Algorithm Acceleration and Data Storage Volume Reduction in Reliability Modeling Within Distribution Network

Investigation and assessment of a system performance was complex. So Equipment reliability modeling may be a proper solution to easier it. Electricity distribution network is among systems reliability of which is of great importance. They are highly complex due to their large size and hence their modeling takes a long time. This study offers a simple algorithm for Electricity distribution networks modeling which enjoys algorithm high speed and data low volume. It is based on a graph search method called coloring. In this innovative method, it is suggested to search minimal path from the beginning of the feeder to the sink points. Also it is applied by saving paths in terms of switches and through logical operations of symmetric difference without searching paths from each input to estimate minimal path between all inputs and outputs. In so doing there will be no need for explicitly defining the source node, as it is easily recognizable via junction matrix composition. This kind of network modeling is so flexible and it is simply applicable to bidirectional or non-planar graphs.


INTRODUCTION
In general, Reliability is as much as systems performance.The more is a system size and complexity, the more its unreliable behavior and costs will be (Chen, 2011).In the past research, the main objective of "spares and reliability optimization problem" in systems is often to maximize total reliability under such constraints as budget and space.As far as components multiple-choice is concerned, mathematical methods were not effective and heuristic and meta-heuristic methods must be applied (Genç, 2012).Many methods for reliability optimization assume that systems components are parallel or series, different plans are available and various methods have been applied to solve this problem (Patelli, 2010).Reliability is referred to as a system capability in realizing its required performance under environmental and functional circumstances and within a certain time span.In the other words, reliability is as much as systems performance (Arya, 2012).To maximize the system total reliability, the required space and costs, and the weight of costs pertaining to maintenance, hiring and firing operator, equipment substitution in different work shifts must be minimized (Yingkui, 2012).In the last decade, researchers have studied reliability optimization problem consisting of Redundancy Allocation Problem and Reliability Assignment Problem.Solving these two problems simultaneously is further more difficult comparing with solving them separately as two different kinds of decision-making variables must simultaneously be determined (Wang, 2012).Since considering these objectives or constraints in the problem makes it closer to the reality and further complicates its solving, it is necessary to provide an innovative and mixed solution.Two types of modeling of multiple-objective reliability problem are as following.
Table1.The first model is a multiple-objective one which aims at maximizing the system reliability and minimizing the system costs and weight.The second model is a binary-objective one whose objectives are maximizing system reliability and minimizing system costs with regard to constraints of cost, weight, volume and reliability low limit for subsystems components and the whole system.System reliability depends upon its components decoration.It means that system reliability with parallel components differs from a system with series components and a system comprising of these two decorations (Khalili-Damghani, 2012).So:  If the system is constituted of series components, then the total reliability equals to:


If the system is constituted of parallel components, then the reliability equals to: If the system comprising of series-parallel components, the reliability is calculated with respect to the system design (regular or complex) and through the mentioned relations (Zhang,2007).In general and regardless of differences, all of the minimum cut-set and tie-set algorithms attempt to offer a model characterizing the network graph and tracing the current path between the network nodes.All tracing methods are divided in two general groups from mathematical modeling and graph search methods perspective.The first group is Breadth First Search, and as the name suggests, it is a level to level graph search.It starts at the root (selecting some node as the root in the graphs or the root-less trees) and puts it in the first level.Then for each of those neighbor nodes in turn, it inspects their neighbor nodes which were unvisited and so on.It is stopped when all neighbors of the final level nodes have been visited.
The second group is called Depth-first search, and as the name suggests, it is a deeper search in the graph.It starts at the root (selecting some node as the root in the graphs or the root-less trees) and explores neighboring nodes via output edges of current node as far as possible along each branch before backtracking.Upon facing with unvisited neighbors, it is implemented recursively for that node.If all neighbors have been already visited, it backtracks and it is continued in a node from which we have come to the current node.It goes deeper and deeper as far as possible.It continues until all accessible nodes will be visited (Moslemi, 2011).Based on the above introduction, the present paper titled as "offering a quick method of modeling and assessing reliability" aims at providing a simple and quick method for network routing through using former methods.This method is based on what is called graph coloring that has not yet been applied in graph search studies.

Literature review
Emphasizing on the effect of the system components location, Karbasian et al. (2009) have provided a model to keep balance between the system reliability and costs of positioning, maintenance and repair, and have solved it via lingo software.Zhao et al. (2007) applied penalty function and local search for solving reliability optimization in series-parallel systems.Yadav et al. (2009) combined two approaches of reliabilitybased design and robust design and optimized it via Taguchi quality loss function.Wang et al. (2009) addressed uncertainty of likely events of failure in spare systems based on Cross Entropy Approach.Gupta et al. (2009) solved the problem of spare allocation in series systems or interval values through unconstrained integer programming and formulated interval coefficients (high and low limits for reliability) and via penalty function and advanced genetic algorithm.Also Ferreira, et.al (2012) used nonlinear binary programming model aiming to minimize the SAIDI and SAIFI reliability indices of a distribution feeder based on the branch-andbound algorithm by identifying types and locations of protection devices and sectionalizing switches.Arya, et.al (2012) optimized distribution Reliability with respect to failure rate and repair time of each section of the distribution.Differential evolution (DE), Particle swarm optimization (PSO) and co-ordinated aggregation based PSO (CAPSO) have been used to develop computational algorithms.

METHODOLOGY
In fact, the routing algorithm presented here, is part of a comprehensive algorithm developed for modeling and predicting the reliability1 .The purpose of its implementation is to produce the required data for the reliability prediction algorithm.Network data including minimal path matrix of each sink point to the source and minimal path matrix of maneuver points are required in order to calculate reliability in open ring networks.It enjoys node/branches concept and constitutes network nodes junction through using raw data of matrix network.Then by examining splits, from each node it extracts the current paths between the source node and each sink point.Figure 1 shows the overall framework of the algorithm.The raw data of nodes junction is applied in constituting network junctions matrix.These data are actually the inputs of the network modeling algorithm comprising of: node number, parent node number, element code, branch code, element status (closed status=one and open status=zero).Hence nodes junction matrix is obtained for each feeders branched from secondary grid substation.

MODELING RESULTS
The algorithm initial idea stems from depth search concept, yet it is a recursive algorithm that searches each time again from the beginning of the feeder which reduces the program speed.To overcome this problem, another idea called Graph Coloring Method has been employed.Simply, graph coloring concept is a method for recognizing ring in the graph based on a particular constraint.Graph coloring is labeling graph vertices so that the neighboring vertices do not possess a label.These labels are called colors.It is said that a graph is kcolorable if it can be colored by maximum k different colors.If a graph is not k-colorable, it has a ring.It is called vertices coloring.Similarly, in coloring the edges, each color is allocated to one edge so that two neighboring edges will not possess an identical color (Yufeng, 2009).Also to compare search speed of BFS and DFS, if we consider each branch search as a time unit, DFS method searches the whole network in 48 time units while graph coloring and BFS methods will search the whole network in only 23 time units.However in determining the minimal path, BFS method occupies 9 times more memory volume than the DFS and graph coloring methods.Consequently it can be asserted that in the real distribution networks with a lot of splits, at least 50% time saving in network search stage is brought about.To determine existing paths between all sinks and their sources, suffice it to search the switches between each sink and the related source.This is applicable to both open ring distribution feeders and secondary grid substation that is a ring network.Search starts at a node and it is studied whether the next node is a switch, another element in the network or a sink.If it is a switch, the node number is saved in a matrix called path matrix.In principle, path matrix is a one that demonstrates which switches exist in each path.It is a matrix (K×P) in which K denotes path matrix numbers  This kind of search increase the program speed and since the paths are merely recognized and saved via existing switches in the source and sink path, the memory calculations and data volume is reduced.Moreover, as this algorithm has been developed for calculating distribution network reliability, network utilization has been accomplished in algorithm implementation; it means that for maneuver operations, after error occurrence and/or DGs junctions to the network to retrieve some of the sinks, minimal paths between maneuver usually-open points and the sinks are required.To accomplish this part and to prevent routing algorithm reimplementation which will be time-consuming for real network dimensions, a very simple and innovative solution based on a simple principle in the graph theory has been suggested.As per figure (3), if we have two paths of AB and ACD, then BCD path is easily obtained via logical operations of symmetric difference.This idea has been employed for reducing searching operation so that through searching the path between the beginning of the feeder and all the nodes and maneuver points and by using logical operations, the path between all DGs or maneuver points and the sinks will be gained without a new searching operation and the program operation speed will significantly be modified as well.Because searching methods are further more time-consuming comparing with logical operations.The algorithm first block (figure 4) relates to constituting the network junctions matrix and determining network topology.In the input, such data as the process has been expressed.The algorithm starts from a choose node and while determining the joined nodes, it always investigates whether there is a ring or not, and if any, an error massage will be appeared.The second block (figure 5) constitutes flowcharts of developing minimal paths on the basis of searching all switches between the source and sink points.In this stage, a code corresponding with a color is allocated to each network branch.Whenever a new switch is found, it is added to a matrix called MP matrix, and if it reaches a sink, the corresponding row has been developed in the matrix.

CONCLUSION
In this paper, a simple and innovative algorithm based on the graph coloring method has been offered for network graph modeling and path searching in the distribution network.In this method, in order to reduce data processing volume and accelerate the algorithm, only the switches existing in the paths are searched rather than the whole elements.In fact, the network is equated based on its switches.In this model, minimal paths with graph searching are calculated directly for ring networks and radial networks.Also it is applied by saving paths in terms of switches and through logical operations of symmetric difference without searching paths from each input to estimate minimal path between all inputs and outputs.In so doing there will be no need for explicitly defining the source node, as it is easily recognizable via junction matrix composition.This kind of network modeling is so flexible and it is simply applicable to bidirectional or non-planar graphs.Besides, there is no constraint in the number of source nodes or final nodes.These may be considered among the offered algorithm advantages.

Figure 1 .
Figure 1.Conceptual Framework of the Network Routing Algorithm node and its joined splits -Examining joined switches of the node Constituting minimal paths matrix in each sink point Intl.Res.J. Appl.Basic.Sci.Vol., 7 (13), 940-945, 2013 and P denotes switch numbers.Then split node becomes the current node and the algorithm continues to search until it reaches the sink point and the path is completed.For example, simple network with three sources, three sinks and five switches shown in figure 2 is studied.

Figure 2 .
Figure 2. Sample Ring Network with a few Switches between 3 Sources and 3 Sinks In the figure, existing paths between source 1 and sink 2 by searching corresponding switches including 9 paths are as below.{2, 3, 5, 1&2, 1&3, 1&5, 4&2, 4&3, 4&5}This kind of search increase the program speed and since the paths are merely recognized and saved via existing switches in the source and sink path, the memory calculations and data volume is reduced.Moreover, as this algorithm has been developed for calculating distribution network reliability, network utilization has been accomplished in algorithm implementation; it means that for maneuver operations, after error occurrence and/or DGs junctions to the network to retrieve some of the sinks, minimal paths between maneuver usually-open points and the sinks are required.To accomplish this part and to prevent routing algorithm reimplementation which will be time-consuming for real network dimensions, a very simple and innovative solution based on a simple principle in the graph theory has been suggested.As per figure (3), if we have two paths of AB and ACD, then BCD path is easily obtained via logical operations of symmetric difference.This idea has been employed for reducing searching operation so that through searching the path between the beginning of the feeder and all the nodes and maneuver points and by using logical operations, the path between all DGs or maneuver points and the sinks will be gained without a new searching operation and the program operation speed will significantly be modified as well.Because searching methods are further more time-consuming comparing with logical operations.

Figure 3 .
Figure 3. Routing of Symmetric Difference Logical Operations between Two Paths The algorithm flowcharts have been shown in figure (4) and (5).

Figure 4 .
Figure 4.The First Block of Modeling Algorithm