DIRECTED GRAPH BASED POWER-FLOW ALGORITHM FOR 1-PHASE RDN

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Darrell Hart
6 years ago
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1 method is used to formulate element-currents during the Forward-Sweep and the inclusion of power variable in the algorithm during Backward Sweep leads to more accurate results. The proposed TT based power-flow solution is verified to be fast converging, with reduced number of iterations for different 1-phase Radial Distribution Network. CHATER 3 DIRECTED GRAH BASED OWER-FLOW ALGORITHM FOR 1-HASE RDN 36

2 3.1 Introduction Modern Distribution System (MDS) need a direct and fast convergence power-flow algorithm. The Directed-Graph (DG) power flow algorithm for 1-phase RDN reduces the intricacy of node-numbering and successfully extract the DG based power-variable information such as: Ordered in-put line-load data Node Load to ath Load [NLL] information which leads to directed-graph power flow solution, when compared to other methods like [17,19]. The new method of indexing the input data and DG based [NLL] are found to be useful for obtaining load, loss and current variables for the network. With these variables, it is easy to find the power-flow solution (voltage & phase-angle). Extending the application of Tellegen s Theorem using Directed-Graph method to build [NLL] matrix, would help in computing load, loss, branch current and power flow solution in matrix form. The performance of algorithm is evaluated in MATLAB for RDN data as in [8]. 3.2 Directed Graph based solution method The Directed Graph (DG) based numbering for a typical RDN is presented in the Fig. 3.1 having n nodes and b (= n-1) elements. The 37

3 nodes in the RDN are numbered path by path from left to right side based on DG. Similarly, it proceeds till the end (node-n) of network Node Numbering and arrangement of input data In Fig.3.1, if the element which lies between i th top and (i+1) th bottom node of radial network, then it is suggested that the element number is the down-stream node number itself. Further, similar thing is applicable to elements. Fig.3.1 ower injection at each node of RDN. 38

4 The input data numbering is done as per above suggestion. The line impedance-data and nodal-power data are arranged in such a way that the receiving end node must be in an ascending order as shown in Table 3.1. Table 3.1 Line-data and Node-data tabulation Line Data Nodal Sending Node Receiving Node Line Impedance ower 1 2 Z 2 S Z 3 S 3 3 i Z i S i r i+1 Z i+1 S i+1 k i+2 Z i+2 S i j n Z n S n where r, k,..j refers to any sending end node for given network. Assigning, i, i+1, i n. in an incremental order to number the receiving end nodes for a given RDN, then specified line-data [ZD] and node-data [SD] with (n-1) 1 sizes can be expressed as Z2 S2 Z3 S3 Zi Si Z S Zi + 2 Si Zn n S i + 1 i + 1 ZD = (3.1) SD = (3.2) 39

5 3.2.2 Directed Graph with paths to find nodal voltage The paths i for IEEE 15-Node [8] are as shown in Fig.3.2. These paths are always traveling along the elements from node 1, to the node at which the voltage is to be calculated. Fig.3.2 ath directed graph of RDN The path lying between nodes 1 and 2 has path impedance Z12 (i.e. Z2). In the same way, path along the nodes 1,2,3 has path impedance. Z2 - Z3. The nodal information about the path is tabulated in Table 3.2. Likewise, path information is identified for path i to n till the end of the network. 40

6 Table 3.2 Directed Graph ath with nodes and elements ath i Nodal-ath information Element-ath information 2 1,2 (2) Z 2 (1) 3 1,2,3 (3) Z 2,Z 3 (2) 4 1,2,3,4 (4) Z 2,Z 3,Z 4 (3) 5 1,2,3,4,5 (5) Z 2,Z 3,Z 4,Z 5 (4) 6 1,2,3,4,5,6 (6) Z 2,Z 3,Z 4,Z 5,Z 6 (5) 7 1,2,3,4,5,6,7 (7) Z 2,Z 3,Z 4,Z 5,Z 6,Z 7 (6) 8 1,2,3,4,5,6,7,8 (8) Z 2,Z 3,Z 4,Z 5,Z 6,Z 7, Z 8 (7) 9 1,2,3,4,5,6,7,8,9 (9) Z 2,Z 3,Z 4,Z 5,Z 6,Z 7,Z 8,Z 9 (8) 10 1,2,3,4,10 (5) Z 2,Z 3,Z 4,Z 10 (4) 11 1,2,3,4,10,11 (6) Z 2,Z 3,Z 4,Z 10,Z 11 (5) 12 1,2,3,12 (4) Z 2,Z 3,Z 12 (3) 13 1,2,3,12,13 (5) Z 2,Z 3,Z 12,Z 13 (4) 14 1,2,3,12,13,14 (6) Z 2,Z 3,Z 12,Z 13,Z 14 (5) 15 1,2,3,12,13,14,15 (7) Z 2,Z 3,Z 12,Z 13,Z 14,Z 15 (6) Construction of Node Load to ath Load matrix Null matrix of order (n-1) (n-1) is created initially. The presence of node-impedance information can be taken as +1 for path i, which is collected from Table 3.2. This necessary information about each impedance [Zi] is used to fill +1 in columns of [NLL] matrix. If total number of 1 s in row of [NLL] matrix indicates node-load or element-loss information in that path sequence Then [NLL] matrix for IEEE 15-Node data [8] can be written as: 41

7 NLL = (3.3) The steps to construct [NLL] Form a [NLL] having null matrix of size (n-1) (n-1). a) Simply place +1 in that column position of [NLL] for any path i, if Zi element is available in that path. b) For the range of node number, i = 2, n, select 1 as incremental value. c) Set +1 in the diagonal position of [NLL]n n. d) The column-wise positions of [NLL] are 2..i,(i+1) th, (i+2) th. n are filled with 1 or 0 according to directed path as per connectivity of the nodes. e) The dimension of the [NLL] is chosen to be (n-1) (n-1), which is suitable to compute (n-1) nodal voltages. 42

8 3.2.4 Backward-Sweep to compute power injections at nodes The power injection [Si] can be expressed at all nodes from k=1 to k=n for RDN using Nodal-Load to ath-load [NLL] information connectivity matrix. Each row in [NLL] gives nodal and element information, which is helpful to compute power injections along the path. Then the nodal power injections are computed by summation of loads and losses in the paths. Here to calculate the power-injections in each path, every relation is expressed in matrix form. The up-stream power-injection is equal to sum of loads and losses in downstream as shown in Fig.3.3. Fig.3.3. Load at nodes and losses in the elements of RDN n-upstream n-upstream [Si] = Node Loads + Branch Losses k=downstream k=downstream (3.4) n-upstream n-upstream [S i] = [SD(k)]+ [EL(k)] k=downstream nodes k=downstream nodes+1 (3.5) where, [SD(k)]=ower Drawn at node-k; [EL(k)]=Element ath Loss 43

9 The diag command of MATLAB can be applied for input node-data of equation (3.2) to keep the load data in diagonal form[sd]. For example node-data [SD] can be expressed as S S S S S S S SD = S S S S S S S 15 a) The information matrix [NLL], when multiplied with load data [SD], it gives directly the loads drawn in the downstream of a path [SD(k)] = [NLL][SD] S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 0 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S S4 S5 S6 S7 S8 S9 S10 S S S S S S S6 S7 S8 S S7 S8 S S8 S SD(K) S S10 S S S S S S S13 S14 S S14 S S15 (3.6a) (3.6b) 44

10 The line-data [ZD] and flat-start voltage matrix [V] are used to calculate the losses in the elements of a network by arranging the elements [S] and [V] in MATLAB as EL abs diag SD./ V diag( SD./ V )) ZD Similarly, when information matrix [NLL] is multiplied with Element Loss [EL], it gives only the Element ath Loss occurring in the downstream of RDN is [EL(k)]= [NLL][EL] (3.7) b) The power-injections [Si] in the upstream is computed using (3.6) and (3.7) as shown below [Si]= [SD(k)]+[EL(k)] (3.8) Forward-Sweep to compute element-current If the injected element-current is taken as conjugate of ratio of the injected power to voltage. Then forward sweep to update most recent current. [ I (k)] = ( [ S (k) ]./[ V (k)])* (3.9) i i In equation(3.9)./ command can be used to compute for element by element division operation in matrix form as per MATLAB. 45

11 3.2.6 Computation of nodal voltage matrix Apply KVL to update the nodal voltage from first level to last level of the radial distribution network shown in Fig.3.6. Fig. 3.4 A typical node voltage level RDN The voltage at (k+1) node is equal to [ V (k+1)] = [ V(k)] - [ I(k+1)][ ZD (k+1)] (3.10) 3.3 erformance evaluation of DG based algorithm The basic power flow methods are analyzed in Table 3.3 as per Forward and Backward Sweep techniques. 46

12 Table 3.3 Comparison of proposed algorithm with other algorithms [24]-Algorithm [17]- Algorithm roposed-algorithm Nodal-Current: The current injection Ii (k) is * Ii(k) = Si( k) V ( k) Backward-Sweep: Expression for element currents (p) I (k 1) i emanating nodes k 1 () In i (k) where In is nodal current Forward sweep: Nodal voltages are computed in forward sweep as Vi(k 1) Vi(k) Z(k 1)Ii(k 1) Nodal-Current: The current injection Ii (k) is * Ii(k) = Si( k) V ( k) Bus-Injection-Bus Current(BIBC): where [BIBC] having 1 s and 0 s converts given nodal-currents information into element currents. Element-Current to Bus- Voltage [BCBV]: Final voltage at bus using [BCBV] [BIBC] and [Ii] is [V(k)] [V(1)] [BCBV][BIBC][I (k)] i Backward-Sweep: The powerinjection Si(k) is [S i] =[ NLL][SD]+[NLL][EL] where [NLL] informative matrix converts the nodal powers and losses into path powers and losses. Injected element-current matrix is the conjugate of the ratio of the injected powers to voltage [Ii] ([Si ]./[Vi]) * Where [I i] upstream injected element-current matrix. [Vi ]is the assumed voltage in matrix form having 1+j*0 Nodal-Voltage [Vi] = [V(k +1) ] [V(k)] [I (k +1) ][ZD(k +1) ] [Vi] (p+1) = updated voltage Eqs. (3.8), (3.9), and (3.11) are to be executed step by step to reach the convergence. The voltage at (p+1) th iteration and then we can find the mismatch voltage can be computed by subtracting previous voltage minus present voltage to obtain correction. [V p+1 ]= [V p ] + [ΔV p+1 ] (3.12) 47

13 Table 3.4 ower-flow results for IEEE -15 bus RDN Volt. [24]-Algorithm [17]-Algorithm roposed-algorithm Mag. Angle Mag. Angle Mag. Angle V V V V V V V V V V V V V V V Table 3.5 erformance comparisons for IEEE 15-node RDN Algorithms Accuracy Iterations Memory NT based method [17] KB LT based method [24] KB roposed Method KB Table 3.6 Results of the for higher values of R/X ratio Vol. 4*(R/X) Line 7*(R/X) Line 10*(R/X) Mag. Angle Mag. Angle Mag. Angle V V V V V V V V V V V V V V V

14 3.4 Results and Discussions The Node to Elemental, which leads path information of RDN are presented in Table 3.2 with known data from Table 3.1. The results of Directed-Graph based method are compared with NT based method of [17] and LN based method of [23] for the Radial Delivery Network data of [8] and comparison of LN and NT are shown in Table 3.3. For high values of R/X the proposed-algorithm convergence results are tabulated in Table 3.4. The proposed-algorithm is found advantageous based on inclusion of element losses along with the nodal powers in equation (3.8). The power injection calculation using [NLL] matrix is helpful for both loads and losses calculation in equation (3.9), which is useful to find nodal voltage obtained from (3.11). The DG-algorithm has been found to be superior in accuracy, number of iterations and efficient as per the results. The collection of DG based power variable information and [Node Load-ath Load] matrix are found useful to obtain thee power flow solution. Table 3.5 indicates the performance test for the set accuracy of The robustness of the algorithm is tested by calculating the voltage and phase angles at nodes of the Radial delivery Network with various values of R/X ratios are presented Table

15 3.5 Conclusions Simple Directed Graph paths are used as information in the proposed-algorithm to obtain ordered and converged power flow solution. The algorithm shows effective way of conversion of directed node-load to path-power and element-losses into path-losses in matrix form using NLL matrix. With the inclusion of element loss during the power-injection computation, it is found that the proposed-algorithm has accurate convergence characteristics when compared to existing algorithms. The algorithm is found to be robust in nature for different values of R/X ratio. The algorithm can be easily extended to solve 3- unbalanced networks also. 50

CHAPTER 2 LOAD FLOW ANALYSIS FOR RADIAL DISTRIBUTION SYSTEM

16 CHAPTER 2 LOAD FLOW ANALYSIS FOR RADIAL DISTRIBUTION SYSTEM 2.1 INTRODUCTION Load flow analysis of power system network is used to determine the steady state solution for a given set of bus loading