WO2004052073A2 - Optimal feeder design in distribution system planning - Google Patents

Abstract

The distribution feeders, which are restricted to `available' areas, having Right of Way (ROW), can be interconnected in several ways, each leading to a different feeder design. Identification of Optimal Feeder Design (OFD), which minimizes length and loss of the feeder layout, is an important NP complete combinatorial optimization problem. Conventional designs like Minimum Spanning Tree (MST) minimizes capital investment, but neglects running cost of feeder losses. Feeder reconfiguration is then used to minimize loss of MST, wherein loss reduction strongly depends upon the existing network. OFD layout, developed using this method does not require any existing network. This method further calculates a Lower Bound (LB) on feeder cost, thereby avoiding exhaustive search for the global minimum. This method also defines a set of cofeeders, each in series with a Normally Open (NO) switch, to transform OFD into Single Fault Tolerant (SFT) OFD layout. The method also completely defines solution space of all feasible networks, including their representation. The method places heavily loaded feeders close to source, thereby ensuring substantial loss reduction at minimal cost of additional feeders. An exemplary embodiment of the invention, when applied to a conventional layout, results in loss reduction by 67%.

Description

OPTIMAL FEEDER DESIGN IN DISTRIBUTION SYSTEM PLANNING

FIELD OF INVENTION

The invention relates to identification of OFD layout for a distribution system, from the plurality of feasible layout, which minimizes the sum of feeder length and feeder loss in the layout. The distribution system comprises of a source node and a plurality of sink nodes positioned on a layout map having some specified area having Right of Way, which is earmarked for laying of feeder. The method of present invention calculates a Lower Bound for feeder loss in the distribution system thereby avoiding exhaustive search for layout with lowest loss. The method also identifies a minimal set of NO switches, which convert OFD layout to SFT OFD. BACKGROUND OF THE INVENTION

A radial system is generally used to distribute electrical power, drawn from a single source, to multiple sinks located on a layout map of the service area of the source. A feeder is an edge connecting two nodes in the distribution system, which is always restricted to 'available' area of the layout map having ROW. Any feeder network has a capital cost given by cost of total length of all feeders in the network and running cost given by cost of total loss in all the feeders. The ownership cost of feeder system is the sum of a capital cost and present worth of the running cost spread over a time period. The feeder designing is a process to decide which nodes are to be connected, as well as, identifying a feeder route for each connection, from a plurality of alternate feasible routes. OFD, which minimizes the ownership cost of feeder network, is an important component of distribution system planning. The complexity of OFD problem, a NP complete problem, is due to large number of discrete switching variables, quadratic objective function and a plurality of feasible network configurations, which are related to geography and distribution requirement of the service area.

A single fault in any feeder splits the radial feeder network in two sections, namely 'an isolated section' and 'a healthy section'. A SFT feeder network is one in which in spite of a fault in any feeder, a path still exits from the source node to every sink node in the distribution system. Any radial feeder layout can be converted into a SFT layout by introducing a minimal set of appropriately located cofeeders, each cofeeder in series with NO switch. A cofeeder as special type of feeder is required for achieving SFT feeder network, such that the cofeeder has non zero distribution only in event of a fault and zero distribution otherwise. For such single fault in the feeder, there is a cofeeder in series with a NO switch, such that by closing of the NO switch, each disconnected sink node in the isolated section gets reconnected to the source node through the cofeeder. In such SFT layout, uniform size of feeder is maintained throughout so that all feeder segments, in the alternate longer route for reverse feed has adequate capacity to carry the increased distribution.

An area incorporating town planning has a grid of roads with blocks placed in between them. Under these conditions, only ROW available is along roads parallel to X or Y axes and hence, the feeders can be routed only along these two axes. Such restrictions on feeder routing lead to a multiple alternate feeder routes between any two nodes on the grid, all routes having same length. Further the feeder length between two nodes on the grid is given by the 'First norm' of the distance between them, which is more than usual ' Euclidian norm'. Thus, incorporation of grid structure in feeder layout results in two problems. The first problem is the increase in the feeder loss due to increased length associated with the use of First norm. The second problem is the increase in the number of alternate feasible feeder layouts due to the existence of multiple alternate feeder routes of equal length.

In a grid based distribution system having nonzero distribution requirement at all nodes, the uniform grid spacing ensures that the connection length of all layouts is the same. Under such constraints of equal connection length and uniform size of feeder, the loss of the feeder network depends only on the interconnection of feeders. In view of equal connection length and hence equal capital cost, the OFD layout for such cases minimizes the loss in the distribution network.

Since, feeder designing for identifying OFD is so complex, in practice the feeder designing problem is generally approached in two ways, namely Greenfield planning and reconfiguration. Greenfield planning, which involves development of new distribution system in an area where old feeder layout does not exist, deals with selection of a particular network from a large number of feasible network configurations. Reconfiguration approach deals with some local changes in an existing network configuration, by using a limited number of available switching options. The loss reduction obtained by reconfiguration method is strongly dictated by the 'goodness' of existing network and available switching options. In view of the complexity of feeder designing problem for identifying OFD, a NP complete problem, the conventional feeder designing methods consider minimization of capital cost as the primary objective. The conventional methods then attempt to reduce losses in such 'pseudo optimum' network by applying reconfiguration methods. Such sequential application of these methods simplifies the feeder designing problem. However, due to dependence of loss on the layout, this sequential approach does not result in minimum ownership cost of feeder system.

A book titled "Power Distribution Planning Reference Book" by H. Lee Willis, Marcel Dekker Inc.1997, chapter 8, discusses the problem of feeder design. It lists four types of radial feeder layouts, namely main trunk, multi branch, feathered and mixed type. It mentions taxicab measurement of distance, grid of road and blocks, considers branching /splitting of feeders, uniform feeder size. However, the disclosed methods do not consider selection of layout based on lower loss, does not calculate lower bound on the distribution loss, can not handle feeders with unequal resistances, does not define complete solution space of feasible layouts, does not give exact number of different layouts and does not provide method to identify optimum feeder layout.

The publication titled " Optimization Applications To Power Distribution", IEEE Computer Applications in Power Oct 1995 Pages 13-17, by H. Lee Willis, Hahn Tram, Michael V. Engel and Linda Finley, addresses three aspects of distribution configuration, namely layout, switching and sizing. It discusses two main styles of layout, namely main trunk and multi branch feeder [Refer Fig 1 and Fig 2]. It defines layout as m anner i n which, the feeders a re s plit a nd resplit from o ne s ource to m any s inks. It also mentions about combinatorial challenges in design, which is the selection of OFD layout from a very large number of feasible layouts. Since all spanning tree layouts for a continuous distribution system on a grid, wherein all sink nodes have nonzero distribution requirement, h ave the same feeder l ength, considerations like reliability or minimum feeder length, do not help in selection of layout based on lower loss. However, the disclosed methods do not consider selection of layout based on lower loss, does not calculate lower bound on the distribution loss, can not handle feeders with unequal resistances, does not define complete solution space of feasible layouts, does not give exact number of different layouts and does not provide method to identify optimum feeder layout.

The publication further states "Most planners and engineers have preference for a particular approach to layout, but there are no firm guidelines within the industry and standard practice vary widely". It also mentions "Thus, structured design procedure such as those that are in use in most utilities, which begin with a recommended layout rule generally fall short of achieving the lowest possible overall cost....

Such procedures result in workable design.... That cost can be reduced between 5 to 10 percent by application of optimization..." These remarks reaffirm that there are no standard procedures or industry guidelines for solving OFD problem.

The p ublication titled " Optimum F eeder Designing In Distribution System Planning Using Dynamic Programming Technique And GIS Facilities" by M.P. Papadopoulos and Nicholas G. Boulaxis, IEEE Transaction on Power Delivery. Jan 2002 Page no 242 to 247, claims optimal feeder designing using Dynamic Programming. This work incorporates grid restricted feeder ways and GIS techniques, considers investment cost as well as cost of losses in its objective function. However, this work does not identify the optimum feeder layout, does not calculate the lower bound on the distribution losses, does not define complete solution space of feasible layouts and does not give the exact number of different layouts. The method uses downsizing of feeders in service area away from the source node. These downsized feeders do not have adequate feeder capacity to carry the increased distribution requirement for feeding a ll sink nodes in the isolated section associated with the fault. Thus the prior art feeder network cannot be converted into SFT network.

The publication titled 'Application Of Evolutionary Algorithms For The Planning Of Urban Distribution Networks Of Medium Voltage' by Cirdas and Miguez, IEEE Transaction on Power Systems Aug 2002, deals with optimization of investment and loss cost for HV and MV networks, by using Genetic Algorithm. In this work, the authors have considered solution space of loop feeder network restricted to urban maps. The genetic algorithm involves selection, crossover and mutation operators. As in the previous studies, this work does not identify OFD, particularly in the absence of layout diagram of service area, does not consider variation in distribution requirement at the sink node, does not ensure unique mapping from the domain of feeder layout to the codomain of chromosomes, does not ensure radial nature of optimum layout corresponding to chromosome solution of Genetic Algorithm, does not calculate lower bound on the distribution loss, does not define complete solution space of feasible layouts, and does not give exact number of different layouts To summarize, the prior art therefore can broadly be classified into Greenfield planning and reconfiguration methodologies. The major challenge in green field planning is to select a design of the new system, from a wide range of alternate layouts. In such cases, the conventional design methodologies refer to some preferred feeder layout design [main trunk, multi branch, feathered and mixed layout etc.]. This preference to preferred layout in feeder designing does not minimize losses. The reconfiguration methodology seeks to reduce loss of existing network, wherein the loss reduction achievable is restricted by limited switching actions and 'goodness' of existing network. The glossary of various relevant terms from prior art is as follows.

1. A distribution system is a set of unconnected nodes, without a single edge, for which OFD is to be identified, wherein the set comprises of a source node and a plurality of sink nodes, each sink node with a distribution requirement, located on a layout map.

2. The layout map is divided into an 'available' area and an 'unavailable' area based on 'Right of Way' consideration, which decides whether or not feeder can be laid in the area.

3. A feeder is an edge connecting two nodes in the distribution system, such that the feeder is always restricted to the available area of the layout map.

4. A feeder network is a set of feeders, wherein the feeders form a tree s panning a ll the s ink nodes, with source at the root, such that there is only one path from the source node to every sink node in the distribution system.

5. A SFT feeder network is a set of feeders and cofeeders, wherein addition of cofeeders to the feeder network ensues that in spite of a fault in any feeder, a path exists from the source node to every sink node in the distribution system.

6. The cofeeder is a special type of feeder, required for converting the feeder network into the SFT feeder network, such that the cofeeder has non zero distribution only in event of fault in a feeder and zero distribution otherwise.

7. A connection length of the feeder network, which represents total length of all feeders in the feeder network.

8. An ownership cost of the feeder network, is sum of 'a capital cost given by cost of the connection length of the feeder network' and 'present worth over a time period, of a running cost given by cost of total loss in the feeder network'.

9. A feeder designing is a process comprising the steps of determining the nodes to be connected and identifying a feeder route for each connection, from a plurality of alternate feasible routes.

10. An OFD problem is identifying a feeder design, from a plurality of alternate feasible feeder designs, which minimizes the ownership cost of the feeder network.

11. A length of the SFT feeder network as sum of the connection length of the feeder network and a reliability length, wherein the reliability length is total length of all the cofeeders in the network.

12. An ownership cost of the SFT feeder network, is sum of the ownership cost of the feeder network, feeder cost of the reliability length and cost of all the NO switches. 13. A minimal set of the cofeeders associated with the feeder network, wherein a. A fault at any feeder in the feeder network, splits the feeder network in two sections,

• 'an isolated section' comprising of a set of downstream sink nodes which, as a result of the fault, are disconnected from the source node and

• 'a healthy section' compromising of rest of the sink nodes, which are still connected to the source node. b. For every fault at any one feeder in the feeder network, there is a cofeeder in series with a NO switch, such that by closing of the NO switch, each disconnected sink node in the isolated section get reconnected to the source node through the cofeeder. c. Addition of the minimal set of cofeeders and associated NO switches to the feeder network converts the feeder network to the SFT feeder network.

SUMMARY OF THE INVENTION

The method gives a procedure to identify OFD layout for the distribution system located on a layout map. The OFD layout is then converted into a SFT OFD by adding a set of cofeeders and associated

NO switches.

The process disclosed in this invention involves the following steps a. defining an OFD problem . b. inducing a plurality of cutsets on the system. c. designating cutset identifier for the cutsets. d. identifying a contour of the cutset . e. decomposing the OFD problem at the contour of the cutset. f. sectioning the system into a plurality of cascaded partitions . g. decomposing the OFD problem at into a set of distinct OFD sub-problems, h. defining ODOF at a sink node i. triggering plurality of feeder expansion stages for all partitions comprising of the followings. identifying invariants of the partition. evaluating the Lower Bound (LB) for the length and loss in the partition.

Identifying fixed layout branches iv. Identifying minimum and maximum outgoing branches at all sink nodes v. Identifying net outgoing flow at all sink nodes vi. compiling a list of several possible outdegree combinations, vii. evaluating the length and loss for each outdegree combination in the above list viii. selecting the outdegree combination having minimum sum of feeder length and feeder loss ix. identifying the feeder layout corresponding to selected outdegree combination with upstream partitions x. connecting every sink node located on the outer contour to a single sink node located on the inner contour. j. identifying the guidelines for combination of outdegree k. ensuring feasibility of ODOF by developing a set of feasible currents

I. positioning of heavily loaded feeders in partitions closest to the source node. m. ensuring maximum loss reduction by using minimal length of express feeders n. identifying minimal set of NO switch and cofeeders such that the OFD layout is converted into SFT

OFD. o. ensuring EC at all sink nodes in the layout p. defining complete solution space of all feasible feeder layouts, q. defining local and global minima r. identifying a procedure to reduce loss of an existing layout s. verifying the existence of distinct local minima with different loss, t. evaluating the Lower Bound (LB) for the length and loss in the system

The glossary of various relevant terms, definitions and results from this invention is as follows. The OFD layout wherein: a. Any change in the OFD feeder layout increases the ownership cost. b. Equality Criteria (EC), wherein the path length from source node to any sink node is equal to the minimum length of shortest path, provided that the overall variation in the distribution requirement at all sinks nodes in the distribution system is not excessive. c. EC, in case of a grid network, equates path length from source node to any sink node to the first norm distance from source node to sink node.

The EC characterization, which substantially reduces size of the solution space of feasible feeder networks for OFD of the distribution system

For every cutset, the contour of the cutset splits the distribution system into two parts, an upstream part is positioned on the source side and a downstream part is positioned on side away from the source side.

The method positions of heavily loaded feeders in lowest partition, wherein providing small additional feeder length for providing express feeders running directly from the sink nodes to the source, avoids summation of flow and reduces total loss in the feeder network substantially.

The method represents the distribution system as a set of cascaded partitions, such that a partition receives power supply from adjacent upstream partition at the nodes in lower cutset and delivers power supply to adjacent downstream partition at the nodes in higher cutset.

Defining a local minimum as a feeder network, wherein any change in the feeder network restricted to a local neighborhood, which is obtained by changing the parent node of any one sink node, does not reduce loss of the local minimum.

Defining a global minimum as a feeder network, wherein every plurality of changes in the feeder network, does not reduce loss of the global minimum. 'iii. A solution space of feasible network for a sink node in the distribution system, comprising of a. a Set of Feasible Parent Nodes (SFPN) for the sink node, wherein each element can be parent of the sink node. b. a selection of one element from SFPN as the parent of the sink node. c. a symbol representing the selection and relative position of the parent node with reference to the sink node d. elements in the SFPN, which depend on location of the sink node in the distribution system and maximum possible length of feeder from the sink node to the parent node. e. a number of alternate feeder networks for the sink node, as the number of elements in SFPN. ix. A solution space of feasible network for the distribution system, wherein a. selection of a parent node for any sink node, is independent of similar selection of a parent node for any other sink node. b. total number of alternate feeder network for the distribution system, as the product of the number of elements in SFPN for all sink nodes in the distribution system. c. a string of symbol representing a feeder network, one symbol for each sink node, wherein the string is the collection of symbols for all sink nodes. x. A LB for loss in the partition, wherein, the loaddownstream is distributed over the branchnumber such that the product of, flow and a value representing resistance to flow, is the same for all feeders in the partition. BRIEF DESCRIPTION OF THE DRAWINGS

In all the figures the source node is located at origin, while the available areas are the grid lines parallel to either axes. The unavailable areas are the square blocks located within grid lines. The distribution system is rectangular or square shaped demarcated by the opposite vertices. The cutset identifier for the cutset are indicated by C1.C2 etc, such that the prefix C stands for cutset and the number is the value of the cutset identifier. The figures also indicate the cofeeder and associated NO switch by using thick dashed lines.

FIG 1 ,2 and 3 represent three feeder layouts for the same distribution system of rectangular shape and bounded by diagonal nodes (1 ,4) and (6,-4) [Willis et.al]. All the fifty four sink nodes have unit distribution requirement.

FIG.1 is a SFT Main Trunk layout from the prior art Willis et.al/. FIG. 2 is a SFT Multi Branch layout from prior art /Willis et.aiy.

FIG. 3 i s a S FT O FD for the s ame d istribution system /^"Willis et.al/, i n a ccordance w ith t he p resent invention.

FIG 4, 5, 6 and 7 represent four different feeder layouts for a typical distribution system of square shape and bounded by diagonal nodes (-4, 4) and (4, 4). All the eighty sink nodes have unit distribution requirement. FIG. 4 is a SFT Main Trunk layout from the prior art.

FIG. 5 is a SFT Multi Branch layout from the prior art.

FIG. 6 is SFT OFD, which is a local minima, in accordance with the present invention.

FIG. 7 is SFT OFD, which is another distinct local minima, in accordance with the present invention.

FIG. 8 is an optimum layout from the prior art, which was proposed by Papadopoulos for the distribution system [Papadopoulos et al].

FIG. 9 is a SFT OFD layout for the distribution system of Papadopoulos in accordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION

The method of present invention is scalable to any number of nodes, flexible enough to accommodate arbitrary shape of distribution area as well as arbitrary source location. Unlike other approaches

[Papadopoulos et al], which generate layouts, which may be infeasible and then reject these layouts after observing their infeasibility, our method ensures that only feasible layouts are considered.

The following paragraph gives two fundamental concepts, which result in substantial loss reduction over the existing layouts.

1. Optimal Distribution of Outgoing Flow (ODOF) at a node

The l oss i n a feeder i s d efined a s the p roduct of s quare of f low through the feeder and the feeder resistance to the flow. A net outgoing flow at a node is divided in a multiple outgoing feeders with different resistances, such that total loss in all outgoing feeders is minimum.

The total loss in all outgoing feeders is minimum for such flow distribution, wherein the product of flow through the feeder and resistance to flow, is equal for all outgoing feeders. The associated flow distribution is defined as ODOF at the node.

The value of corresponding minimum loss is given by loss due to net outgoing flow in a single equivalent feeder; wherein the resistance of feeder is equivalent is equal to resistance of parallel combination of all feeders.

In a special case of uniform conductor size and uniform grid spacing, which is mostly observed in practice, the resistances of all feeders are equal. In such cases, the ODOF is simply equal distribution of net outgoing flow over all feeders.

Example: - Compare the distribution of net outgoing current at node (1 ,0) in three feeder layouts namely

Main Trunk, Multi Branch and OFD as shown in figure 1 , fig 2 and fig 3. In all the three cases loaddownstream [fifty-three units] is distributed over number of feeders given by branchnumber [three feeders]. Due to equal resistances of all feeder segments, loss is calculated as the square of current.

The current distribution and loss in these three layouts is as shown below.

The most equitable current distribution and lowest total loss is obtained in OFD layout as shown in fig 3.

2. Equality Criteria

The path length from source node to any sink node is equal to the minimum length of shortest path, provided that the overall variation in the distribution requirement at all sinks nodes in the distribution system is not excessive.

EC, in case of a grid network, equates path length from source node to any sink node to the first norm distance from source node to sink node.

The invention consists of the following steps:

1. Divide the layout map into two parts, namely available and unavailable areas. Earmark available area for laying of feeders.

2. Induce multiple cutsets on the distribution system as follows: The cutset is a set of sink nodes provided that a. Each cutset contains a plurality of sink nodes and every sink node belongs to only one cutset. b. Joining all the sink nodes in the cutset forms a contour of the cutset. c. When all the contours of all the cutsets in the system are identified, the system is fragmented into multiple consecutive convex contours of the cutsets, which are concentric around a source node. d. The contour of every cutset always lies in area, which does not have a right of way. Hence, any two sink nodes located on the contour of the cutset can not be connected by a feeder e. A numerical cutset identifier is assigned to each cutset in the system. f. The contour of the cutset splits the distribution system into two parts, an upstream part positioned on the side of the contour closer to the source node and a downstream part positioned on side of the contour which away from the source node. g. Designating a first cutset identifier representing the cutset containing only the source node. h. Designating a second cutset identifier representing the cutset containing the sink nodes that are geographically closest to the source node, i. Designating a l ast c utset i dentifier representing the c utset containing t he sink nodes, which are geographically farthest away from the source node, j. Designating a numeric value of the cutset identifier of the cutset located adjacent to the source node to be lower than a numeric value of the cutset identifier of the cutset located at a distance from the source node, k. Ensuring that the sink node in the cutset represented by a lower cutset identifier is geographically no farther from the source node than the sink node in the cutset represented by a higher cutset identifier; and I. Ascertaining that the contour of the cutset represented by a lower cutset identifier lies within the contour of the cutset represented by a higher cutset identifier. For all the three test cases, which are a special case of electrical distribution system on a grid involve a. Positioning a plurality of feeders along a horizontal axis or along a vertical axis; b. Designating a cutset identifier identifying each cutset such that the cutset identified by a number K represents a set of all sink nodes located at first norm distance of K units of grid spacing from the source node. c. Joining the sink nodes in the cutset to form a diamond shape contour, which is centered around the source node, and consists of maximum four unity slope diagonal line segments. d. Ensuring that the contour of every cutset always lies in area other than a horizontal line or along a vertical line, such that any two sink nodes of the cutset cannot be connected by a feeder.

For each cutset calculate the following data a. Number of nodes in the cutset b. Coordinate of all nodes in the cutset c. Cutset preset distribution requirement, which is the sum of distribution requirement of all sink nodes in the cutest

3. Induce multiple partitions on the distribution system such that: a. A partition is a set of two cutsets having consecutive cutset identifiers and the feeders connecting the sink nodes in both of the cutsets; b. A partition has an inner contour identified by lower cutset identifier and an outer contour identified by higher cutset identifier; c. Every sink node located on the o uter contour of the p artition h as o nly o ne i ncoming feeder from only one sink node located on the inner contour of the partition; d. The partition splits the distribution system into upstream partitions positioned closer to the source node and downstream partitions positioned away from the source node; e. Flow conservation is satisfied at each sink node on the inner contour of the partition; f. The partition is identified by the value of the cutset identifier of the outer cutset; g. For a case of d istribution system o n a g rid, d esignating a p artition i dentifier K which identifies a partition comprising of two cutset identified by cutset identifier K-1 and K, for the inner and outer contour respectively.

The nodes in the lower cutset of the partition being currently considered for feeder expansion are termed as active sources. These nodes cease to be active sources as soon as immediate downstream partition is considered for feeder expansion, when the new set of nodes in the lower cutset take over as active sources.

For each partition, initialize the following data a. Cutset identifier for the inner and outer cutset. b. Number of feeders = Number of nodes located on the outer contour. c. Number of active sources = Number of nodes located on the inner contour. d. For each active source, the net outgoing flow as difference of incoming flow and distribution requirement. e. loaddownstream = Sum of distribution requirement for all nodes located on the outer contour as well as all nodes in downstream partitions. f. For every nodes located on the inner contour, calculate its maximum outdegree, which is the maximum number of outgoing feeders from the node. Due to radial feeder network, out degree of a node located on the inner contour, is the number of nodes located on the outer contour that are connected to it.

4. In the feeder expansion a pproach, d evelopment of feeder i s i nitiated from the s ource n ode a nd proceeds downstream to farthest sink nodes. In such approach, feeder expansion is considering in terms of ascending order of outer contour of partitions. In such an approach, at the time of deciding feeder layout for any partition, the optimum feeder layout for all upstream partitions is already known.

5. Decomposition Of Feeder Layout Problem

Any feeder network can be expressed as cascaded configuration of all consecutive partitions. In such arrangement partition receives the feeder flow from the adjacent upstream partition at the nodes on the inner contour and delivers the feeder flow to the adjacent downstream partition at nodes on the outer contour. Such partitioning decomposes the feeder layout problem for the entire network, initiated at the source node, into a set of decomposed sub problems, one sub problem for each partition.

The minimum loss in any partition is subject to feasibility of ODOF, which in turn depends upon appropriate n etwork configuration downstream partitions. In such cases, the feeder expansion is carried out assuming that appropriate feeder network corresponding to ODOF can be found out in downstream partitions. If such assumption is found invalid at later stage, then best feasible flow distribution closest to ODOF is chosen.

6. For first partition initialize the following data a. Inner contour and outer contour b. Number of feeders = Number of nodes in the cutset 1 c. Number of sources = one, since source is the only node in the cutset 0 d. loaddownstream = Total distribution requirement of the distribution system e. All n odes i n cutset 1 a re connected to s ource by feeders of appropriate resistances, depending upon the feeder length and feeder size.

7. Solve the feeder design problem for the first partition as specified above, by using ODOF as follows. Divide the total distribution in the system amongst the feeders in the first partition such that, the product of flow and a value representing resistance to flow, is equal for all feeders in the first partition, wherein for electrical distribution system, Optimum Distribution of Outgoing Current (ODOC) such that , the product of current and resistance is equal for all feeders in the first partition.

8. Working With Each Partition

Starting from first partition up to a partition with increasing branchnumber , for each of the partitions, the method involves the following seven steps from i to vii.

Initialization of Outgoing flow

For each node located on the inner contour of the partition, initialize the net outgoing flow as the difference between corresponding input over the feeder of the immediate upstream partition and distribution requirement at the sink node.

Invariants of Partitions

For any partition, irrespective of its actual feeder layout, the following parameters are invariants under all its feasible layouts that satisfy EC. Identify the values of the invariant of partitions a. loaddownstream b. branchnumber c. Minimum and maximum outdegree of all sink nodes

The various feasible layouts, which satisfy EC, differ only in the distribution of loaddownstream over the branchnumber.

Calculation of LB a. LB for the loss in the partition is calculated as square of loaddownstream multiplied by equivalent resistance of parallel combination of all outgoing feeders. b. In case the multiple feasible feeders within the partitions, having different resistances, arrange the feeders in ascending order of resistance to flow. Consider the resultant resistance of the first branchnumber feeders connected in parallel. c. The loss of any feasible layout within a partition is always higher than [or in most rare cases equal to] the corresponding LB. d. The calculation of LB is independent of exact feeder layout and depends only upon the grid geometry and distribution requirement at all sink nodes. e. The calculation of LB finds out most equitable distribution of flow, while neglecting constraints of conservation of flow. Hence, the feeder layout corresponding to LB is not generally feasible.

Combinations of outdegree

In the feeder expansion stage for any partition, at each node located on the inner contour the following data is known a. Its net outgoing flow as per initialization step above. b. Its maximum out degree . c. Unless out degree is zero, [when grid geometry prohibits it or when net output at the sink node is zero] the minimum out degree is considered one.

The out degree of each node located on the inner contour can be varied in many ways up to the maximum value, such that the sum of out degree at all nodes located on the inner contour is equal to branchnumber of the partition. All such feasible combinations are listed.

Ownership cost for each combination

For any partition, the following observations are valid. a. Every node located on the inner contour acts like a root of sub tree, wherein the tree has number of branches equal to outdegree at the node. Hence, any partition has as many disjoint sub trees as the number of nodes located on the inner contour. b. For each disjoint sub tree, the concept of ODOF is applied, leading to optimum flow distribution and associated minimum loss. For each sub tree, the ownership cost is calculated.

The above step is carried out for all sub tree associated of a feasible combination. The ownership cost for this feasible combination is calculated as sum of ownership costs for all its sub trees.

Selection of a combination

The combination in the above list having minimum ownership cost is selected as optimum feeder layout for present partition. Minimum ownership cost for the present partition is the ownership cost associated with the feasible combination.

Fusion and initialization

This a bove s elected p artition 1 ayout i s m erged w ith t he e xisting n etwork i n o rder t o obtain optimum network up to and including present partition.

9. Guidelines for outdegree combination

The following are the guidelines regarding distribution of out degree at nodes in the lower cutset. a. Subject to integer solution of splitting ratio, achieve splitting at all nodes located on the inner contour such that the ratio of outgoing flow to splitting is same for all the nodes . For example consider outgoing flow and splitting for a node k be IR and n_k . In this case, try to achieve splitting such that the ratio of outgoing flow to splitting at all nodes is IJ n_k b. Define a set consisting of number of nodes located on the contour each cutset as its elements. Calculate the G.C.D. of the elements of this set. If this G.C.D, say k, is more than one, get a revised set by dividing all elements of old set by k.

Divide the grid in k sections such that a. Each section has continuous and exclusive area. b. The number of nodes in each Cutset is given by the element of new set.

This process leads to k totally decoupled independent sections, which have source as only common node. The OFD problem of entire grid is resolved in k sub problems, one per each section.

Here the G.C.D. of all nodes is four. Hence, this OFD problem can be decomposed into four decoupled sub problems having the following data.

The four sections consist of one quadrant and one of its axes, namely first quadrant and positive X-axis and so on. c. Splitting should start at a node having largest outgoing flow. In other words, select the nodes for splitting in descending order of outgoing flow. 10. Creation of set of feasible flow by BOTTOMS UP approach This calculation helps in identifying best feasible flow distribution as follows.

Starting from highest partition, calculate the set of feasible flows up to partition with not fewer branchnumber. The logic is based on the following steps a. Leaf node is a node, which does not has any child node. A feeder connecting a leaf node is called leaf feeder. b. Each partition has some leaf feeders and some carried over feeders. c. The leaf feeders do not have any carried over flow and the feeder flow is equal to distribution requirement of leaf node. d. The carried over feeder have some flow carried over from downstream partition and the feeder flow is equal to sum of distribution requirement of sink node and carried over flow. e. Minimizing the feeder flow in the partition and at the same time meeting the distribution requirement at all sink nodes, within all partitions considered develops the set of feasible flow. f. Route the braches carried over from earlier partition, through the node having minimum distribution requirement so as to minimize increase in flow in the network. g. If one feeder is to be added without disturbing the symmetry, then two feeders with smallest flow can be clubbed together. This reduces the number of feeders by one; so two new feeders can be added, resulting in net increase of feeders by one. Refer the calculation for partition six given below. h. The set of feasible flows as developed above helps to find a unique feeder layout in all the downstream partitions. The logic for finding this unique layout is based on the following steps, i. The grid layout leads to some feeders having fixed layout, particularly at the boundary of service area and feeders on X and Y axis, j. The set of feasible flows as developed above is compared with the output of the highest partition developed in step 9. The feeder layouts developed at step 9 and feeder layout of this step, are fused at the contour of the cutset. Example - Test case 1, page 29

The branchnumber from partition 10 up to partition 5 is nondecreasing. Hence, develop the set of feasible current for these partitions using BOTTOMS UP approach.

In the network expansion approach from partition 7 to partition 8, there is an increase of branchnumber by one. I n order t o add one feeder without d isturbing the symmetry, the two feeders with smallest current of one each, are clubbed together resulting in a single feeder having current of three units.

Let us compare the elements of set of feasible current with net outgoing current of partitions 5, the highest partition developed as per step 9.

It becomes obvious that outgoing current of 7 at both nodes namely (2,2) and (2,-2) has to be split in the ratio of 5:2, rather than in the best ratio 4:3. There is no splitting of current at other nodes, as the outdegree of other nodes is one.

11. Beyond OFD Layout

In a distribution feeder, any increase in the feeder length, which does not form loops, always leads to splitting of flow in existing feeders and hence reduces losses. However, any linear increase in capital cost always results in a nonlinear decrease in running cost. For a given increase in the feeder length, finding the location corresponding to maximum loss reduction and evaluation of corresponding reduction in loss is a difficult task.

In the layout obtained by the process of present invention, a single feeder connects two nodes, a restriction which forces heavy flows over a few feeders in the lower partitions. These heavily loaded feeders compromise feeder loss as well as reliability, and represent weakness of the feeder layout. The process of present^* invention identifies such critical feeders and further reduces losses as well as improves reliability, by providing disjoint additional feeder. These feeders avoid flow summation and associated heavy losses. The earliest splitting of flow ensures that these feeders are always located in the lowest partition. Hence, the additional feeder length of parallel disjoint feeder up to source is always small, but achieves a large reduction in losses. Generally, the incremental cost for providing additional parallel feeders is expected to be less than the capitalization of reduction in loss, and represent strength of the feeder layout. Thus, the process of present invention converts a weak point of feeder layout to a strong point.

The strength of OFD layout is that it ensures favorable cost benefit ratio for such parallel disjoint feeders.

In general, to prevent summation of k Feeders at a node (x, y) having distribution requirement of s units, the additional length required is (k-1)* |x]+|y| units. If the X1 , X2, X3 are the outgoing flows in ascending order at the node (x, y), then the net loss reduction is given by {(X1+ X2+ X3+s)²-(X1+ s) -(X2)²-

(X3)²}*[|x|+|y|]

If X1=X2=X3, the associated loss reduction is 66.66% of earlier loss, an illustrative conversion of weakness to strength.

12. Reliability Considerations

This section gives a procedure convert OFD into SFT OFD. The scheme involves addition of minimal set of cofeeders and associated NO switches to ensure SFT feeder layout. Starting from the highest partition, carry out the following steps a. Add cofeeders to the feeder layout such that a loop is formed. Now all nodes in the loop have two paths from the source. Hence, the entire load in loop can be served in spite of a single fault in the loop. b. Mark all the feeders in the path between two ends of the cofeeders as covered from reliability consideration. c. Continue this process until all the feeders in the layout are covered from reliability consideration. d. Check whether any cofeeders are redundant. If so, try to remove the redundant cofeeders. e. Starting this procedure at highest partition along with EFO, ensures that maximum feeders are covered at the cost of minimum numbers of cofeeders. f. In case of any single fault, there is discontinuity in the load downstream the fault. In such case, after isolating the faulty section form both ends, the corresponding NO switch is closed to restore the supply to the load downstream the fault.

13. Characterization Of OFD

The feeder layout corresponding to optimum network has some/interesting characteristics as follows. a. Every proper rooted sub tree of OFD TREE is also OFD TREE. b. Any feeder in the optimum layout is always from lower cutset node to next higher-cutset node. c. EQUALITY CRITERIA: For movement restricted to grid, the minimum source sink path length for every node (x, y) from source at (0,0) , is equal to its first norm given by |x| + |y|. d. There are a large number of equal length alternate paths from source at (0,0) to node (x, y), given by [ l^xi ⁺ ly|]/[ l^xl ^* |y|]- F°^r nodes on the X-axis or Y-axis, there is only one path to source namely X- axis or Y-axis. This follows from putting x = 0 or y = 0 in the above equation. These nodes on either axes lead to fixed feeders. e. There are nodes in the higher cutset that can be reached from only one node in the immediately lower cutset. These cases lead to fixed feeders.

14. EQUALITY CRITERIA

Case 1 : Violation of EQUALITY CRITERIA

The multi branch layout of Fig 2 [Willis et. al] has 10 nodes, for which the path length from source is more than the first norm path length by two units. These are five lower nodes of the top feeder and five upper nodes of the bottom feeder. Due to these violations of equality criteria, this layout is not a candidate for OFD layout.

Case 2: Violation of EQUALITY CRITERIA

In the Papadopoulos layout in Fig 8 [Reference: 3], the path length from source to node (6,0) on the X- axis is more than the first norm path length by two units. Due to this violation of equality criteria, this layout is not a candidate for OFD layout.

15. Defining Complete Solution Space

The aim is to evaluate the total number of feasible layouts for a distribution system, a number likely to be too large for exhaustive search. Another aim is to give a unique string of symbols to represent every feasible OFD layout.

Let K denote the maximum feeder length in terms of grid spacing, which connects a child node to parent node. There are 2*K*(K+1 ) feasible parent nodes located within K units of first norm distance from the child node. The child node has a set of feasible parents, comprising of 2^*K*(K+1 ) elements. Thus, a set of 2*K*(K+1 ) distinct symbols can completely specify the relative location of any parent node with respect to the child node. Depending on the selection of a particular parent node from the set of feasible parent nodes, corresponding symbol is used to represent the connection between the child node and parent node. There exists one to one correspondence between domain of a symbol string and co domain of feeder layout. Further, selection of exactly one parent node for all nodes in the distribution system ensures that the resulting network is always radial.

Selection of parent node at a sink node is independent of selection of parent node at another sink node. Further, the number of different layouts for a sink node is the number of elements in corresponding to SFPN. Hence, the number of different layouts for the entire distribution system is the product of number of elements in corresponding to SFPN for every node in the distribution system. Application of EC leads to substantial reduction in the solution space as follows. For any child node on the distribution system, only those feasible parent nodes that lie within or on the rectangle, having source and the child at opposite vertices are considered. There are K*(K+3)/2 such feasible nodes satisfying EC. For other nodes in the set of feasible parent nodes, the source child node path length is more than the first norm of the child node. The equality criteria rejects such parent nodes and thus reduces the number of probable parent node by the ratio of .25 [1 + [2/(K+1)]]. For large K, the reduction approaches 25% value. For normal case of K=1 , the EC reduces the solution space by 50 %. Consider earlier example of distribution system of Fig. 1 ,2 and 3. [Willis et.al] Let origin be located at source node and consider feeders of unit length. The grid geometry of service area dictates the following data for layouts meeting EC and other layout not meeting EC. Following table gives symbolic representation for any feasible feeder layout for the distribution system, which does not satisfy EC. The symbols represent relative position of the parent node with reference corresponding sink node as follows.

U- Up; L- Left; R- Right and D- Down.

The table gives relative position of the parent node with reference to all the 54 sink nodes.

No Of Feasible Parents

No Of Feasible Layouts

Number of nodes → 21 27 Λ 1*2 * 21 „ .27

Number of nodes with multiple parents Applying EQUALITY CRITERIA

Following table gives symbolic representation for any feasible feeder layout for the distribution system satisfying EC. The symbols represent relative position of the parent node with reference corresponding sink node as follows. U- Up; L- Left; R- Right and D- Down. The table gives relative position of the parent node with reference to all the 54 sink nodes.

Any arbitrary string of forty symbols, along with sixteen fixed orientation feeders, given by sixteen nodes in the above table having a single parent node completely and uniquely defines corresponding spanning tree layout. For example, if all forty symbols are set to represent vertical and horizontal feeders, leads to main trunk layout having main trunk in X and Y axis respectively. Improving an existing feeder layout

Let the distribution requirement of all sink nodes in the sub tree having root at a sink node be S. For each of the multiple paths from the sink node to the source, calculate ΔL, the incremental increase in loss as follows.

ΔL = S² * Σ an _feed_ers [feeder length] + 2 * S * Σ _an _feeders [feeder flow * feeder length] The feeder flows in the above equation are the quantities before adding load S to the feeder layout. Select a path that has least incremental increase in loss.

In case of feeder layout on a grid, the first term is equal for all paths, while the feeder length in the second term is always equal to grid spacing. Under these assumptions the criteria for incremental increase in loss is modified as follows.

ΔL = Σ an feeders [feeder flow]

The feeder flows in the above equation are the quantities before adding load S to the feeder layout.

Select a path that has least incremental increase in loss.

The following process is applied to an existing feeder layout for reducing the loss. a. Start the changes in the BOTTOMS UP approach, from highest partition towards first partition. b. For all nodes in the higher cutset of the partition, check whether the loss associated with existing path is lower than loss of every path associated with each element in SFPN. If not, change the parent from existing node to the corresponding element in SFPN. Thus at any stage, the feeder layout being considered for possible change, is restricted to one partition.

The proposed change in this partition affects entire upstream network, but does not affect the downstream network. The network obtained at the end of this method has loss in close vicinity of OFD loss.

OBJECTIVE OF CASE STUDY

The objective of test case is to illustrate the method of present invention to a few non-limiting cases of distribution system. In all the test cases the source location is taken as origin (0,0).

1. Continuous & uniform load distribution - In both cases, all nodes in the distribution system have equal load. For case 1 and 2, any spanning tree layout has length equal to 54 and 80 units of grid spacing respectively. Hence, a layout with minimum loss is OFD. Case 1: - It has rectangular service area with source at center of longer side. In this prior art by

Willis et. al, two main types namely main trunk and multi branch layout are mentioned. The losses for both the layouts are calculated and it has been shown that our methodology reduces loss substantially.

Case 2: - It has square service area with source at center of square, a most general case. Two conventional layouts, namely main trunk and multi branch layout are compared with OFD.

Discontinuous load distribution - In this case, not all nodes in the distribution system have load requirement. There are some nodes without any load requirement.

Case 3: - It has rectangular service area with source almost at center of smaller

It has been shown that by applying our methodology, loss and cost are substantially reduced as compared with the prior art results.

The summary of the results is as shown below. The last column denotes the substantial improvement in loss, achieved at the cost of minimal feeder length over the length given in column

5. [Refer Claim 15]

As a result of SFT feeder design, the feeder size is assumed to uniform. Further due to uniform grid spacing, the distance between any pair of adjacent nodes is equal. As a result, the resistance of any feeder is constant and hence, the feeder loss is calculated as square of feeder current. In all the three cases, the calculations give the following details for each partition a. Invariants of partitions - branchnumber and loaddownstream b. List of fixed layout feeders - Due to constraint of EC, some nodes in the higher cutset must be connected to particular nodes in the lower cutset. The list mentions such feeders with fixed layout. c. Input to the partition - consists of the following data for each node in the lower cutset a. Incoming current, shunt load at the node b. Net outgoing current = Incoming current - shunt load at the node c. Min and Max outdegree d. The outdegree at each node in the lower cutset can be varied within the Min and Max outdegree, such that the sum of outdegree at all nodes in the lower cutset adds up to branchnumber. There are a limited number of such combinations of outdegree. e. For each s uch combination, the m inimum l oss i s calculated based o n O DOF. A table g ives the details of ODOF and Min loss. The combination associated with the lowest loss is selected. f . The solution of the partition is the ODOF associated with the minimum loss as per above step g. The LB for the loss in the partition and the minimum loss is compared Continuous load distribution: Case 1 FIG 3

Figure 1 and 2 shows Main trunk, Multi Branch layout from prior art, for the rectangular shaped service area being served from source at (0,0) and bounded by diagonal nodes (1 ,4) and (6,-4). [Publication by Willis et al.]. Figure 3 shows the OFD layout for the same service area achieved by applying the method of the present invention.

Uniform load equal to base value is assumed at all the fifty four sinks nodes in the distribution system. Further due to blocks within the grid lines, the available area are restricted to lines parallel to x or y axes. In such case, any spanning tree has same capital cost given by connection length of fifty four units of grid spacing. Thus, a layout with minimum loss is would minimize owner ship cost and would be the OFD layout. Hence, the method finds out the layout with lowest loss, preferably in close vicinity of LB. Calculation of LB

The following table gives the calculation of LB for the distribution system. The first column indicates partition, while the next three columns indicates corresponding 'Nodes in the Higher cutset', 'branchnumber' and 'loaddownstream' respectively. The last column indicates the minimum loss in the partition given by square of loaddownstream divided by branchnumber. The LB is the sum of Min loss for all the partitions. The table also mentions loss of OFD layout and compares it with LB

The branchnumber increases from partition 1 to partition 5, while the branchnumber is nondecreasing from partition 10 to partition 5. Hence, the source centric incremental feeder expansion in TOPS DOWN approach is carried out till the output current at all nodes in the lower cutset of partition 5 is obtained. The set of feasible current using BOTTOMS UP approach is developed from partition 10 up to partition 5. The elements in the set of feasible current are compared with net outgoing current in the lower cutset of partition 5.

In all partitions, the outdegree combinations is with reference to ordered set of the sink nodes as per table titled 'OUTDEGREE AND CURRENT' PARTITION 1

INVARIENTS: loaddownstream = 54, branchnumber = 1 FIXED LAYOUT BRANCHES

From To

0,0 1 ,0

OUT DEGREE AND CURRENT

SOLUTION OF PARTITION

PARTITION 2 INVARIENTS: loaddownstream = 53, branchnumber = 3 FIXED LAYOUT BRANCHES

From To

1 ,0 1 ,1

1 ,0 2,0

1 ,0 1, -1

OUT DEGREE AND CURRENT

The current distribution as per first row is selected over other two cases, due to its symmetry of feeder layout across X-axis. SOLUTION OF PARTITION

LB on loss = 936.66, Actual lowest loss = 937. PARTITION 3

INVARIENTS: loaddownstream = 50, branchnumber = 5 FIXED LAYOUT BRANCHES

OUT DEGREE AND CURRENT

The outdegree combination of the third row has minimum loss and hence corresponding layout is selected- SOLUTION OF PARTITION

LB on loss = 500, Actual lowest loss = 546. PARTITION 4

INVARIENTS: loaddownstream = 45, branchnumber = 7 FIXED LAYOUT BRANCHES

OUT DEGREE AND CURRENT

The outdegree combination of the last row has minimum loss and hence corresponding layout is selected.

SOLUTION OF PARTITION

LB on loss = 289.28, Actual lowest loss = 301.

PARTITION 5

INVARIENTS: loaddownstream = 38, branchnumber = 9 FIXED LAYOUT BRANCHES

4,0 5,0

1 ,-3 1 , -4

OUT DEGREE AND CURRENT

This opens up forty-two cases as explained below. The branchnumber in the higher cutset of the partition is more than the branchnumber in the lower cutset by two, resulting in the splitting at two nodes. Let there be splitting in two feeders at any node out of seven nodes. Then there are six other nodes where remaining splitting in two feeders can take place. This gives us total of forty-two cases.

However, the lowest loss is achieved if splitting takes place at a node having highest outgoing current, i.e. node (2,2) and (2, -2). For the node sequence given above, the desired distribution of out degree is

1 ,2,1 ,1 ,1 ,2,1.

SET OF FEASIBLE CURRENT BY BOTTOMS UP APPROACH

The branchnumber from partition 10 up to partition 5, which are partitions arranged in the BOTTOMS

UP approach, is nondecreasing. Hence, develop the set of feasible current for these partitions using

BOTTOMS UP approach.

In the BOTTOMS UP approach from partition 7 to partition 8, there is an increase of branchnumber by one. In order to add one feeder without disturbing the symmetry, the two feeders with smallest current of one each, are clubbed together resulting in a single feeder having current of three units. Let us compare the elements of set of feasible current in partition with net output current in partitions 5 as follows.

It becomes obvious that O/G current of 7 at node (2,2) and (2, -2) has to be split in the ratio of 5:2 [loss = 29], rather than in the best ratio 4:3[loss = 25], the cost of additional loss of four units [29-25 =4] is cost for ensuring feasibility of the network. There is no splitting of current at other nodes, since the out degree at these nodes is one. It also shows that OFD layout is the best feasible layout, which tries to approximate ODOF without violating flow conservation. SOLUTION OF PARTITION

The set of feasible currents as developed above helps to find a unique feeder layout in all the downstream partitions.

BEYOND OFD LAYOUT

The layout obtained until now is a spanning tree layout; there are no parallel feeders anywhere in the grid. To eliminate current summation of k feeders at a node with level I, the additional feeder length required is given by l*(k-1 ) units. Consider provision of parallel feeders to avoid current summation up to level 2 as follows.

Thus, at a 10 % additional cost of six units of feeder, given by (60-54), the loss decreases from 5089 to

2497 by 50.93%.

If this process is further continued for eliminating current summation at nodes in higher cutset, the cost increases while benefits decreases.

Reliability Considerations

Set Of NO Switch And Feeder Segment By Bottoms Up Approach

The above set of six elements, each of single NO switch and associated cofeeder, ensure that all the loads in the system are served in spite of a single fault at any arbitrary location, except in feeder connecting (0,0) to (1 ,0).

Due to the peculiar shape of distributions system, SFT layout covering fault in the feeder connecting (0,0) to (1 ,0) is not possible. The SFT layout covering fault in any feeder automatically gets achieved, when two parallel disjoint feeders connecting (0,0) to (1 ,0) are introduced. These feeders not only reduce loss of the layout but also achieve SFT layout. Results Compared

Continuous load distribution: Case 2

Figure 4 and 5 shows Main trunk, Multi Branch layout, for the square shaped service area being served from source at (0,0) and bounded by diagonal nodes (4,4) and (-4,-4) . Figure 6 shows the OFD layout for the same service area achieved by applying the method of the present invention. Uniform load equal to base value is assumed at all the eighty sinks nodes in the distribution system. Further due to blocks within the grid lines, the available area are restricted to lines parallel to x or y axes. In such case, any spanning tree has same capital cost given by connection length of fifty four units of grid spacing. Thus, a layout with minimum loss is would minimize owner ship cost and would be the OFD layout. Calculation of LB

The following table gives the calculation of LB for the distribution system. The first column indicates partition, while the next three columns indicates corresponding 'Nodes in the Higher cutset', 'branchnumber' and 'loaddownstream' respectively. The last column indicates the minimum loss in the partition given by square of loaddownstream divided by branchnumber. The LB is the sum of Min loss for all the partition. The table also mentions loss of OFD layout and compares it with LB

Here the G.C.D. of number of nodes in all cutsets is four. Hence, this OFD problem for the square grid can be decomposed into four independent OFD sub problems one for each section having 25% of entire service area. The four sections consist of one quadrant and one of its axes, as shown below. Four sections of the Distribution System

The following solution is worked out for first quadrant and positive X-axis. This solution is repeated for the three sets of remaining three quadrants and corresponding positive or negative axis.

The branchnumber increases from partition 1 to partition 3, while the branchnumber is nondecreasing from partition 8 to partition 4. Hence, the source centric incremental feeder expansion in TOPS DOWN approach is carried out till the output current at all nodes in the lower cutset of partition 4 is obtained.

The set of feasible current using BOTTOMS UP approach is developed from partition 8 up to partition 4.

The elements in the set of feasible current are compared with net outgoing current in the lower cutset of partition 4.

In all partitions, the outdegree combinations is with reference to ordered set of the sink nodes as per table titled OUTDEGREE AND CURRENT

PARTITION 1:

INVARIENTS: loaddownstream =20, branchnumber =1

FIXED LAYOUT BRANCHES

OUTDEGREE AND CURRENT

SOLUTION OF PARTITION

PARTITION 2:

INVARIENTS: loaddownstream = 19, branchnumber = 2 FIXED LAYOUT BRANCHES

OUT DEGREE AND CURRENT

COMBINATIONS OF OUTDEGREE

Outdegree O/G Current O/G Current Loss

O/G current Combination to (1 ,1 ) to (2,0) value

1 :1 19 10 181

1 :1 19 10 181

Without loss of generality, the current distribution as per first row is se ected. SOLUTION OF PARTITION

PARTITION 3

INVARIENTS: loaddownstream = 17, branchnumber = 3 FIXED LAYOUT BRANCHES

OUT DEGREE AND CURRENT

COMBINATIONS OF OUTDEGREE

SOLUTION OF PARTITION

PARTITION 4

INVARIENTS: loaddownstream = 14, branchnumber = 4 FIXED LAYOUT BRANCHES

OUT DEGREE AND CURRENT

Set of feasible current by BOTTOMS UP approach

The branchnumber from partition 8 up to partition 4, which are partitions arranged in the BOTTOMS UP approach, is nondecreasing. Hence, develop the set of feasible current for these partitions using BOTTOMS UP approach.

Let us compare the elements of set of feasible current in partition with net output current in partitions 4 as follows.

It becomes obvious that O/G current of 7 at node (3,0) has to be split in the ratio of 5:2 [loss = 29], rather than in the best ratio 4:3[loss = 25], the cost of additional loss of four units [29-25 =4] is cost for ensuring feasibility of the network. There is no splitting of current at other nodes, since the out degree at these nodes is one. It also shows that OFD layout is the best feasible layout, which tries to approximate ODOF without violating flow conservation. It also shows that OFD layout is the best FEASIBLE layout, which tries to approximate LB current distribution without violating flow conservation. COMBINATIONS OF OUTDEGREE

SOLUTION OF PARTITION

LB on loss = (14)²/4=196 ; Actual lowest loss = 4*54=216. The set of feasible currents as developed above helps to find a unique feeder layout in all the downstream partitions as shown in Fig 6. BEYOND OFD LAYOUT

The layout obtained until now is a spanning tree layout, there are no parallel feeders anywhere in the grid. Consider provision of parallel feeders to avoid current summation up to level 2 as follows.

Thus, at a 15% additional cost of feeder (twelve units given by 92-80), the loss decreases from 3160 to 1960 by 37.94%. This kind of cost benefit represents the conversion of weak point into strong one. If this process is further continued for eliminating current summation at nodes in higher cutset, the cost increases while benefits decreases. RELIABILITY CONSIDERATIONS set of NO SWITCH AND COFEEDERS

The above table gives only one set for the feeder layout in the first quadrant. For feeder layout in the other quadrants, there is symmetrically placed set as shown in Fig 6. The above set of 3 elements, each of single NO switch and associated cofeeder, ensure that all the loads in the system are served in spite of a single fault at any arbitrary location except in feeder connecting (0,0) to (1 ,0). In order to achieve SFT layout covering fault in the feeder connecting (0,0) to (1 ,0), cofeeder and NO switch is added at a location as follows.

The SFT layout covering fault in any feeder automatically gets achieved, when two parallel disjoint feeders connecting (0,0) to (1 ,0) are introduced. Hence after addition of these feeders, the above mentioned cofeeder and NO switch in the partition 5 can be removed. The feeders from (0,0) to (1 ,0) not only reduce loss of the layout but also achieve SFT layout. RESULTS COMPARED

Discontinuous load distribution: Case 3 FIG 8

For this case, DISCONTINIOUS NON-UNIFORM LOAD DISTRIBUTION is considered.

Figure 8 shows layout proposed by Papadopoulos, for the rectangular shaped service area being served from source at (0,0). Figure 9 shows the OFD layout for the same service area achieved by applying the method of the present invention.

Due to blocks within the grid lines, the available area is restricted to lines parallel to x or y axes.

Calculation of LB

The following table gives the calculation of LB for the distribution system. To simplify calculations, load of 100kva was taken as one unit. This case has total load of 28units. The first column indicates partition, while the four columns indicates corresponding 'Nodes in the Higher cutset', 'LoadNodeDown-Stream',

'branchnumber' and 'loaddownstream' respectively. For discontinuous load distribution branchnumber is calculated as minimum of 'Nodes in the Higher cutset' and 'LoadNodeDown-Stream'. The last column indicates the minimum loss in the partition given by square of loaddownstream divided by branchnumber. The LB is the sum of Min loss for all the partition. The table also mentions loss of OFD layout and compares it with LB

The branchnumber increases for partition 1 , while the branchnumber is nondecreasing from partition 8 to partition 2. Hence, the source centric incremental feeder expansion in TOPS DOWN approach is carried out till the output current at all nodes in the lower cutset of partition 2 is obtained. The set of feasible current using BOTTOMS UP approach is developed from partition 8 up to partition 2. The elements in the set of feasible current are compared with net outgoing current in the lower cutset of partition 2.

Maximum out degree of source node is three. Thus, the three feeders from cutset 0 to cutset 1 should have almost equitable current distribution.

In all partitions, the outdegree combinations is with reference to ordered set of the sink nodes as per table titled 'OUTDEGREE AND CURRENT

PARTITION 1 :

INVARIENTS: loaddownstream =28, branchnumber = 3

FIXED LAYOUT BRANCHES

OUTDEGREE AND CURRENT

SOLUTION OF PARTITION

0,0 -1 ,0 2(2, -2) 2(4, -2) 5(6, -2)

This is the most equitable load distribution among the three feeders.

LB on loss=3*(28/3) =261.33 ; Actual lowest loss =[8²+11²+9²]=266.

PARTITION 2:

INVARIENTS: loaddownstream = 28, branchnumber = 4

OUT DEGREE AND CURRENT

COMBINATIONS OF OUTDEGREE

Due to the typical arrangement of load nodes, the best splitting of outgoing current at (-1 ,0) can be done in ratio of 2:7 and not 4:5. SOLUTION OF PARTITION

SET OF FEASIBLE CURRENT BY BOTTOMS UP APPROACH

The set of feasible currents as developed above helps to find a unique feeder layout in all the downstream partitions as shown in Fig 9. BEYOND OFD LAYOUT

The layout obtained until now is a spanning tree layout, there are no parallel feeders anywhere in the grid. Consider provision of parallel feeders to avoid current summation up to level 2 as follows.

Thus, at a 4.16 % additional cost of one unit of feeder, given by (25-24), the loss decreases from 1162 to 1102 by 5.16 %. RELIABILITY CONSIDERATIONS SET OF NO SWITCH AND FEEDER SEGMENT by BOTTOMS UP approach

The above set of four elements, each of single NO switch and associated cofeeder, ensure that all the loads in the system are served in spite of a single fault at any arbitrary location. Results Compared

Claims

ClaimsWhat is claimed is

1. A method of providing a Single Fault Tolerant (SFT) and Optimum Feeder Design (OFD) for a distribution system which is a set of unconnected nodes without a single edge, said method comprising the steps of: i. defining an OFD problem as selecting a feeder layout minimizing a feeder length and a feeder loss in the system from a plurality of feasible feeder layouts; ii. inducing a plurality of cutsets on the system, such that each cutset contains a plurality of sink nodes and every sink node belongs to only one cutset; iii. identifying a contour of the cutset by joining all the sink nodes in the cutset, such that when all the contours of all the cutsets in the system are identified, the system is fragmented into a plurality of consecutive convex contours which are concentric around a source node; iv. sectioning the system into a plurality of cascaded partitions around the source node where a partition is a set of feeders located between two adjacent contours such that both end points of the feeder are located on the sink nodes positioned on the adjacent contours; v. decomposing the OFD problem at the contours of the cutsets into a set of distinct OFD sub- problems, such that an OFD sub-problem is a part of the OFD problem which only seeks to identify the OFD for the given partition and the number of sub-problems in the system equals the number of partitions in the system; and vi. triggering for each partition, a feeder expansion stage centered around the source node, wherein all the feeders in the partition are assigned a flow magnitude, until OFD layout for each partition is identified.

2. The method of claim 1 , wherein the step of inducing the cutsets further comprises: i. assigning a cutset identifier for each cutset in the system, where the cutset identifier numerically represents the cutset in the system; ii. splitting the distribution system into two parts by using the contour such that an upstream part is positioned on the side of the contour closer to the source node and a downstream part positioned on the side of the contour away from the source node; iii. ensuring that the contour of every cutset always lie in an 'unavailable ' area which does not have right of way; and iv. connecting the sink nodes in the cutset to the sink nodes in the cutset positioned on the immediate upstream part.

3. The method of claim 2, wherein the step of designating the cutset identifier further comprises: i, designating a first cutset identifier for the cutset containing only the source node; ii. designating a second cutset identifier for the cutset containing the sink nodes that are geographically closest to the source node; iii. designating a last cutset identifier for the cutset containing the sink nodes which are geographically farthest away from the source node; iv. designating the cutset identifier for the cutset located adjacent to the source node, a numeric value lower than a numeric value of the cutset identifier for the cutset located at a distance from the source node; v. ensuring that the sink node in the cutset represented by a lower cutset identifier is no farther from the source node than the sink node in the cutset represented by a higher cutset identifier; and vi. ascertaining that the contour of the cutset represented by a lower cutset identifier lies within the contour of the cutset represented by a higher cutset identifier.

4.The method of claim 1 wherein the step of decomposing the OFD problem at the contour of the cutset, further comprises verifying that a selection of any layout design in the downstream part, from the plurality of feasible feeder design, does not affect a feeder flow and the feeder length of the already developed layout design in the partitions located in the upstream part.

5. The method of claim 1 , wherein the step of sectioning the system into a plurality of cascaded partitions further comprises: i. positioning the source node and the plurality of sink nodes on a layout map of the distribution system; ii. dividing the layout map into an available area and an unavailable area based on right of way consideration; iii. connecting the source node to the plurality of sink nodes, via a common radial distribution layout employing the plurality of feeders where each sink node has a preset distribution requirement; iv. passing a feeder flow of preset magnitude through the feeder; and v. detecting an optimal feeder layout, which minimizes the total feeder length and the feeder loss in the system by using the feeder flow / and a resistance r of the feeder whereby the feeder loss L is determined from the equation: L = i²*r.

6. The method of claim 1 wherein the step of decomposing the OFD problem into a set of distinct OFD sub- problems further comprises: i. defining the partition as a set of two cutsets having consecutive cutset identifiers and the feeders connecting the sink nodes in both of the cutsets; ii. locating an inner contour of the partition which is the contour of the cutset identified by lower cutset identifier and locating an outer contour of the partition which is the contour of the cutset identified by higher cutset identifier; iii. providing an incoming feeder such that for every sink node located on the outer contour of the partition, such that only one incoming feeder is supplied from only one sink node located on the inner contour of the partition; iv. splitting the distribution system after creating the partition into a plurality of upstream partitions located closer to the source node and a plurality of downstream partitions located away from the source node; v. adjusting a feeder flow over the feeders in the partition while satisfying a flow conservation requirement at each sink node on the inner contour of the partition; vi. conditioning the OFD problem decomposition upon availability of an outgoing flow of the assigned magnitude at the sink nodes in the lower cutset, the assignment made in the feeder expansion stage; vii. receiving the feeder flow for the partition from the adjacent upstream partition at the nodes on the inner contour; viii. delivering the feeder flow from the partition to the adjacent downstream partition at nodes on the outer contour; ix. defining a sub-network providing OFD for the each partition and for each OFD sub-problem in the distribution system; and x. expressing the OFD for the entire distribution system, as a set of cascaded configuration of the sub-networks of the partitions.

The method of claim 1 wherein the step of triggering feeder expansion further comprises: i. considering a source centric feeder expansion where the partitions are arranged in an ascending order of a plurality of numeric identifiers; ii. identifying a branchnumber which equals the number of feeders located within the partition and a loaddownstream which is the total distribution requirement of the sink nodes located on the outer contour and all the sink nodes located in the downstream partitions; iii. evaluating a lower bound for the feeder length and the feeder loss in the partition; iv. identifying an outdegree of the sink nodes which equals the number of outgoing feeders at the sink node such that outdegree of the sink nodes vary within a preset minimum and maximum value; v. compiling a list of a plurality of possible outdegree combinations at all the sink nodes, such that for each outdegree combinations the sum of outdegree of all sink nodes equals branchnumber; vi. evaluating the feeder length and the feeder loss for each outdegree combination in the list; vii. selecting an optimum feeder layout corresponding to the outdegree combination from the list having a minimum feeder length and a minimum feeder loss; viii. connecting at each stage every sink node located on the outer contour of the partition to a single sink node located on the inner contour of the partition based on the optimum feeder layout, and fusing the optimum sub-network with optimum feeder layout in the upstream partitions and i. entering into a plurality of feeder expansion stages from the source node, one stage for each partition and continuing the feeder expansion stages until OFD layout for each partition is identified.

8. The method of claim 7 comprising step of defining a branchnumber and a loaddownstream, as an invariants of the partition, such that all feasible layouts satisfying equality criteria (EC) differ only in the splitting of total flow in the partition given by the loaddownstream, over total number of feeders in the partition given by the branchnumber.

9. The method of claim 1 wherein the method employs an equality criteria (EC), which requires that a path length from any sink node to the source node be equal to the shortest distance between the sink node and the source node.

10. The method of claim 9 wherein for the_distribution system which is a grid, the path length from any sink node to the source node is equal to the first norm of the distance between them the sink node and the source node.

11. The method of providing SFT of claim 1 where if an equality criteria is satisfied for the sink node, an optimum feeder layout associated with a local minima always remains optimum regardless of any increase or decrease in the distribution requirement at any one sink node in the system.

12. The method of providing SFT of claim 1 wherein if an equality criteria is not satisfied for the sink node, an optimum feeder layout associated with a local minima always remains optimum for a specific increase in the distribution requirement at any one sink node in the system.

13. The method of claim 1 further comprises the following steps for each sink node in the distribution system: i.enlisting a Set of Feasible Parent Nodes (SFPN) for the sink node; ii.selecting an element from the SFPN; iii.changing the connection of the sink node from the existing parent node to the selected SFPN element; iv.evaluating the changes in step iii; v.accepting the changes in step iii. if the change reduces feeder loss; vi. repeating steps from ii to v for all elements in SFPN, and maintaining then existing feeder layout as a local minima if the feeder loss cannot be further reduced in spite of the above procedure .

14. The method of claim 13 further comprises: i. locating a plurality of local minima with different feeder losses for a distribution system; ii. defining a global minima which is a feeder layout having a minimum sum of total feeder length and total feeder loss in the system; and iii. avoiding the need for exhaustive search of a solution space containing all feasible feeder layouts in the system.

15. The method of claim 1 further comprising the step of: i. positioning a plurality of heavily loaded feeders in the lowest partition; and ii. avoiding a summation of flow and substantially reducing the total feeder loss in the system by employing a minimal length of express feeders running directly from the sink nodes to the source node.

16. A method of providing a Single Fault Tolerant (SFT) Optimum Feeder Design (OFD) for a distribution system which is a set of and unconnected nodes without a single edge, the method employs an Optimum Distribution Of Outgoing Flow (ODOF) at a sink node, wherein a product of a feeder flow and a value representing a resistance to flow is same for all of the outgoing feeders at the sink node.

17. The method of claim 16 wherein the distribution system is an electrical distribution system comprising an electrical current and a resistance representing a feeder flow and a value representing a resistance to flow, where the assigned magnitude of current is adjusted to achieve Optimum Distribution Of Outgoing Current (ODOC) such that the product of the electrical current and the resistance is same for all outgoing feeders at the sink node.

18. A method of providing a Single Fault Tolerant (SFT) Optimum Feeder Design (OFD) for a special case of electrical distribution system on a grid comprising: i. positioning a plurality of feeders along a horizontal axis or along a vertical axis; ii. ^' placing a plurality of cutsets on the electrical distribution system such that each cutset contains a plurality of sink nodes and every sink node belongs to only one cutset; iii. designating a cutset identifier identifying each cutset such that the 'K'th cutset identifier represents a cutset, which is a set of all sink nodes located at first norm distance of K units of grid spacing from a source node; iv. joining the sink nodes in the cutset to form a diamond shape contour which is centered around the source node, the contour consisting of maximum four unity slope diagonal line segments; and v. ensuring that the contour of every cutset always lies in area other than a horizontal axis or along a vertical axis.

19. A method of identifying a minimal set of the cofeeders associated with the feeder layout wherein i. the cofeeder is a special type of feeder, required for converting the feeder layout into the SFT feeder layout ii. for every fault at any one feeder in the feeder layout, there is a cofeeder in series with a Normally

Open (NO) switch, such that by closing of the NO switch, each disconnected sink node get reconnected to the source node ; iii. addition of the minimal length of cofeeders and associated NO switches to the feeder layout converts the feeder layout to the SFT feeder layout.

20. A method of identifying a Lower Bound (LB) for a feeder loss in a distribution system, comprising the steps of: i. sectioning the system into a plurality of partitions; ii. identifying the LB for the feeder loss in the partition; iii. calculating the LB for the feeder loss in the system, which is equal to the sum of the LB for the feeder loss in all the partitions in the system; iv. verifying that the LB for the feeder loss in the system is less than or equal to the feeder loss of any feeder layout, including the feeder layout global minima; v. avoiding exhaustive search for an optimal feeder layout with lowest loss by using the LB for the feeder loss in the system as a reference point; and vi. identifying an infeasibility of feeder layout corresponding to LB, wherein the constraints of flow conservation is neglected.