Object-oriented load flow for radial and weakly meshed distribution networks

IEEE TRANSACTIONS ON POWER SYSTEM,VOL.18,NO.4,NOVEMBER 20031265Object-Oriented Load Flow for Radial and WeaklyMeshed Distribution NetworksArturo Losi ,Member,IEEE,and Mario Russo ,Member,IEEEAbstract—Object-oriented load flow modeling is presented forboth radial and weakly meshed distribution systems.An OO algo-rithm based on the Newton–Raphson technique is proposed.In the object oriented formulation,some approximations to the full Jaco-bian matrix are introduced.Consequently,a detailed study of the convergence characteristics of the proposed object oriented algo-rithm is presented and some sufficient conditions for convergence are derived.Particular attention is paid to the relationship between the electrical parameters of the distribution system and the math-ematical parameters that influence the convergence properties of the algorithm.The numerical results obtained in the case of some test systems give evidence of the features of the algorithm.Index Terms—Convergence of numerical methods,load flow analysis,Newton–Raphson method,object-oriented methods,object oriented programming,power distribution.I.I NTRODUCTIONLOAD flow analysis is a basic function in modern distribu-tion management systems (DMSs)[1].In literature,many methods and solving algorithms have been proposed for dis-tribution load flow analysis.They can be essentially classified into three categories:direct methods,backward/forward sweep methods,and Newton–Raphson (NR)-based methods.Direct methods [2],[3]require the construction of an impedance matrix and the direct solution of some equations in theformdegli Studi di Cassino,Cass Digital Object Identifier 11266IEEE TRANSACTIONS ON POWER SYSTEM,VOL.18,NO.4,NOVEMBER2003the previously presented OO-based load flow modeling is ex-tended to account also for weakly meshed distribution systems. The distribution load flow equations are revised to include both real and imaginary components of bus voltages.An additional component is introduced to model looping switches.An OO al-gorithm based on the NR technique is proposed.In the OO for-mulation,some approximations to the full Jacobian matrix have to be introduced.Consequently,a detailed study of the conver-gence characteristics of the proposed OO algorithm is presented and some sufficient conditions for convergence are derived.Par-ticular attention is paid to the relationship between the electrical parameters of the distribution system and the mathematical pa-rameters that influence the convergence properties of the algo-rithm.The numerical results obtained in the case of some test systems give evidence of the features of the algorithm.II.O BJECT O RIENTED D ISTRIBUTION L OAD F LOW M ODELING Classes are the key concept in the OO modeling of a system. They are set of objects that share the same attributes,properties, and behavior.In the load flow problem modeling,the base class is an ab-stract class named connection.It has ports and it represents some relations between port variables.Each port is characterized by four variables,which represent the active and reactive powers and the real and imaginary components of the voltage at the port.A connection port is referred to be incoming or outgoing according to whether the conventional direction of powers is as-sumed to be inwards or outwards oriented.From the connection class,all of the concrete classes needed to load flow problem modeling are derived:•branch is a connection with a single incoming port anda single outgoing port and it models a physical line or atransformer;•root is a connection with a single outgoing port and it models the supplying system at higher voltage level;•fork is a connection with a single incoming port and multiple outgoing ports and it models a zero-impedance busbar;•terminal is a connection with a single incoming port and it models the“dead”end of the distribution system;•switch is a connection with two incoming ports and it models a looping switch.The diagram in Fig.1shows the class hierarchy with a higher generalization level(the class connection)and a lower special-ization level(the concrete classes).A specific distribution system is modeled by an oriented graph of object instances.The association of instances is obtained by connecting outgoing ports to incoming ports.The graph is built according to the following constraints:•an outgoing port can be connected to only one incoming port;•an incoming port can be connected to only one outgoing port;•no port can be left unconnected.Consequently,only root objects can be the origins of the ori-ented graph and only terminal or switch objects can be the ends of thegraph.Fig.1.Class hierarchy for distribution load flowmodeling.Fig.2.Simple distribution system:(a)one-line diagram;(b)oriented graph of instances in the OO modeling.The oriented graph of object instances is divided into sec-tions.A section is defined as a sequence of connections,starting with either a root or a fork,ending with either a fork,or a ter-minal or a switch,and with only branches in between.Let us assume that the networkhasreferred tothe section.For example,Fig.2(a)shows the one-line diagram and the re-lated oriented graph of instances[Fig.2(b)]for a simple distri-bution system.The graph is composed of a main section(section #0)which starts with the root and ends with a fork.Two subse-quent sections(sections#1and#2)starts with the fork and end start with a switch.In this simplesystem,,,and.In the following,the equations that characterize each class are analyzed.They refer to steady-state balanced operation of distribution systems.A.BranchBranches are used to model physical lines and transformers, including loads and shunt compensating devices at the ending busbar.Fig.3shows the equivalent electrical circuit assumed forthe section.The incoming port variablesare,,and,which represent, respectively,the incoming active and the reactive powers and the real and imaginary parts of the voltage,all referring to the starting bus.The outgoing port variablesareLOSI AND RUSSO:OBJECT-ORIENTED LOAD FLOW FOR RADIAL AND WEAKLY MESHED DISTRIBUTION NETWORKS1267Fig.3.Branch equivalentcircuit.(5)(6)whereand are,respectively,the real and theimaginary components of the slack bus voltage phasor,whichare constant values.C.ForkThe fork class represents a zero-impedance busbar that con-nects an incoming port to two or more outgoing ports.It is theending connection of the incoming section and the starting con-nection of the outgoing sections.With reference to a fork whichis located afterthe branch at the end of the section andwhich supplies the sections that belong to thesetbranch at the end ofthe section,the equations that are forced on the port vari-ablesare(11)(12)E.SwitchThe switch class is used to represent a looping switch indistribution systems with meshed topology.A switch hastwo incoming ports and two states—open or closed.In theopen(closed)state,it represents an infinite impedance(a zeroimpedance)connecting the two ports.Any other logical devicecan be represented either by a combination of two or moreswitch objects or by a further specialization of the class switch.Let us consider the switch that connectsthe branch atthe end ofthe section tothe branch at the end of thesection.In the open state,the switch model yields,for eachsection,a couple of equations of the same type as the terminalones(11),(12).In the closed state,the switch is modeled by thefollowing relations between the two sets of portvariables:1268IEEE TRANSACTIONS ON POWER SYSTEM,VOL.18,NO.4,NOVEMBER 2003Also,in the case of NR techniques,the construction of a system matrix,namely the Jacobian matrix,is required.How-ever,in the following,it is shown how some approximations to the Jacobian matrix allow it to easily implement an OO algo-rithm based on the NR technique.In addition,the NR technique usually converges to solution in a lower number of iterations than the backward/forward sweep methods [13],[16].Such feature is particularly important be-cause the OO programming increases the computing burden of each iteration,due to message passing among objects.In conclusion,the OO distribution load flow algorithm is chosen to be based on the NR technique.The main features of the related OO program are local processing,message passing among objects,and graph traversing techniques.In the scope of the present paper,only the features of the OO algorithm are studied,while readers are referred to [23]for more details concerning the OO software implementation.A.NR AlgorithmThe classical NR algorithm is composed of two phases:1)evaluating the current values of all the variables and,for such values,the errors and the Jacobian matrix of the equations;2)correcting the values of the variables using the errors and the Jacobian matrix.The OO algorithm is derived from the OO problem modeling presented in Section II.In the problem domain,each class has been assigned some equations.Then,each class contributes to the algorithm with a specific method for each one of the above two phases.For the sake of conciseness,vector notation is adopted;all vectors are column vectors.The values of the outgoing port vari-ables ofthesection duringthe iteration is performedby a downstream graph traversing.Starting from the root,com-putation is performed by each object using new values of the incoming port variables.After computation is completed,new values of the outgoing port variables are passed downstream to-ward the next connection.In this way,the computation of the first section (section #0)is carried out.Then,the computations related to the other sections are performed in sequence by prop-agation from the starting connection,through the branches,to the ending connection.Let us refer tothesection:its computation is initiated byevaluating,the starting connection is a root and (5)and (6)are used to evaluate thevoltages.In other cases,the starting connection is a fork.Let us assume,according to (7)–(10),thatthesection is one of the sections that are connected to the end of theing the values available from thesection,in particular thevalues are evaluated by (9)and (10),respectively.In all cases,the root and the forks correct the values of activeand reactive powers using theresultssection carry out their com-putations in sequence.The load flow equations (1)–(4)are eval-uatedforto obtain thevaluesto the Jacobian matrix re-lated tothesection is evaluated by each branch.In partic-ular,thesection receives the Jacobianmatrixfrom the preceding branch,evaluates the Jacobianmatrix(20)and sends it downstream to the next connection.Eventually,the Jacobianmatrixsec-tion is available at the ending connection of the section.The following partition of the matrix into four2sectionto thesection from (13)–(16),the error is obtainedassection is a forkwhich supplies the sections that belong to thesetiteration is per-formed by an upstream graph putation starts from the terminal and the switch objects of the last sections.Forthesection,thecorrections are eval-uated by the ending connection,as described in the following,and sent upstream to the starting connection.Then,computa-tion continues with the previous sections and,eventually,with the first section.LOSI AND RUSSO:OBJECT-ORIENTED LOAD FLOW FOR RADIAL AND WEAKLY MESHED DISTRIBUTION NETWORKS 1269If the ending connection ofthe section is a terminal or an open switch,it evaluates the corrections by [see (21)and(22)]section tothesection,it evaluates the corrections by [see (21)and(23)](26)wheresection is a forkwhich supplies the sections that belong to thesetare the corrections available fromthe downstream sections supplied by the fork.B.Simple System ApplicationLet us consider the simple system shown in Fig.2as a working example.In the following,the approximations to the full NR technique introduced by the proposed algorithm are analyzed in the two cases of radial and meshed topologies,respectively,according to whether the switch is in the open or closed state.1)Radial Topology:In the case of open switch,the OO modeling [see Fig.2(b)]yields the following load flow equa-tions written in a compact formulation.From (7)–(10),the following equations arederived:(29)(32)where the componentsof,(38)whereis given by (24)for,by (22)for(40)the null matrix.Let us now consider the proposed OO algorithm applied to the simple system.Corrections are obtained by (25)specifiedfor,(41)with(43)with1270IEEE TRANSACTIONS ON POWER SYSTEM,VOL.18,NO.4,NOVEMBER2003 fork object,(29)–(32)still stand.Concerning the closed switch,(13)–(16)can be written as[similar to(35)and(36)](44)(45)(46)given by(37).Applying a full NR solving technique to such nonlinearproblem requires evaluating thecorrections to beappliedto(48)beingby(23)(specifiedfor),and where(29),(30),and(44)–(47)hold,and(28)for,is a2m a t r i x g i v e nb y(49)w i t h t h e f o l l o w i n g a p p r o x i m a t e d J ac o b i a n m a t r i x:(52)w i th4matrix for eachsection,see(21)and(25)—(28).On the other hand,the proposed OO algorithm is essentiallyan approximated NR technique.In both the cases of radial andweakly meshed topologies,the approximation can be expressed,see for example(43)and(52),in terms ofdifferenceand thematrixstep of the iterative solution.Then,it is important to study the convergence characteristics ofthe OO algorithm.A.Sufficient Conditions for Local ConvergenceAs well known from numerical analysis[24],the full NRtechnique assures convergence to a solution with an order whichis at least equal to two in the case of quadratic convex prob-lems.Due to(1)–(4),the distribution load flow problem is notquadratic.Yet,the above convergence characteristics are guar-anteed if the starting point belongs to a closed convex set thatincludes the solution.Various methods have been proposed inliterature[25]to define sufficient conditions for the existenceof such a region and to determine its dimension.Following the trace of classical numerical analysis of the fullNR algorithm,assuming that the Jacobianmatrixiteration of the proposed approximated NRalgorithm:(53)where is the2-norm(the Euclidean length forvectors),,that is the square root of the biggest eigenvalue of thesymmetric non negative definitematrix.From(53),the following can be easily derived:Theorem1:If the Jacobianmatrix is Lipschitzwithconstant,then there ex-ists—to be satisfied atthe,if thesolutionand,beingLOSI AND RUSSO:OBJECT-ORIENTED LOAD FLOW FOR RADIAL AND WEAKLY MESHED DISTRIBUTION NETWORKS 1271If the sufficientconditions of Theorem 1holds,then comes the problem of choosing a starting point of the al-gorithm which is “sufficiently close”to the solution according to the concept of region of convergence in Theorem 1.In prac-tice,it is not a very difficult problem.The choice of an initial startingvalue[see,for example,(43)and (52)],it can be easily proved [26]thatandisstrongly dependent on the values of the electrical parameters of the distribution system which affect the componentsofis the 1-normof,and where -normofandand,consequently,the higher the order ofconver-Fig.4.Index of Convergence (IC )of the algorithm for the radial topology:a)increasing loads;b)increasing the impedances of sections #1and #2;and c)increasing the impedance of the supplying transformer (section #0).gence.The right-hand side of (54)can be directly related to the electrical parameters of the distribution system that affect the absolute values of the elementsof(57)Obviously,theis not strictly the order of convergence,but it is a useful index of the actual convergence performance of the algorithm.With reference to Fig.4,in case a)the loads of the two sec-tions #1and #2are increased,yielding an increase of active and reactive losses along the two sections as well as an increase of1272IEEE TRANSACTIONS ON POWER SYSTEM,VOL.18,NO.4,NOVEMBER 2003the voltage drop on the supplying transformer.In case b),the line impedances of the two sections #1and #2are increased,yielding an increase of active and reactive losses.Finally,in case c),the impedance of the supplying transformer is increased,yielding an increase of the voltage drop along section #0.In all three cases,due to the changes of the values of the elec-trical parameters,the value ofof the algorithm decreases almost linearly in a logarithmic scale.It should be noted that the highest values ofstill guarantee conver-gence of the algorithm (in fact,in all cases.2)Meshed Topology:In the case of a meshed topology (closed switch in Fig.2),reference is made to (52).Concerning the termin (52)presents another submatrix,thatis,which is a further contributionto the value ofand theLOSI AND RUSSO:OBJECT-ORIENTED LOAD FLOW FOR RADIAL AND WEAKLY MESHED DISTRIBUTION NETWORKS1273Fig.6.Graphical OO representation of the 69-branch test system (arrows evidence additional switches).TABLE I S WITCH POSITIONTABLE IIA LGORITHM P ERFORMANCE W ITH O PEN SWITCHESTABLE IIIA LGORITHM P ERFORMANCE W ITH C LOSED SWITCHESFrom Tables II and III,it is apparent that:•in all cases the order of convergence is at least unitary;•increasing the load level worsens the performance in terms of convergence rate (see Section IV-C);•closing loops worsens the convergence performance with respect to radial topology for a given load level (see Sec-tion IV-C);•closing all of the switches allows to significantly increase the loading capability of the distribution system.VI.C ONCLUSIONThe OO paradigm has been applied to the load flow problem for radial and weakly meshed distribution systems.A robust OO algorithm based on NR technique has been proposed,which proves to reach convergence in few iterations,even in extreme operating conditions of weakly meshed topologies.Each iter-ation does not require heavy computations,although the OO approach introduces a message passing overhead.On the other hand,the OO-based modeling gives new features to the soft-ware,such as flexibility,extensibility,easiness of maintenance,and can best exploit the advantages of distributed computing in open systems architectures of modern DMSs.Future work will concern the inclusion of the models of voltage-dependent loads,of distributed generation and of other components,as well as the extension to unbalanced distribution systems.A PPENDIXIn the following,equation (53)is derived for a generic ap-proximated NR algorithm.Atthe.Then,(58)can be writtenas1274IEEE TRANSACTIONS ON POWER SYSTEM,VOL.18,NO.4,NOVEMBER2003 Concerning the second term in(60),it is given by[28](62)moreover,sinceit can be expressed as(63)and substituting(63)foris assumedto have Lipschitz derivative with constant。