Power distribution system configured as a radial network

Abstract

A power distribution system configured as a radial network includes buses having respective voltages, and distribution lines having respective currents. The radial network interconnects the buses with the distribution lines in a tree-like manner. A bus has a link to at least two distribution lines. The bus voltages and distribution line currents are determined by a processing circuitry configured to receive a Branch Matrix (BM), iteratively determine currents for the distribution lines and voltages for each of the buses until a difference is below a predetermined tolerance, and output final bus voltages and final distribution line currents. The circuitry iteratively determines the currents by determining a current matrix (CM) using the BM, and by determining the currents for the plurality of distribution lines in a zig zag manner over the matrix elements in the CM. The system finds a solution using fewer iterations than the backward forward sweep method.

Claims

1. A power distribution system including, in electrical connection, an electricity generation plant, an electricity transmission grid, and an electricity distribution grid having a plurality of power distribution lines, wherein the power distribution system is configured as a radial network comprising: a plurality of buses including a root bus having respective voltages; the plurality of distribution lines having respective currents and impedances, wherein the radial network interconnects the plurality of buses with the plurality of distribution lines beginning with the root bus in a tree-like manner, where the number of buses between a bus and the root bus defines a layer, wherein subsets of the plurality of distribution lines are of different lengths, wherein a voltage of a bus at one of the layers is different from a voltage of a bus at a different one of the layers, wherein at least one of the buses has a link to at least two of the distribution lines, wherein the bus voltages and distribution line currents are determined by a processing circuitry configured to: receive a Branch Matrix (BM) having a row for each distribution line and a column for each bus, wherein the BM contains values indicating buses and associated distribution lines, iteratively determine currents for the plurality of distribution lines and voltages across each of the buses until a difference between a bus voltage and a previous iterated bus voltage at each bus is below a predetermined tolerance and output final bus voltages and final distribution line currents, and assign the final voltages to each respective bus and the final currents to each respective distribution line, wherein the circuitry iteratively determines the currents by determining an initial current matrix (CM) using the BM by multiplying a node current with corresponding matrix elements in the BM, wherein the node current is a ratio of a complex power at a bus and a voltage of the bus, and determining the currents for the plurality of distribution lines in a zig zag manner over the matrix elements in the CM′ and records the determined currents in a final current matrix (CM), the circuitry iteratively determines the voltages by determining the voltage across each bus based on a voltage drop between buses, starting from buses upstream and moving to buses downstream in a direction of voltage flow.

2. The system of claim 1, wherein the circuitry determines the currents for the plurality of distribution lines in the Zig Zag manner by iteratively performing, over all elements of the initial current matrix CM′, applying a Zig Zag formula to calculate CM elements I.sub.ij in accordance with $I_{\mathrm{ij}}=\underset{n=1}{\overset{M}{.Math.}}\left[\frac{I_{ij}^{\prime }}{K_i}+\underset{k=1,2,.Math.,N}{\max }{I}_{ik}\right]\times B_{ij}$ wherein K.sub.i is a number of branches from a source side, B.sub.ij represents the existence of a bus in a certain branch, in a vertical movement, adding the matrix element in the first row, Nth column, to the next row in the Nth column, and continuing until the last element in the last row, last column is reached; in a horizontal movement, moving to column before N−1; and working backwards through all other columns in the matrix CM.

3. The system of claim 1, wherein the circuitry determines the voltage across each bus based on a voltage drop between buses in accordance with
V.sub.i=V.sub.j−I.sub.ij Z.sub.ij where i is a specific bus, the symbol j is a bus with a former index (j<i) that is directly connected to i, and I.sub.ij and Z.sub.ij are the current and the impedance, respectively, of the distribution line between buses i and j.

4. The system of claim 1, wherein the radial network is an extended radial network having multi-terminal distribution lines, wherein the number of links connected to the bus is the number of multi-terminal distribution lines connected to the bus.

5. The system of claim 1, wherein the radial network has an X/R ratio less than 10, wherein the X/R ratio is the amount of reactance X divided by the amount of resistance R.

6. The system of claim 1, wherein each bus is a bus bar, wherein the circuitry iteratively determines voltages across each bus bar.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) A more complete appreciation of this disclosure and many of the attendant advantages thereof will be readily obtained as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings, wherein:

(2) FIG. 1 illustrates an exemplary power grid;

(3) FIG. 2 illustrates a multi-terminal power line;

(4) FIG. 3 is a diagram of an exemplary radial distribution system;

(5) FIG. 4 is a block diagram of the computer system, in accordance with exemplary aspects of the disclosure;

(6) FIG. 5 is a flowchart of a backward/forward sweep (BFS) load flow method;

(7) FIG. 6 is an exemplary 5-bus circuit;

(8) FIG. 7 is a flowchart of a Zig-Zag load flow method, in accordance with exemplary aspects of the disclosure;

(9) FIG. 8 is a flowchart of a method for finding the currents for the branches in a Zig Zag manner, in accordance with exemplary aspects of the disclosure;

(10) FIG. 9 is a graph of a voltage profile of a 5 node radial distribution network as a comparison of BFS and Zig Zag techniques;

(11) FIG. 10 is a graph of a voltage profile of a 7 node radial distribution network as a comparison of BFS and Zig Zag techniques;

(12) FIG. 11 is a graph of a voltage profile of a 11 node radial distribution network as a comparison of BFS and Zig Zag techniques;

(13) FIG. 12 is a graph of a voltage profile of a 25 node radial distribution network as a comparison of BFS and Zig Zag techniques;

(14) FIG. 13 is a graph of a voltage profile of a 28 node radial distribution network as a comparison of BFS and Zig Zag techniques;

(15) FIG. 14 is a graph of a voltage profile of a 30 node radial distribution network as a comparison of BFS and Zig Zag techniques;

(16) FIG. 15 is a graph of a voltage profile of a 33 node radial distribution network as a comparison of BFS and Zig Zag techniques;

(17) FIG. 16 is a graph of a voltage profile of a 34 node radial distribution network as a comparison of BFS and Zig Zag techniques;

(18) FIG. 17 is a graph of a voltage profile of a 69 node radial distribution network as a comparison of BFS and Zig Zag techniques; and

(19) FIG. 18 is a flowchart in accordance with exemplary aspects of the disclosure.

DETAILED DESCRIPTION

(20) In the drawings, like reference numerals designate identical or corresponding parts throughout the several views. Further, as used herein, the words “a,” “an” and the like generally carry a meaning of “one or more,” unless stated otherwise. The drawings are generally drawn to scale unless specified otherwise or illustrating schematic structures or flowcharts.

(21) Furthermore, the terms “approximately,” “approximate,” “about,” and similar terms generally refer to ranges that include the identified value within a margin of 20%, 10%, or preferably 5%, and any values therebetween.

(22) Backward/Forward Sweep (BFS) and Zig Zag power flow techniques in this disclosure both achieve accurate distribution load flow results in radial distribution systems. Aspects of this disclosure are directed to a Zig Zag-based load flow approach that achieves the same or better distribution load flow accuracy using fewer iterations to converge than the BFS power flow analysis approach. The Zig Zag-based load flow is applicable to extended radial distribution systems having multi-terminal lines.

(23) The backward/forward sweep method for the load flow computation is an iterative method in which, at each iteration, two computational stages are performed. The first stage is for calculation of the power flow through the branches starting from the last branch and proceeding in the backward direction towards the root node. The other stage is for calculating the voltage magnitude and angle of each node starting from the root node and proceeding in the forward direction towards the last node.

(24) A radial distribution network interconnects the buses with the distribution lines beginning with the root bus in a tree-like manner. The number of buses between a bus and the root bus defines a layer. For example, a bus that is the third bus in the load direction from the root is in a second layer of the network. The distribution lines may be of different lengths, and thus will have corresponding impedance and voltage drop. Also, the voltage of a bus at one of the layers may be different from a voltage of a bus at a different one of the layers. Any bus may have a link to at least two of the distribution lines. The radial configuration entails that there are no connected loops in the system and each node is linked to the source via one way. Loops might exist but separated via a normally open breaker. A distribution system combines single-, two-, and three-phase circuits, delta-connected and unbalanced loads that are considered challenging. It is the most economical power supply and has an acceptable reliability index to certain types of customers. The radial distribution configuration is typically used in sparsely populated areas and in the remote scattered hydrocarbon facilities.

(25) Embodiments of the present invention create a nodal current matrix and, using the nodal current matrix, find the currents for the system branches in a Zig Zag manner. The zig zag manner of finding currents for system branches replaces the backward sweep stage of the BFS method and greatly reduces the number of iterations needed for convergence.

(26) FIG. 18 illustrates an embodiment of the present disclosure in flowchart form. FIG. 18 shows steps of a method for load flow analysis of a power distribution system. The method begins at S1801 that shows receipt of a Branch Matrix (BM) having a row for each power distribution line and a column for each bus in which the BM contains values indicating buses and associated power distribution lines. This is followed by S1802 including determining currents for the plurality of power distribution lines and voltages across each of the buses until a difference between a bus voltage and a previous iterated bus voltage at each bus is below a predetermined tolerance. Determining the current (S1802) includes (S1803) determining an initial current matrix (CM′) using the BM by multiplying a node current with corresponding matrix elements in the BM in which the node current is a ratio of a complex power at a bus and a voltage of the bus and (S1804) determining the currents for the plurality of power distribution lines in a zig zag manner over the matrix elements in the CM′ and records the determined currents in a final current matrix (CM). S1805 shows determining the voltage across each bus based on a voltage drop between buses, starting from buses upstream and moving to buses downstream in the direction of voltage flow. The bus voltages are displayed on a graph for each bus (S1806).

Computer System

(27) The disclosed methods may be performed in a computer system, such as a laptop computer, desktop computer, or a computer workstation. The disclosed method may also be performed in a virtual computer environment such as a cloud service, data center or virtualized server network.

(28) FIG. 4 is a block diagram of a basic computer system. In one implementation, the functions and processes of the computer system may be implemented by a computer 426. Next, a hardware description of the computer 426 according to exemplary embodiments is described with reference to FIG. 4. In FIG. 4, the computer 426 includes a CPU 400 which performs the processes described herein. The process data and instructions may be stored in memory 402. These processes and instructions may also be stored on a storage medium disk 404 such as a hard drive (HDD) or portable storage medium or may be stored remotely. Further, the claimed advancements are not limited by the form of the computer-readable media on which the instructions of the inventive process are stored. For example, the instructions may be stored on CDs, DVDs, in FLASH memory, RAM, ROM, PROM, EPROM, EEPROM, hard disk or any other information processing device with which the computer 426 communicates, such as a server or computer.

(29) Further, the claimed advancements may be provided as a utility application, background daemon, or component of an operating system, or combination thereof, executing in conjunction with CPU 400 and an operating system such as Microsoft® Windows®, UNIX®, Oracle® Solaris, LINUX®, Apple macOS® and other systems known to those skilled in the art.

(30) In order to achieve the computer 426, the hardware elements may be realized by various circuitry elements, known to those skilled in the art. For example, CPU 400 may be a Xenon® or Core® processor from Intel Corporation of America or an Opteron® processor from AMD of America, or may be other processor types that would be recognized by one of ordinary skill in the art. Alternatively, the CPU 400 may be implemented on an FPGA, ASIC, PLD or using discrete logic circuits, as one of ordinary skill in the art would recognize. Further, CPU 400 may be implemented as multiple processors cooperatively working in parallel to perform the instructions of the inventive processes described above.

(31) The computer 426 in FIG. 4 also includes a network controller 406, such as an Intel Ethernet PRO network interface card from Intel Corporation of America, for interfacing with network 424. As can be appreciated, the network 424 can be a public network, such as the Internet, or a private network such as LAN or WAN network, or any combination thereof and can also include PSTN or ISDN sub-networks. The network 424 can also be wired, such as an Ethernet network, or can be wireless such as a cellular network including EDGE, 3G and 4G wireless cellular systems. The wireless network can also be WiFi®, Bluetooth®, or any other wireless form of communication that is known.

(32) The computer 426 further includes a display controller 408, such as a NVIDIA® GeForce® GTX or Quadro® graphics adaptor from NVIDIA Corporation of America for interfacing with display 410, such as a Hewlett Packard® HPL2445w LCD monitor. A general purpose I/O interface 412 interfaces with a keyboard and/or mouse 414 as well as an optional touch screen panel 416 on or separate from display 410. General purpose I/O interface also connects to a variety of peripherals 418 including printers and scanners, such as an OfficeJet® or DeskJet® from Hewlett Packard®.

(33) The general purpose storage controller 420 connects the storage medium disk 404 with communication bus 422, which may be an ISA, EISA, VESA, PCI, or similar, for interconnecting all of the components of the computer 426. A description of the general features and functionality of the display 410, keyboard and/or mouse 434, as well as the display controller 408, storage controller 420, network controller 406, and general purpose I/O interface 412 is omitted herein for brevity as these features are known.

Backward/Forward Sweep

(34) Rupa describes a backward/forward sweep method that uses an objective function in terms of real and reactive power to find power flow as:

(35) $$\begin{gathered} P_{k+1}=P_k-P_{\mathrm{loss},k}-P_{Lk+1} \\ {Q}_{k+1}=Q_k-Q_{\mathrm{loss},k}-Q_{Lk+1} \end{gathered}$$
where P.sub.k is the real power flowing out of a bus; Q.sub.k is the reactive power flowing out of the bus; P.sub.Lk+1 is the real power at bus k+1; Q.sub.Lk+1 is the reactive power at bus k+1, where k is an identifier assigned to a bus and k+1 refers to a bus at the other end of a line in the load direction. During backward propagation, effective real and reactive power flows of all branches are computed. During forward propagation, node (bus) voltages and phase angles are updated. The power loss P.sub.loss,k or Q.sub.loss,k may be computed in the line section connecting busses k and k+1. The total power loss of the feeder may be determined by summing up the losses of all line sections of the feeder.

(36) FIG. 5 is a flowchart of a Backward/Forward Sweep method. The method in FIG. 5 has been implemented for purposes of comparison with the Zig Zag method of the present invention. As mentioned above, BFS is one of the robust techniques that is widely used in a distribution network analysis for load flow in radial distribution systems. This is because of its effectiveness for radial and weak meshed systems.

(37) In S501, the BFS method reads network configuration data. In S503, the BFS method initializes voltage V and current I. The BFS method then performs the backward sweep S505 to calculate currents. The currents are calculated across each branch, starting with the last layer in the system. In the backward sweep, the BFS method moves backward to the branches connected to the Swing bus, which is usually bus no. 1. The currents are calculated using the following formula:

(38) $\begin{array}{cc}{I}_i={\left(\frac{S_i}{V_i}\right)}^{*}+\underset{k=a,b,c,.Math.\mathrm{etc}}{.Math.}{I}_k& \left(1\right)\end{array}$

(39) Where i is a specific branch in which the current is calculated. The symbols a, b and c, etc. represent the branches connected to Branch i from the load side. For purposes of this disclosure, real and reactive power are represented as the symbol S for complex power in a per unit quantity.

(40) Next, the BFS method performs a forward sweep S507 to calculate the voltages. The voltage across each bus is calculated in a forward manner. In the forward sweep, the BFS method starts with the Swing bus and covers all the other buses in the network. The calculation is performed using the following voltage drop formula.

(41) $\begin{array}{cc}{V}_i=V_j-I_{ij}{Z}_{ij}& \left(2\right)\end{array}$

(42) Where i is a specific bus that is under calculation. The symbol j is a bus with a former index (j<i) that is directly connected to i (load side). I.sub.j and Z.sub.ij are the current and impedance between buses i and j.

(43) The BFS method is performed iteratively until a convergence criteria, S509, is met. In some BFS approaches, convergence may be achieved when a voltage mismatch (difference between voltage V and a previous voltage Vat bus i) is less than a tolerance of about 0.0001.

(44) When the BFS method meets the convergence criteria (YES in S509), in S511, the computer system outputs final branch currents and final bus voltages V.sub.i. The final bus voltages may be displayed as a graph for each bus, or a selected range of busses.

Zig Zag-Based Load Flow

(45) Disclosed embodiments relate to an improvement over the BFS method. It has been determined that a specific order of processing over a matrix may lead to fewer number of iterations to converge. In particular, the disclosed embodiments involve processing over a matrix in a Zig Zag order. The matrix that is processed in this manner may have several hundred to several thousand rows and columns. The Zig Zag-based load flow method may be implemented as a computer program in a programming language such as Fortran or C, or in a numerical computing environment such as MatLab, Mathematica, or scripting language such as Python, to name a few. The computer program may be executed in a computer system such as that in FIG. 4.

(46) FIG. 7 is a flowchart of a Zig Zag-based load flow method, in accordance with exemplary aspects of the disclosure. The Zig Zag method calculates currents using a zig zag pattern over a nodal current matrix instead of using backward sweep as in the BFS method. The Zig Zag-based method is applicable to extended radial distribution systems where multi-terminal lines exist. The Zig Zag-based load flow method begins with, S701, reading in network configuration data and, in S703, initializing voltage V and current I. Configuration data involves labeling of branches and nodes in the radial distribution network. As noted above, a branch is a distribution line between two nodes. A branch will have a current flow and a voltage drop due in part to the length of a branch. A node (bus), also referred to as a bus bar, distributes power to distribution lines, which fan out to other buses or customers. The bus will have a voltage. Also, a load flow analysis may be used for designing a new distribution network or expansion of an existing distribution network. The branches and nodes represent distribution lines and buses for a new distribution network or expansion in an existing network. For purposes of simplifying a representation of a network, branches may be represented as single lines and busses may be represented as a connection point between branches.

(47) After initialization, the Zig Zag-based method includes the following steps: 1) S705, creating a branch matrix (BM) as shown below where the rows reflect the circuit branches and the columns represent circuit buses.

(48) $$\begin{gathered} \begin{array}{cc}\mathrm{BM}=\left[\begin{array}{ccc}{B}_{11}& .Math.& B_{1N}\\ .Math.& \ddots & .Math.\\ B_{M1}& .Math.& B_{MN}\end{array}\right] \\ {B}_{ij}=\{\begin{array}{cc}1& \mathrm{if}\mathrm{bus}j\mathrm{exists}\mathrm{in}\mathrm{Branch}i\\ 0& \mathrm{if}\mathrm{bus}j\mathrm{does}\mathrm{not}\mathrm{exist}\mathrm{in}\mathrm{Branch}i\end{array}& \left(3\right)\end{array} \end{gathered}$$ B.sub.ij represents the existence of a bus in a certain branch. It takes either 1 (bus exists) or 0 (bus does not exist). M and N reflects the number of branches and buses in a circuit respectively. FIG. 6 illustrates an example 5-bus circuit with five buses and two branches. A BM matrix for the example 5-bus circuit shown in FIG. 6, is as follows:

(49) $\begin{array}{cc}\mathrm{BM}=\left[\begin{array}{ccccc}1& 1& 1& 1& 0\\ 0& 0& 1& 0& 1\end{array}\right]& \left(4\right)\end{array}$ A value of 1 in the BM matrix indicates a bus associated with the line. The above mentioned system has two lines (Rows 1 and 2) and five buses (Columns 1-5). Line 1 (Row 1) has four buses (Buses 1-4) represented by a value of one at Rows1-4. Line 2 (Row 2) has two buses (Bus 3 and 5) represented by a value of one at Rows 3 and 5. A value of 0 indicates no bus at the specific line (row). 2) S707, developing a new matrix called initial current matrix (CM′).

(50) $\begin{array}{cc}{\mathrm{CM}}^{\prime }=\left[\begin{array}{ccc}{I}_{11}^{\prime }& .Math.& I_{1N}^{\prime }\\ .Math.& \ddots & .Math.\\ I_{M1}^{\prime }& .Math.& I_{MN}^{\prime }\end{array}\right]& \left(4\right)\end{array}$ Where I′.sub.ij represent the load current of Bus i given by this formula:

(51) $\begin{array}{cc}{I}_{ij}^{\prime }={\left(\frac{S_i}{V_i}\right)}^{*}& \left(5\right)\end{array}$ Where S.sub.i represent complex power at Bus I, V.sub.i represent bus voltage. For the 5-bus test circuit displayed in FIG. 6, the CM matrix is given as follows:

(52) $\begin{array}{cc}C{M}^{\prime }=\left[\begin{array}{ccccc}{I}_1^{\prime }& I_2^{\prime }& I_3^{\prime }& I_4^{\prime }& 0\\ 0& 0& I_3^{\prime }& 0& I_5^{\prime }\end{array}\right]& \left(6\right)\end{array}$ 3) In S709, the method determines the currents for the system branches in a Zig Zag manner over the CM matrix and records them in the final matrix denoted by CM matrix. Each bus will have its own current tagged with the branch.

(53) $\begin{array}{cc}\mathrm{CM}=\left[\begin{array}{ccc}{I}_{11}& .Math.& I_{1N}\\ .Math.& \ddots & .Math.\\ I_{M1}& .Math.& I_{MN}\end{array}\right]& \left(7\right)\end{array}$$\begin{array}{cc}{I}_{\mathrm{ij}}=\underset{n=1}{\overset{M}{.Math.}}\left[\frac{I_{ij}^{\prime }}{K_i}+\underset{k=1,2,.Math.,N}{\max }{I}_{ik}\right]\times B_{ij}& \left(8\right)\end{array}$
I.sub.ij is called the Zig Zag formula which calculates the elements of the CM matrix in a Zig Zag manner. The symbol K.sub.i reflects the number of links connected to a specific Bus i (i.e., K is the number of branches from a source side in the case of multi-terminal lines). The CM matrix for the 5-bus test circuit presented in FIG. 6 is formulated as follows:

(54) 0$\begin{array}{cc}CM=\left[\begin{array}{ccccc}{I}_1& I_2& I_3& I_4& 0\\ 0& 0& I_3& 0& I_5\end{array}\right]& \left(9\right)\end{array}$ 4) In S711, the voltage will be identified using the CM matrix and is calculated using the voltage drop formula, starting from upstream through downstream.

(55) The above steps are repeated iteratively until, in S713, the deviation between the load voltage and the previous voltage is below a specified tolerance for the process to converge. The tolerance may be a predetermined value that is within the precision of the computer system used and the likely accuracy achievable by the load flow analysis.

(56) When the Zig Zag-based method meets the convergence criteria (YES in S713), in S715 the computer system outputs final bus voltages V.sub.i. In some embodiments, the final bus voltages may be displayed as a graph for each bus, or a selected range of busses. From the bus voltages (angle and magnitude) all other power circuit parameters can be derived using power flow formulas. This includes active and reactive load flow, voltage drop, losses, etc.

(57) FIG. 8 is a flowchart of a method for finding the currents for the branches in a Zig Zag manner, in accordance with exemplary aspects of the disclosure.

(58) The zig zag method is performed as follows: 1) In S801, initialize the current matrix (CM) by setting all the elements to 0. 2) In S803, apply the Zig Zag formula developed above to calculate all CM elements 3) In S805, in the current matrix (CM), perform vertical movement: starting from 1×N (First Row×Last Column) moving to Element M×N (Last Row×Last Column), where ‘x’ represents position, adding the matrix element in the first row, Nth column, to the next row in the Nth column, and continuing until the last element in the last column is reached.

(59) In S807, perform horizontal movement: move to the column before (N−1).

(60) Repeating S803 to S807, working backwards through other columns, till, in S809, it completes the analysis for the whole matrix. In S811, the CM matrix is saved and used in determining the voltages.

(61) This Zig Zag methodology is applicable to extended radial with multi-terminal lines distribution system. The matrices approach will cater for any extension in the system by expanding the CM dimension and reflecting the number of terminals in the K factor.

Results

(62) Implementations of the BFS and Zig Zag methods have been developed and tested using 5, 7, 11, 25, 28, 30, 33, 34, and 69 —bus distribution systems. The simulation results using the implementations are tabulated in Table, Table 2 and Table 3.

(63) TABLE-US-00001 TABLE 2 DEVIATION OF THE LAST TWO ITERATIONS (10.sup.−6) System BFS Zig Zag  5 5.61 2.85  7 2.69 2.69 11 2.95 2.95 25 7.75 4.10 28 9.53 6.45 30 9.35 2.13 33 7.93 0.22 34 9.23 3.63 69 7.87 1.44 Average 6.99 2.94

(64) Accuracy of each of the methods is identified based on the differences between last two iterations. Table 2 shows that the Zig Zag method is superior in accuracy by 58% compared to the BFS method.

(65) TABLE-US-00002 TABLE 1 Required Iterations to Convergence System BFS Zig Zag 5 9 7 7 8 8 11 9 9 25 42 7 28 141 23 30 44 7 33 25 4 34 46 7 69 50 7 Average 42 9

(66) The average iterations required by the BFS method to converge is approximately five times the iterations required by the Zig Zag method as demonstrated in Table 1 (42 iterations for BFS compared to 9 for Zig Zag). The number of iterations associated with each case study for the two methods are presented in Table 1. For example, the results for the 5-bus test circuit converged after seven iterations for the Zig Zag method and resulted in a difference between the last iteration and the one before of 2.85E-06. This shows the robustness and the effectiveness of the Zig Zag method.

(67) TABLE-US-00003 TABLE 3 Elapsed Time to Converge System BFS Zig Zag  5 0.024248 0.024681  7 0.020715 0.031041 11 0.021658 0.035335 25 0.027870 0.034460 28 0.039134 0.059562 30 0.029935 0.034840 33 0.028292 0.027398 34 0.038784 0.032420 69 0.041999 0.049108 Average 0.030293 0.036538

(68) In terms of elapsed time, though the Zig Zag method can provide comparatively outstanding results when compared to the BFS method, the time taken by the Zig Zag method is pretty much the same as the BFS method as observed in Table 3, and also the quality of the solution is better. As a result, a conclusion can be made that the Zig Zag method is the best technique for the studied systems and results in considerably better solutions. Also, better convergence time may be achieved from the Zig Zag method by further optimization of the algorithm.

(69) The voltage profiles of the simulated systems for 5, 7, 11, 25, 28, 30, 33, 34, and 69—bus distribution systems are displayed in FIG. 9 to FIG. 17. The two methods are shown in each figure to compare the load flow results. The figures prove the effectiveness of the Zig Zag method as it has provided exactly the same voltage profiles as obtained by the BFS method.

(70) Numerous modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described herein.