optimal network reconfigurations in distribution systems

7/27/2019 Optimal Network Reconfigurations in Distribution Systems

1/8

1902 IEEE Transactions on Power Delivery, Vol. 5 , No. 4, November 1990Optimal Network Reconfigurations in Distribution Systems:Pa r t 1 : A New Formulation an d A Solution Vlethodology

Hsiao-Dong Chiang, RenC Jean-JumeauSchool of Electrical Engineering

Cornel1 University, Ithaca , NY 14853

AbstractA new formulation of the network reconfigurationproblem for both loss reduction and load balancingtaking into considerat ions load constraints and op er ational constraints is presented. The number of switch-on/switch-off operations involved in network reconfig-urat ion is put into a constraint . T he new formulat ionis a constrained, multi-objective an d non-differentiableoptimizat ion problem with both equali ty and inequal-i ty constraints . A two-stage solution methodologybased on a modified simulated annealing technique andthe -con straint metho d for general multi-objective op-timization pro blems is developed. A salient feature ofthe solution methodology is that it allows designersto find a desirab le, global non-inferior solution for th eproblem. An effective scheme to speed up the solutionmethodology is presented and analyzed.

IntroductionThere are two types of switches in primary distr ibut ionsystems: normally closed switches which connect linesections, and normally open switches on the tie-lineswhich connect t w o primary feeders, or t w o substat ionsor loop-type laterals . Th e former are termed section-nlizing switches and the lat ter are referred to as ti esw i tches (see Fig. 1). These switches are designed forboth protect ion ( t o isolate a fault) and configurationmanagement (to reconfigure the network).

Network reconfiguration (or feeder reconfiguration)is the process of altering the topological structures ofdistribution feeders by changing the open/closed sta-tus of the sectionalizing and tie switches [l]. Duringnormal operat ing condit ions, an important operat ionproblem in configuration management is network re-configuration. As operat ing condit ions change,

9d Y i i 16L+-L+ ?URDby the I d 3 Transmission and Distribution Cormitteeof the I E ,E Power Engineering Society for presentationa t th e I X X / P C S 1990 Winter ;:ectin&, Atlanta, Georzia,February 1, - 8 , 1990. :lanuscript subnittedSepteinber 1 , 1989; imde avai lable for print in2Tjecembor 27, l 9S9.

A paper recoinmended and a!i:x-ovcci

0 normaly opened 0rrki-occ substation

node U oad pdnt

SS 1

s s 2

Figure 1: A sample distr ibut ion system

networks are reconfigurated fo r two purposes: (1) t oreduce t he sys tem real power losses and ( 2 ) to relieveoverloads in the network. Th e former is referred toas network reconfiguration for loss reduction an d thelat ter as load balancing. Another configuration man-agement operat ion involves the restorat ion of serviceto as many customers as possible during a restorativesta te following a fault. This problem is called servicerestorat ion.

In this paper, we consider the network reconfigu-rat ion problem fo r both loss reduction and load bal-ancing. Conceptual ly, this problem belongs to the so-called .minimal spanning tree pro blem: Given a graph(i.e. nodes of the system), f ind a spanning tree ( i .e .a radial configuration ) such that a desired ol>ject,ivefunction is minimized while certain system constraintsare satisfied. Recently, this problem has been formu-lated as a nonlinear optimization problem with a dif-ferent iable object ive funct ion. All the solution algo-ri thms proposed in the l i terature for solving the prob-lem employ various techniques belonging to the classof greedy search technique, which accepts only searchmovements t hat produce immediate improvement . A sa result, these solution alg orithms usually achieve localopt imal solut ions rather than global opt imal solut ions.

An early work on network reconfiguration for lossreduction is presented by Merlyn and Back [ 2 ] . Theirsolution scheme star ts with a meshed distr ibut ion sys-tem obtained by first setting all switches closed and

0885-8977/90/1100-1902$01.00 Q 1990 EE E


2/8

then opening successively the switches to eliminateth e loops in the system. Later, the solution schemeis improved by Shi rmohammad i and Hong [3]. In [4] ,two solution algorithms are presented for network re-configuration based on some performance indices tomeasure the degree of cOnstraint violations as well asto check the op timality for loss reduction through net-work reconfiguration [l].The y derive a simple formulabased on some simplifying assumptions to calculateth e loss reduct ion due t o a branch exchange. Aoki etal. formulate the problem of both service restorationand load balancing taking into account capaci ty andvoltage constrain ts as a mixed-integer non-linear op-t imizat ion problem an d converts the probiem into aseries of continuous quadrati c progra mming subprob-lems [5,6]. Baran and Wu formulate the problem forloss reduction and load balancing as an integer pro-gramming problem [7]. Two efficient load flow meth-ods with varying degree of accuracy are presented an dincorpora ted int o the solution alg orithm which followsthe approach proposed in [l].

The cont r ibut ions of this paper are summarized asfollows. Firstly, in order to truly reflect the objectiveof load balancing, we propose a system load balanc-ing index which is a Chebyshev norm of each branchload balancing index. Th e purpose of load balancingis then realized via solving a min-max optimizationproblem. Secondly, owing to the fact tha t these twoobject ive funct ions - loss reduction and load balancingare incommensura ble, we formulate th e network recon-figuration problem as a constrained, multi-objectiveand non-differentiable optimization problem with bothequal i ty and inequal i ty constraints . This is a step to-ward practic al formulati on of the network reconfigura-t ion problem. Thirdly, we develop a two-stage solutionmethodology for general multi-objective optimizationproblems. This new solution methodology allows de-signers to find a desirable, global non-inferior solutionfor the problem. Forthly, we present a modified simu-lat,ed annealin g te chnique t o solve multi-objective op-t imizat ion problems with the at t ract iv e feature thatit can yield a global non-inferior solution rather thanjust a local non-inferior solution. An effective schem eto speed up the simulated anneal ing technique is pre-sented an d analyzed. Lastly, given a desired numberof switch-on/switch-off operation s involved in networkreconfiguration, the proposed solution algorithm canidentify th e most effective operation s. In contr ast,most of existing solution algorithms make use of allof the control variables in solving the problem. Thiskind of solution is impractical because the number ofoperations could be too large to be executed in actualimplement at on.

Problem FormulationIn this section, a new formulation of the network re-

"00I vo 1 'ok %k +l1903

"Wb

\+ ..5Figure 2 : One-line diagram of a main feeder with lat-er als

configuration problem is presented. This formulationis an extension of the problem formulations proposedby many researchers via truly reflecting the objectiveof load balancing.Load ConstraintsConsider a main feeder with a lateral branching outfrom the main feeder as shown in Fig. 2 . To simplifythe presentat ion, the system is assumed to be a bal-anced 3-phase system. We summarize the load flowequation formulated in [SI. Please refer to [SI for de-tails. We refer to node k as a branching node indicatingtha t there is a lateral . In the f igure, lV,l represents thesubstat ion bus vol tage magnitude and is assumed tobe constant . Th e distr ibut ion l ines are modeled as se-ries impedanc es q = rl + j z l . Load demand at bus i ismodeled as a constan t power sink, S L ~ P L ~ Q L * -In general, the real and reactive power flowing a tthe receiving end of branch i+ l , P,+I, Q1+l and thevoltage magnitude at the sending end lq+l[ an beexpressed by t he following recursive set of equations

Equations ( 1 ) - ( 3 ) , called the branch potu equation, can b ewritten in compact formx,+1= f ,+l (X,) ( 4 1

where X i = [P,, Q;,Kl']. Th e branch flow equations atbranching node k of the main feeder ca n then be writtenas

POk = P O k ( x 0 k - 1 ) - P k O ( 5 )Q o k = Q o k ( X o k - 1 ) - Qlio ( 6 )

w o k 1 = %lOk(XOk--l) ( 7 )There are several boundary conditions to be satisfied:

1. at the s ubstation; the voltage magnitude \Vool s spec-ified.

2 . at t h e en d of th e main feeder;PO,= P o , ( X O , ~ - - I ) 0 an d Qo , = Qo,(XO,-I) = (13. at the en d of lateral k; k n = 0 an d Q l c n = 0


3/8

1904For this system (one main feeder with one lateral), sincet.he substat ion voltage magnitud e is specified, knowing th evariables (P oo , oo, P k o , Q k o ) is sufficient to determine therest of the system variables.

We now consider a general radial distribution systemconsisting of a main feeder (with n branches) with m lat-erals (with n1, n2,...,nm branches on laterals 1,2 ...,m re-spectively). The branch flow equation (1-3) also holds forevery node of the system. For each lateral we choose thereal power and reactive power injected from the ma in feederto the lateral as variables. For instance, for lateral k, thevariables ar e zk o := P k O , Q t o ] . Introducing these two vari-ables enables us to reduce the total number of branch flowequations by performing an elimination process and to de-rive th e following equations:

P O n = . i )On(ZOOr Zlo,..., Z k O , IvOOl) = 0 ( 8 )Q O n = QOn(Z00j ~10,.., Z ~ O , V O O ~ )0 (9)

P k n k = p k n k ( Z O O , Z lO , . . . , 210, IvOOl) = 0 (10)Q k n k = & k n t ( Z O O , z10!...> J O , IvOOl) = 0 (11)

J _< k , a n d k = 1 , 2 ...,mor in compact form

F ( z ) = 0 (12)where z := (P io , Qio, ..., Pmo, Qmo, poo, Q o o ) ~ .Th e distribu tion power flow equatio n (13) is a set ofnonlinear algebraic equations. T he question regarding theexistence and the uniqueness of solution has been investi-gated in [ 8 ] ,where it shows tha t the load flow solution withfeasible voltage magnitude for practical radial distributionnetworks always exists an d is unique.Op e r a t i o n a l Co n s t r a i n t sThe voltage magnitude of each node and line current ofeach branch must lie within a permissible range. Here abranch can be a transformer, a line section or a tie-linewith a sectionalizing switch.

or in compact formG(z) 5 0 J = 1, ..., L (13)

Due to some practical considerations, the re could be a con-straint on the number of switch-on/switch-off operationsinvolved in th e network reconfiguration. Let th e desirednumber be nd and the original network configuration beexpressed as go , then we express t his constraint as follows:

N ( g O , g * ) 5 nd (14)Ob j e c t i v e Fu n c t i o nTh e objective function of the networ k reconfiguration prob-lem comprises tw o terms. Th e first term is concerned withloss reduction and the second term is related to load bal-ancing.Loss reductionThis objective is to minimize the total real power losses

arising from line branches, which can be calculated as fol-lows:

where nb is the total number of branches in the systemLoad balancingTo consider a load balancing index of the ent ire system, wefirst use t he following function as a branch load balancingindez, which serves a measure of how much branch i isloaded

where S, is the a pparen t power at th e sending end of hranclii and ,Far is its KVA capacity. We then define the loadbalancing index of the entire system, termed system loadbalancing indez as the Chebyshev norm (i.e. 11 ) - norm ofeach branch load balancing index

Smaz of [+,i = 1, .., b ]In comparison with the existing system load balancing in -dex which is defined as the 12-norm of each branch loadbalancing index, the proposed index is more reflective ofthe objective-load balancing. Th is is because minimizingthe &norm of each branch load balancing index may re-sult in one very small and another very large branch loadbalancing index even though the total sum of the squaresof each index is minimized. On the oth er hand , minimizingthe Chebyshev norm of each branch load balancing indexleads to adjusting each branch load balancing index, thu sachieving the objective of load balancing. Of course, froma mathematical point of view, the 12-norm is more desir-able because it is differentiable but the Chebyshev norm isnot.

Now, we discuss the issue of coordin ating th e above twoobjective functions; namely (1) loss reduction and ( 2 ) oadbalancing. Since these two objective functions are incom-mensurable (or even incompatible), it is not appropriateto combine them together in to a single objective functionsuch as a linear combinatio n of these two functions. Wepropose the following multi-objective formulation for thecost function of the problem

In summary, given a radial distribution system of IInodes, we seek an optimal radial configuration g' (i.e. anoptimal spanning tree) among aU possible radial configu-rations g; by changing the open/closed s tat us of the sec-tionalizing and tie switches such that both loss reduct.ionand load balancing are optimized while load constrailitsand opera tional constra ints are satisfied. In mathelllaticalterms, this problem is expressed asP;" + Q?min ET,-9. IV*2nb*= 1


4/8

G(r,g,) 5 0 (23)N ( g O , g v ) 5 n d (24)The above formulation of the network reconfigurationproblem is a constrained, multi-objective and non-

differentiable optimi zation problem. A solution algor ithmto solve the problem will be presented in a companion pa-per [lo].

M u l t i - o bj e c t i v e Op t i m i z a t i o n ProblemsWe consider t he following general multi-objective optimiza-tion ( M O ) problem.

min f l (z )

such that

A major distinction between a M O problem and a tradi-tional single objective problem is the lack of a completeordering of th e fa's, i = 1,2, ..,m. In th e single objec-tive problem, say c(x), a point z* s a global minimumif c(z*) 5 c(z) for all x lying in the region of interest. Inth e MO problem, a point x such that every component off simultaneously reach its global minimum is usually non-existent. Thi s occurs especially when some compon ents off compete so that one component of f decreases while an-other increases. In oth er words, when objectives competethere is no "optimal solution" t o the MO problem. In thiscase, the concept of non-inferior (also known as eficiency,Pareto optimality) is used to characterize the solution toth e MO problem.Defi niti on: Th e feasible region R is the set of state vec-tors x ha t satisfy t he constraints, i.e. fl = { x : F(x) = 0,Definiti on: A point i E 51 is a local non-inferior pointif there exists an e > 0 such that in the neighborhoodN(? ,E) of 5 , there exists no other point x such that (1 )f t ( z ) 5 f I (Z) , i = 1 ,2 ...,m and (2 ) f J ( z )< f J ( ? ) , forsome j E {1,2,...,m}.In other words, i is a local non-inferior point if there ex-ists a neighborhood N(i . , c ) such that for any other points E N ( i . , c ) , at least one component of f will increase itsvalue relative t o its value at i or f , ( z )= f,(i),=1,2 , ...,m.Definiti on: A point i E R is a global non-inferior point ifthere exists no other point x such tha t (1) fI(z) 5 fa(?),= 1, 2,...,m and (2 ) f J ( z )< f J ( 5 ) ,or some j E {1 ,2,...,m}.

G(x) L 0 1.

In general there a re an infinite number of (global) non-inferior points for a given MO problem. Th e collection ofsuch points is called th e non-inferior set . We call the imageof the non-inferior set trade-ofl (or non-inferior) surface.From a design point of view, a non-inferior point corre-sponds to an optimum trade-off design where attemp ts toimprove any ob jectiv e will lead to a degradation in at leaston e of the other objectives. Thus, the availability of th enon-inferior s et will allow a designer to reach a final design

1905based on a pre-defined, desired priority.

Th e most widely used meth od of generating non-inferiorpoints is t o minimize a non-negative convex combination ofthe functions f,, = 1, 2 ...,m, i.e. minimize a2f,},where a, 2 0 an d a, = 1 . This method s u f -fers from the drawback that it can not generate the en-tire non-inferior set. Particularly, t,he non-inferior poiot,swhose images are on the non-convex part of the trade-offsurface can not be found irrespective to what values a, reused. On the other hand, three methods that can gener-at e the entire non-inferior set have been developed: the-constraint method [ ll] , he weighted minimax method[12] and the shifted minimax function method [13]. Sincethe -constraint meth od will be incorporated i nto a solutionmethodology to be developed for the network reconfigura-tion problem, we briefly discuss this method.Th e -Cons t ra in t M e t h o dFor this method one of the objective functions is chosenas the primary objective function and all other object.ivefunctions as constraints. T o be specific, suppose that f, sthe primary objective function, then the method solves thefollowing constrained optimization problem:

min f,(z)t

such thatf J ( z ) e,, j = 1 , 2 ,...,m ; j # i

i q z ) = 0G(z) 5 0

For different values of , ' S . A systematic variation of E , 'Swill generate the entire non-inferior set.

It should be pointed out that the computational cost tofind the entire non-inferior set is usually very high, if notimpossible. Also, current methods (e.g. the above threemethods) to find th e entire non-inferior set are based onthe presumption that a method which can find the globaloptimal point for single objective problems is available.

From a practical p oint of view, a more desirable methodfor the MO problem would be the one that allow the de-signer to find a desirable, global non-inferior solution in areasonable computation cost. T o this end, we propose thefollowing two-stage solution methodology.

Stage 1: Find a global non-inferior point for the multi-objective optimization problem using the modifiedsimulated annealing technique to be presented be-low. (Let the point obtained at this stage be z * , andc, = f,(z'), j = l ,? ...,m. Also, suppo se the designerchoose fI be the primary objective function).

Stage 2: Apply the simulated annealing technique (for sin-gle objective function) to th e following problem whichis obtained by the application of the -constraintmethod to (27).

min fz(z) (17)such that

f j ( z ) 5 C J + ~ C J , = 1,?,. . . ,m;.? 1F ( z ) = 0G(x ) 5 0


5/8

r906where 6c, > 0 is a trade-off tolerance made by thedesigner.

S i m u l a t e d An n e a l i n gAn optimiza tion technique based on simulated annealing

for finding a global non-inferior point for an MO problemwill be developed in the nex t section. Therefore, we brieflydiscuss the simulated annealing in this section.

Simulated annealing is a powerful general-purpose tech-nique for solving combinatorial optimization problems.This technique is based on the analogy between the sim-ulation and the annealing process used for crystallizationin physical systems. Annealing is the physical process ofheating up a solid to a melting point, followed by coolingit, down until it crystalizes into a state with a perfect lat-tice. In this technique, a parameter called temperature isdefined, which is of the same dimension as the cost. Ju stas in a physical system, the temperature is closely relatedto the freedom with which the entities of the system canmove around. Th e system to be optimized starts at a hightemperature, and all the entities of the system move aboutfreely. T he tempe ratur e is then gradually lowered until thesystem freezes, at which point all the entities of the sys-tem are virtually fixed. This frozen configuration will beclose to the lowest energy (or cost) configuration.

At each te mperat ure, t he Metropolis algorithm derivedfrom statistical physics is employed to simulate the system[14].A set of moves is selected with which the system canalter from one st at e of configuration to another. Movesare chosen randomly. Moves tha t reduce th e system cost(i.e. improve the configuration) are called downhill an dmoves that increase the system cost are called uphill. Inthe algorithm, all downhill moves are accepted and u phill moves are accepted with a probability of e z p ( - % ) ,where A C is the increase in cost and T is the temperature.More specifically, acceptance for uphill moves is treatedprobabilistically in the following way: the Boltzman fac-tor e z p ( - q ) is first calculated and a random number runiformly distributed in the interval [0,1) is then chosen.If r < e z p ( - q ) , the (uphill) move is accepted; otherwisethe move is discarded and the configuration before thismove is used for th e next step. Physically, this means thatthe syst em will accept uphill moves with a reasonable prob-ability as long as these moves do not increase th e cost morethan T. Thus, a t very high te mperat ure the algorithm ac-cepts al moves and moves freely in th e configuration space(searching broad areas); at very low temperature the al-gorithm only accepts downhill moves and behaves like agreedy algorithm (see Fig. 3). It is due to th e probabilisticselection rule that the process can always get out of a localminimum in which it could get trapped and proceed to thedesired global optimum. This feature makes the simulatedannealing different from the greedy search approach.

Th e simulated annealing at each temperatur e can be rep-resented in the following pseudo-coderepeat

1. move2 . evaluate AC3. accept/updateuntil st op criterion = true

(a) Structure of Objective functionc

(b) High entropy.I-

(c) Decreasing entropy.r

DecreasingTemperatureI

(d) Lo w entropy.Figure 3: Aspects of search pattern during annealingprocess

Theoretically, this technique converges asymptotically tothe global opt imum solution with probability one, providedthat certain conditions on the cost function, the moves,and the annealing schedule are satisfied [13]. This tech-nique has been applied to VLSI systems , image processingsystems, antenna systems, neural networks and computersystem design with successful results [13-151.Th e simulated annealing can b e adapted to constrainedoptimization problems. One simply has to make sure thatall the moves generated from the Metropolis.algorithm con-tinue to satisfy the constraints of the system. This is imple-mented by adding a feasibility checking function betweenstep 2 (evaluate AC ) and step 3 (accept/update) of thepseudo-code. This added function serves to monitor thesatisfication of all the equality an d inequality constraints.Simul a ted Anneal ing for Mul t i -objec t ive ProblemsThe network reconfiguration problem can be casted intothe following constrained multi-objective optimizationproblem (witho ut loss of generality, two objective functionsare considered)

min CI(ZI,Z~) (28)min C 2 ( ~ 1 , ~ 2 ) ( 2 9 )

2 1

2 1

In this section we present a modified simulat,ed anneal-ing technique to find a global non-inferior point fo r multi-objective optimization problemsThe key point to find a global non-inferior point usingthe simulated annealing is the coordination of both [)er-

turbation scheme and the acceptance criterion. In order to


6/8

1907This error tolerance property can be used to reduce tl1ecomputation efforts by calculating cost functions appros-imately instead of exactly-at least at high temper ature .Another idea of appro xiniat e cost function calculat,ions isexplored in [15] to replace the Boltznian factor exp(-F)by -9,nd they report that this scheine may speed I I ~the solution algorithm by approximately 30 %.

The impact of the idea of approximate calculatioiis 011the network reconfiguration problem can be manifested i nthe calculation of (18), which itself requires a load flowstudy. The above analysis of error tolerance in cost fiinc-tion therefore suggests that in order to achieve a fast so-lution algorithm, an approximate load flow algorit.hiu beused at high temperature and a exact load flow algorithmbe used at low tem perat ure for the calculation of (18).repeat

1. move2 . compute AC

approzimotely at high temperatureezactly at low temperature

3. accept/updateuntil st op criterion = t rue

Conclusion

handle multiple objectives, we propose to coord inate bothperturbation mechanisms and the acceptance criterion inthe following way. At each tempe rature , the following pro-cedure is repeatedly executed.1. Apply a perturbation mechanism z; + z:+'.2 . Compute xi+' from F(z;+',z2)= 0 .3 . Check the feasibility

If G(zf+',zi+') 5 0, o to 4;Otherwise go t o 1.

4 . Se t z; = z:+' if one of the following conditions issatisfiedAC,(z') 5 0 an d ACz(z') 5 0;min [e-+, e-+] 2 random [0 ,1) .A C ( z ' ) A C ( = I )

Otherwise, leave it unch anged.Note that th e search direction of the abov e procedure do esnot require t hat th e differentiability property be associatedwith the objective functions and the constraint functions.Hence, the modified simulated annealing technique is a pplicable to multi-objective a nd non-differentiable optimiza-tion problems.

Computational Co n s i d e r a t i o n sWhile the advantages of simulated annealing are i ts poten-tial to find th e global optima l solutions, its general appli-cability and its flexibility, its main disadvantage is the po-tentially burdensome amount of computation time requiredt,o converge to a near-global optim al solution. Admittedly,the amount of computation time required strongly dependso n the nature and the size of the problem at hand. Thesituation with respect to th e computational effort becomesworse as problems increase in size. Therefore, it is im-porta nt to investigate possibilities of speeding up solutionalgorithms based o n simulated annealing in order t o keepcomputation times within desirable limits.

In general, speeding up solution algorithms can beachieved by means of the following three approaches:

1. design of a fast (sequential) solution algorithm,2. hardware acceleration, an d3 . design of parallel solution algorithms.

Hardware acceleration can be achieved by employing dedi-cated hardware for evaluating time-consuming parts in thealgorithm. Design of a parallel simulated annealing algo-rithm is a promising ap proach t o speeding u p the execu-tion of the algorithm. Bu t it is by no means a trivial taskt.0 design a parallel version of a simulated annealing algo-rithm mainly due to the intrinsic sequential nature of th ealgorithm. In this paper we focus on the design of a fast(sequential) solution algorithm.

iFrom the acceptance criterion of the simulated anneal-ing, one would intuitively expect that as long as the errori n the cost function calculation is much less than the tem-perature, the solution algorithm would still work. Th is isbecause the probability of accepting an uphill move, whichis e z p ( - Y ) , would no t change significantly if AC were tobe replaced by (ACf6) , where (6


7/8

1908IEEE Transactions on Power Dehvery, vol. 3, No. 3,July 1988 , pp . 1217-1223.[ A. hferlin and H. Back. Search for a Minimal-Loss Operat ing Spanning Tree Configuration in UrbanPower Distribution Systems, Proc. of 5t h Power Sys-tems Comp. Con., Cambridge, U.K., Sept. 1-5, 1975.

[J ] D. Shirmohammadi and H. W . Wong, Reconfigura-tion of Electric distribution Networks for ResistiveLine Losses Reduction, IEEE Transactions on PowerDelivery, vol. 4, No. 2, April 1989, pp.1492-1498.[4] D. W. ROSS, M. Cars on, A. Cohen et al., Develop-ment of Advanced Method s for Planning Electric En-ergy Distribution Systems, D E0 final report no SCI-5263, Feb 1980.

[5] K. Aoki, H. Kuwabara, T. Satoh, and M. Kaneza-shi, An Effecient Algorithm for Load Balancing ofTransformers and Feeders by Switch Operation inLarge Scale Distribution Systems, IEEE PES Sum-mer Meeting, 1987, paper no: 87SM 543-2.

[6] K. Aoki, T. Satoh ,M. Itoh et al., Voltage Drop Con-strained Restoration of Supply by Switch Operationin distribution Systems, IEEE PES Summer Meet-ing, 1987, paper no: 87SM544-0.

[7] M. E. Baran, F. F. Wu, Network Reconfigurationin Distribution Systems for Loss Reduction and LoadBalancing, IEEE Transactions on Power Delivery,vol. 4, No. 2, April 1989, pp . 1401-1407.

[8] H.D. Chiang an d M. E. Baran, On the Existence andUniqueness of Load Flow Solution for Radial Distri-bution Power Networks, to ap pear in IEEE Trans.on Circuits and Systems, Vol. CAS-37, 1990.

[9] M.R. Lightner and S.W. Director, Multiple CriterionOptimization for the Design of Electronic CircuitsIEEE Trans. on Circuits and Systems Vol. CAS-28,Mar. 1981, pp. 169-179.

[lo] Y.Y. Haimes, W.A. Hall and H.T. Freedman, Multi-objective Optimization in Water Resources Systems,New York: Elsevier Scientific, 1975.

[ l l ] D. Mitra, F. Romeo and A. Sangiovanni-Vincentelli ,Convergence and Finite-Time Behavior of SimulatedAnnealing, Proceeding of 24th Conference on Deci-sion and Control, Dec. 1985, pp. 761-767.

I131 U. Faigle and R. Schrader, On the Convergence ofStationary Distributions in Simulated Annealing Al-gorithms, Information Processing Letters, Vol. 27,

[13] P.J.M. Laarhoven and E.H.L. Aarts, Simulated An-nealing: Theory and Applications, Reidel, Dordrecht,1987.

[14] S. Kirkpatrick, C. D. Gelatt Jr., M. P. Vecchi, Opti-mization by Simu lated Annealing, Science, vol. 220,1983, pp . 671.

[15] D.S. Johnson, C.R. Aragon, L.A . McGeoch and C.Schevon, Optimization by Simulated Annealing: anExperimental Evaluation: Part I, AT&T Bell Labo-ratories, Murray Hill, preprint.

[IG! H.D. Chiang and R.M. Jean-Jumeau, Optimal Net-work Reconfigurations in Distribution Systems: Part2: A Solution Algorithm and Numerical Results ,submitted to IEEE PES Winter Meeting 1990.

1988, pp. 189-194.

Hsiao-Dong Chiang received the B.S. and M.S. degreesin electrical engineering from National Taiwan U n i v e r s i t y ,Taipei, Taiwan. He spe nt 1982-1983 as a faculty member atChung-Yuan University i n Taiwan. He received the PI1.D.degree in electrical engineering and computer sciences fromthe University of California at Berkeley in 1986, and the11worked at the Pacific Gas and Electricity Company 011 aspecial project. In 1987, he joined the faculty of CornellUniversity, where he is currently an Assistant Professorof School of Electrical Engineering. In 1988 the NationalScience Foundation named Chiang a recipient of an En-gineering Research Initiation Award. In 1989, Dr. Chi-ang received a Presidential Young Investigator Award fromthe National Science Foundation. He is a member of thetechnical committee on Nonlinear Circuits a nd Systems, amember of th e technical committee on Power Systems andPower Electronics and Circuits, IEEE Circuits and SystemsSociety. His research interes ts include nonlinear systems,power systems , control systems an d optimization theory.Red Jean-Jumeau studied Electromechanical Engi-neering at the State University of Haiti, Port-au-Prince,

Haiti, where he obta ined his B.S. He was awarded a grad-uate fellowship by the Organization of American Stateswhich enabled him to atten d Cornell University. He iscurrently in the Ph.D. program in Electrical Engineeringat Cornel1 University and s tudying with Professor Chiang.His research int erests involve power s ystems, optimizationtheory and nonlinear systems.


8/8

DiscussionRoss Baldick, University of California, Berkeley, CA 94720.This paper applies simulated annealing to t he network reconfigu-ration problem, for which there does not seem to be any fast deter-ministic algorithms. This is a practical approach t o an otherwiseintractable problem and can potentially integrate many aspectsof distribution system control into a single framework. On theother hand , reconfiguration without consideration of, for exam-ple, switched capacitor settings may result in unnecessary losses:consider th e elementary distribution system shown in figure 1. Itconsists of a load th at can be fed from either of two resistive linesand two switched capacitors conn ected to the ends of th e lines.The losses will be calculated under various switch configurations;the initial configuration is indicated in figure 1. For ease of cal-culation, the loads are shown as current sinks, although similarresults could be found with any sort of load-voltage depend ence.Furthermore, assume that load balancing is not a limiting factor.

Substation7Figure 1: Initial configuration of system.

The initial losses are 2 kW . First suppose that capacitorswitching is not included in the reconfiguration optimization:then the only feasible alternative is to close switch Sz and openswitch S3, yielding losses of 4 kW. However, a superior solutionincluding capacitor switching is to close switches S and Sz andopen switches S3 and S4, yielding losses of only 1 kW . This ex-ample indicates the importance of capacitor coordination in dis-tribution system loss minimization.

Capacitor switching is treatable within the simulated anneal-

1909ing framework just as simply as branch interchange, but wouldsignificantly increase the dimension of the problem if there aremany switched capacitors in the system. Instead, I suggest thatth e configuration space of t he simulated annealing algorithm belimited to actual branch exchanges and tha t capacitor optimiza-tion be performed as part of th e evaluation of th e losses for eachconfiguration: fast algorith ms are available to perform t he capac-itor optimization and evaluate the minimum losses, given a net-work configuration [A]. In this way, fast algorithms can be usedon those parts of the problem tha t are tractable, while still treat-ing the difficult pa rts of th e problem using the general simulatedannealing algorithm presented in this paper.

Reference[A] Ross Baldick and Felix F. Wu. Efficient Integer Optimiza-tion Algorithms for Optimal Coordination of Capacitors and Reg-ulators. Paper 90 W M 175-0 PWRS presented at th e IEEEPower Engineering Society 1990 Winter Meeting, Atlanta, Geor-gia, February 4-8, 1990.

Manuscript received March 5, 1990 .

Hsiao-Dong Chiang and RenC Jean-Jumeau: We would like to thankM r. Baldick for an interesting discussion. First, it should be pointed outthat much of the research in the optimal operation and design of electricpower systems consists of two stages of efforts: ( 1 ) problem formulationand (2 ) development of solution algorithms. These two are inter-related. Itis quite common to approximate problem formulations to make the prob-lems under study solvable with available optimization techniques. This canbe exemplified by the work on the optimal power flow problem as well assome of the work on distribution systems. The generality of the simulatedannealing technique makes it possible to decouple these two inter-re-lated process. The technique allows us in the process of formulatingproblems to adhere to normal principles of power system operation so thatthe problem is expressed in a realistic way. In this paper we havepresented a new formulation of the network reconfiguration problem forboth loss reduction and load balancing with load and operational con-straints. The new formulation is a multi-objective, non-differentiableoptimization problem with equality and inequality constraints. Second, itis strongly believed that multiple local optimal solutions are likely to existin many power system problems. A solution algorithm based on simulatedannealing can achieve global optimal solutions rather than just localoptimal solutions. In practical implementation, the solution algorithmstarts with any initial condition and converge to a near-global optimalsolution. In this paper we have developed a two-stage solution methodol-ogy based on a modified simulated annealing technique for general multi-objective optimization problems and applied it to the network reconfigura-tion problem.It is always desirable to be able to optimally coordinate all the availabledevices, such as network reconfiguration, capacitor switching, voltageregulators adjustment, to reduce real power losses in distribution systems.M r. Baldicks suggestion of using the simulated annealing technique todetermine network reconfigurations and using fast algorithms developedby Baldick and Wu in [A ] to determine capacitor switchings appearsattractive. Further assessments of the effectivenessof both this suggestionand the scheme that uses the simulated annealing technique to determineboth network reconfigurations and capacitor switchings on practical distri-bution systems may prove rewarding.Manuscript received April 6, 1990.

optimal network reconfigurations in distribution systems

Documents