Fig. 5 and 6, respectively, Fig. 7(36) and Fig. 9 represent the relative percentage error e",u

a)

3.2 Implementationof the Reduction Methodology

where PAlB and QA/B represent the terminal activeand reactive load either at A or B end bus.

V. Step IV. is repeated until all the branches havebeen visited.

F3

PI P2P3P4 r, PI P2P3 P4 r, PI P2P3P4 Ps PI P2 P1 P4 Ps

Load PI P, P3 P4 Ps Qc

400 kW 300 kW 350 kW 550 kW 410 kW 250 kvar

c)

b)

Fig. 5. Sample feeders for t1Uerrors evaluationa) Elementary feeder with five lumped loads and one

capacitor bank (F I)

b) Two elementary feeders one following the other (F2)

c) Four elementary feeders one following the other (F3)

(37)/I

Q,. = I Qi + QA/ B 'i~l

The procedure described in the section above hasbeen implemented by means of a simple program, whichhas been used to easily evaluate errors that can be madeusing equivalent models. The program has been imple­mented in Pascal (Turbo 7.0) usingforthe radial networkrepresentation a dynamic data structure that allows, witha small memory occupation, to outline a tree network.Each node of the equivalent model is then related to a setof parameters, which are constant within the consideredoptimisation procedures. The main advantage in the use

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a)

b)

c)

I

I

II

I

I

0.8

%

0.6

0.5

i 0.4;:, 0.3

~ 0.2

0.1

o .~/' / / / r : ~1.000.95090 / / rN -

o85 0.80 0.75 0 70 I... cos rp . 0 65 0.60

Fig. 9. Percentage L'lUerrors for not compensated radialnetworks represented in Fig. 6

i kL = 0.5

• kL = 1.0

%

D.')

0.7

0.6

A 0.5

I 0.4~ 0.3" D.2 /' :..- _-

/' / -.L-. _ L.- -

D.I ". / / / /". 7" "j...,"" --":'-...L. _ L>--D ,._/ / / / N

1.00 0.95 Ol);~L-.// IN 2. 0.85 0.80 0.75 0.70 I

... COs rp 0.65 0.60

Fig; 10. Percentage L'lUerrors for compensated radialnetworks represented in Fig. 6.

course for not compensated systems. Diagrams in Fig. 8and Fig. 10 report the error course for compensated testsystems.

From these figures the following observations canbe made:

- The error values are in all cases quite low « 0.8 %).

The entity of the errors increases as the number ofloads, the lines length, the number of branches andthe load factor grow.

- The entity of the errors is higher for low values ofcos tp and for cos qJclose to one.

The presence of capacitor banks turns to reduce theerror value at low values of the loads power factor andto make it higher for higher power factor values.

Within the range of the most common power factorvalues (0.7 to 0.9), the error value is quite negligible«0.33%).

Referring to the main field of application of theproposed equivalent models, that means optimisationprocesses in automated distribution networks with it­erative methodologies, requiring the repeated evalua-

o kL = 0.5

• kL =1.0

0.20

0.30

%0.25

~ 0.15

Fig. 6. Sample networks for L'lUerrors evaluation (the genericbranch is an elementary feeder as F1 in Fig. Sa)a) Network with 3 branches (N I)b) Network with 7 branches (N2)

c) Network with 15 branches (N,)

kL = 0.5

• kL = 1.0

;:, 0.10 F,~ ~

0.05 < / F 1

o IF, F,

1.00 0.95 D')D - / II -o85 0.80 0 75 F I

... . . 070 065 060cos rp

Fig. 7. Percentage L'lUerrors for not compensated feedersrepresented in Fig. 5

Fig. 8. Percentage L'lUerrors for compensated feedersrepresented in Fig. 5

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only once and terms where explicitly appear the vari­ables associated to the particular optimisation prob­lem examined: the capacitor banks status and the en­tity of the loads below each branch for each configu­ration.

That is quite important when heuristic proceduresare used to get optimal solutions in real time for the op­eration of automated distribution networks and conse­quently the state evaluation should be repeatedly exe­cuted.

ss

33

32

45 46 47 48 5

21

I 2 6 7 8 10 II 12 14 15 1617 18 20 3

19

4

SS

Fig. 11. Sample network 6 List of Symbols and Abbreviations

L\UL\U,.

t.; t.,

UMJo, L\UcL\Pl;.eq, lLq,

active and reactive loads derived at thei-th node of the feederreactive power of the capacitor bank atthe i-th node of the feederending buses of the feederbinary variable assuming value I, whenthe feeder is supplied from the B (A) endbus and 0 when it is supplied from the A(B) end bustotal active power flow supplying theloads below the A (B) nodetotal reactive power tlow supplying theloads below the A (B) noderesistance and reactance of one of thesegments composing the feeder taken asa reference (base)equivalent lengths of each segment of thefeedervoltage drop in the feedervoltage drop in the feeder, considered aspurely resistive, due to the active powercontributions of loadsvoltage drop in the feeder, considered aspurely reactive, due to the reactive powercontributions of loadsdistances at which the sum of intermedi­ate loads, respectively, active and reac-tive, must be derived, so as to obtain thesame voltage drops (superscript D) thatcan be calculated in the real feederrated voltage of the networkconstant terms in voltage drop expressionpower losses in the feederdistances from feeding bus A, or B, atwhich the sum of intermediate loadsmust be derived, so as to obtain the samepower losses (superscript L) that can becalculated in the real feeder

L\Q reactive power variations in the feederkL load factorcos <p power factorPA' QA (P B, QB) active and reactive power flows

through A (B) node; they are directedtowards the areas below, when thefeeder is supplied at the B (A) end bus

AI,A2,A3,MQ.i(i= I, 11),

BI , B2, B3, NQ. i (i = 2, 11 + I),D1, D2, D3, D4 constant terms in power losses ex­

pression

tion of the network state, the use of the proposed reduc­tion methodologies strongly decreases the calculationtimes.

A solution algorithm applied to the real networkhas a calculation time depending on the real number ofload nodes, the reduced network has a lower number ofnodes and therefore its solution turns to be faster. As aprove of the effectiveness of the proposed reductionmethodology, a minimum losses reconfiguration andcompensation problem for the network depicted inFig. 11, has been considered. This network has foursubstations HV/MV and is made of 4 I load nodes, 48branches and 12 capacitor banks located at 9 loadnodes. In Fig. I I, the tie-switches and the capacitorsswitches are highlighted, the state connected/discon­nected of which are the control variables in the optim­isation problem. For the solution of this problem anheuristic technique has been apptied using a tabusearch strategy (applied by the authors in [3J). Thesame optimisation strategy has been applied with theobjective function evaluation either using the reducedmodel of the network (the number of branches is re­duced to 18) or using the real network.

The same optimal solution has been obtained in lessthan half time, as we expected.

5 Conclusions

An equivalent feeder model that can be used for dis­tribution networks in which the load can be simulatedwith a constant current model and in which, for man­agement requirements, the supply ending bus maychange during the normal operating state, has been de­veloped.

The equivalent model allows the direct calculationof voltage drops and the total power losses in the feeder.

The approximation introduced in the voltage dropevaluation produce quite low errors, especially withinthe range of the load power factor values that can benormally found in distribution networks.

The use of the model for all the branches of a radi­al network allows huge savings in calculation time andmemory occupation, the benefit of which is even moreevident as the size of the network grows and, more­over, as the number of loads on each branch increases.Indeed the proposed procedure allows to find relationsholding constant terms, which have to be calculated

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References

[I] Vempati, N., Shoults, R. R., Chen, M.S.. Schwobel, L.: Sim­plified feeder modeling for load flow calculations. IEEETrans. on Power Syst. PWRS-2 ( 1987) no. I. pp. 168 - 174

[2J Chen. T. H.. Wang. S. W: Simplified bidirectional-feedermodels for distribution-system calculations. lEE Proc. ­Generation. Transm.. Distr. 142 (1995) no. 5. pp. 459-467

[3J Augugliaro, A., Dusonchet, L.. Riva Sanseverino, E.: Ge­netic. Simulated Annealing and Tabu Search Algorithms:Three Heuristic Methods for Optimal Reconfiguration andCompensation of Distributed Networks. ETEP Eur. Trans,on Electr. Power 9 (1999) no. I. pp. 35 -41

[4] Augugliaro, A., Dusonchet, L., Mangione, S.: Load influ­ence on minimum loss recontiguration of automated dis­tribution networks. "Stockholm Power Tech", Int. Sym­pos. on Electr. Power Engng .. Stockholm/Sweden 1995.Proc. of 'Power Systems'. pp. 771- 776

Acknowledgement

This work was supported by the Italian Ministry of Universityand Scientific and Technological Research.

Manuscript received on November 13, 1997

The Authors

Antonino Augugliaro ( 1(49) receivedthe Doctor degree in Electrical Engi­neering from the University of Paler­mo/ltaly in 1975. From 1978 to 1994he has been Associate Professor andnow he is Full Professor of ElectricalPower Generation Plants at the Facul­ty of Engineering of the University ofPalermo. His main research interestsare in the following fields: simulationof electrical power system; transmis­sion over long distances; mixed three-

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phase/six-phase power system analysis; optimization methodsin electrical distribution- system design and operation; distri­bution automation. (Dipartimento di Ingegneria Elettrica,Universita di Palermo. Viale delle Scienze, 1-90128 Palermo/Italy, Phone: + 39 91/6566241, Fax: + 39 91/488452, E-mail:luini@cuc.unipa.it)

Luigi Dusonchet (1948) received theDoctor degree in Electrical Engineer­ing from the University of Paler­mo/ltaly in 1975. From 1978 to 1990 hehas been Associate Professor and nowhe is Full Professor of IndustrialElectrical Systems at the Faculty of En­gineering of the University of Palermo.His main research interests are in thefollowing fields: simulation of electri-cal power system; transmission over

long distances; mixed three-phase/six-phase power systemanalysis; optimization methods in electrical distribution­system design and operation; distribution automation. (Dipar­timento di Ingegneria Elettrica, Universita di Palermo. Vialedelle Scienze, 1-90128 Palermo/Italy. Phone: + 39 91/489856.Fax: +3991 /6566242. E-mail: luini@cuc.unipa.it)

Eleonora Riva Sanseverino ( 1971) re­ceived the Doctor degree in ElectricalEngineering from the University ofPalermo/Italy in 1995. Since 1995 shehas been working in the ResearchGroup of Electrical Power Systems.Now she is a Ph.D. student in Electri­cal Engineering at the same Univer­sity. Her main research interest is in thefield of optimization method onelectrical distribution-system design,

operation and planning. (Dipartirnento di Ingegneria Elettri­ca. Universita di Palermo. Viule delle Scienze, 1-90128 Paler­mo/Italy. Phone: +3991/6566205. Fax: +3991/488452;E-mail: e.riva@dielectricslab.diepa.unipa.it)

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