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International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.9, pp. 1539-1544 ISSN 2078-2365 http://www.ieejournal.com/ Backward/Forward Sweep Based Distribution Load Flow Method S. Sunisith1, K. Meena2 [email protected], [email protected] Abstract—The function of an electric power system is to connect the power station to the consumer loads by means of inter connected system of transmission and distribution networks. Therefore, an electric power system consists of three principal components: The power station, the transmission lines and the distribution system. The power flow method is a fundamental tool in application software for distribution management system. In this paper, a method to solve the distribution power flow problem has been introduced. The reason, why the convergence of these widely used methods deteriorates when the network becomes radial, is also well analyzed. Subsequently, a theoretical formulation of backward/forward sweep distribution load flow method is described. This proposed method has clear theory foundation and takes full advantage of the radial structure of distribution systems. The numerical test proved that, this method is very robust and has excellent convergence characteristics. Index Terms—Backward/Forward Sweep, Distribution load flow method. I. CONNECTION SCHEMES OF DISTRIBUTION SYSTEM All distribution of electrical energy is done by constant voltage system. In practice, the following distribution circuits are generally used i) Radial System: In this system, separate feeders radiate from a single substation and feed the distributors at one end only. Fig. 1.1(i) shows a single line diagram of a radial system for dc distribution where a feeder OC supplies a distributor AB at point A. Obviously, the distributor is fed at one end only i.e., point A is this case. Fig. 1.1(ii) shows a single line diagram of radial system for ac distribution. The radial system is employed only when power is generated at low voltage and the substation is located at the centre of the load. Fig. 1.1 (i) & (ii) Radial System This is the simplest distribution circuit and has the lowest initial cost. However, it suffers from the following drawbacks a) The end of the distributor nearest to the feeding point will be heavily loaded. b) The consumers are dependent on a single feeder and single distributor. Therefore, any fault on the feeder or distributor cuts off supply to the c) Consumers who are on the side of the fault away from the substation. d) The consumers at the distant end of the distributor would be subject to serious voltage fluctuations when the load on the distributor changes. Due to these limitations, this system is used for short distances only. ii) Ring main system: In this system, the primaries of distribution transformers form a loop. The loop circuit starts from the substation bus-bars, makes a loop through the area to be served, and returns to the substation. Fig. 1.2 shows the single line diagram of ring main system for ac distribution where substation supplies to the closed feeder LMNOPQRS. The distributors are tapped from different points M, O and Q of the feeder through distribution transformers. The ring main system has the following advantages a) There are less voltage fluctuations at consumer’s terminals. b) The system is very reliable as each distributor is fed via two feeders. In the event of fault on any section of the feeder, the continuity of supply is maintained. For example, 1539 S. Sunisith et. al., Backward/Forward Sweep Based Distribution Load Flow Method International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.9, pp. 1539-1544 ISSN 2078-2365 http://www.ieejournal.com/ suppose that fault occurs at any point F of section SLM of the feeder. Then section SLM of the feeder can be isolated for repairs and at the same time continuity of supply is maintained to all the consumers via the feeder SRQPONM. Fig.1.2 Ring main System (iii) Interconnected system: When the feeder ring is energized by two or more than two generating stations or substations, it is called inter-connected system. Fig. 1.3 shows the single line diagram of interconnected system where the closed feeder ring ABCD is supplied by two substations S1 and S2 at points D and C respectively. Distributors are connected to points O, P, Q and R of the feeder ring through distribution transformers. performance of a power system and for analyzing the effectiveness of alternative plans for system expansion to meet increased load demand. These analyses require the calculation of numerous load flows for both normal and emergency operating conditions. The load flow problem consists of the calculation of power flows, and voltages of a network for specified terminal or bus conditions. A single phase representation is adequate since power systems are usually balanced. Associated with each bus are four quantities: the real and reactive power, the voltage magnitude and the phase angle. Three types of buses are represented in the load flow calculation and at a bus, two of the four quantities are specified. It is necessary to select one bus, called the slack bus, to provide the additional real and reactive power to supply the transmission losses. At this bus the voltage magnitude and phase angle are specified. The remaining buses of the system are designated either as voltage controlled buses or load buses. The real power and voltage magnitudes are specified at a voltage controlled bus. The real and reactive powers are specified at a load bus. The static load flow equations are given by Pp Ep Eq Ypq cos(pq q p) n q 1 Qp - Ep Eq Ypq sin(pq q p) n q 1 The above equations are said to be non linear in nature because of involvement of trigonometric terms. Direct solution is not possible, we need to apply iterative Techniques to solve the equations. Those are i. Gauss -Seidal Method ii. Newton- Raphson Method iii. Newton’s Decoupled Method iv. Fast Decoupled Method. Fig. 1.3 Inter Connected System The interconnected system has the following advantages: (a) It increases the service reliability. (b) Any area fed from one generating station during peak load hours can be fed from the other generating station. This reduces reserve power capacity and increases efficiency of the system. II. LOAD FLOW STUDIES Load flow calculations provide power flows and voltages for a specified power system subject to the regulating capability of generators, condensers and tap changing under load transformers as well as specified net interchange between individual operating systems. This information is essential for the continuous evolution of the current III. BACKWARD/FORWARD SWEEP BASED DISTRIBUTION LOAD FLOW METHOD Our studies have shown that, typically, only a few iterations were required for the solution of distribution networks using this power flow solution technique. For the weakly meshed transmission networks the number of iterations was higher, due to the additional nonlinearities introduced by generator buses (PV nodes). In all the cases studied this power flow technique was significantly more efficient than the Newton-Raphson power flow algorithm while converging to the same solution. In this chapter a brief discussion of Breadth First Search method and its applications are provided. Then we emphasize the application of the backward/forward sweep based 1540 S. Sunisith et. al., Backward/Forward Sweep Based Distribution Load Flow Method International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.9, pp. 1539-1544 ISSN 2078-2365 http://www.ieejournal.com/ distribution load flow method to the distribution networks and few practical considerations concludes the Paper. In our algorithm, regardless of its original topology, the distribution network is first converted to a radial network. Hence, an efficient algorithm for the solution of radial networks is crucial to the viability of the overall solution method. The solution method used for radial distribution networks is based on the direct application of the KVL and KCL. For our implementation, we developed a node and branch oriented approaches using an efficient numbering scheme to enhance the numerical performance of the solution method. We first describe this node numbering scheme. branches connected to the root node, the current in branch L, JL is calculated as JL(k)= - IL2(k) + ∑ (currents in branches emanating from node L2) L=b, b-1 ,…….,1 Where IL2(k) is the current injection at node L2. This is the direct application of the KCL. IV. BREADTH FIRST SEARCH METHOD (BFS) In graph theory breadth-first search (BFS) is a graph search algorithm that begins at the root node and explores all the neighboring nodes. Then for each of those nearest nodes, it explores their unexplored neighbor nodes, and so on, until it finds the goal. Fig. 5.1 single line diagram of IEEE 15 bus system using BFS and Branch numbering scheme Fig. 4.1 Example of breadth first search method BFS is a uniformed search method that aims to expand and examine all nodes of a graph systematically in search of a solution. In other words, it exhaustively searches the entire graph without considering the goal until it finds it. It does not use a heuristic. V. SOLUTION METHODOLOGY Given the voltage at the root node and assuming a flat profile for the initial voltages at all other nodes, the iterative solution algorithm consists of three steps 1. Nodal current calculation: At iteration k, the nodal current injection, Ii(k), at network node i is calculated as, Ii(k) = (Si / Vi(k-1))* - YiVi(k-1) i=1,2,…….,n where ,Vi(k-l) is the voltage at node i calculated during the (k-l)th iteration and Si is the specified power injection at node i. Yi is the sum of all the shunt elements at the node i. 2. Backward sweep: At iteration k, starting from the branches in the last layer and moving towards the For IEEE 15-Bus system the branch currents from the Fig. 5.1 are calculated as given below J[14]= -I[15]; J[13]= -I[14]+(J[14]); J[12]= -I[13]; J[11]= -I[12]; J[10]= -I[11]; J[9]= -I[10]; J[8]= -I[9]+(J[13]); J[7]= -I[8]+(J[10]+J[11]+J[12]); J[6]= -I[7]; J[5]= -I[6]; J[4]= -I[5]+(J[9]); J[3]= -I[4]+(J[7]+J[8]); J[2]= -I[3]+(J[5]+J[6]); J[1]= -I[2]+(J[2]+J[3]+J[4]); 3. Forward sweep: Nodal voltages are updated in a forward sweep starting from branches in the first layer toward those in the last. For each branch, L, the voltage at node L2 is calculated using the updated voltage at node L1 and the branch current calculated in the preceding backward sweep. 1541 S. Sunisith et. al., Backward/Forward Sweep Based Distribution Load Flow Method International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.9, pp. 1539-1544 ISSN 2078-2365 http://www.ieejournal.com/ VII. FLOW CHART VL2(k) = VL1(k) – ZL JL(k) L=1,2,….,b Where, ZL is the series impedance of branch L. This is the direct application of the KVL.. For IEEE 15 bus system the nodal voltages from Fig. 5.1 are calculated as given below v[2]=v[1]-(z[1]*J[1]); v[3]=v[2]-(z[2]*J[2]); v[4]=v[2]-(z[3]*J[3]); v[5]=v[2]-(z[4]*J[4]); v[6]=v[3]-(z[5]*J[5]); v[7]=v[3]-(z[6]*J[6]); v[8]=v[4]-(z[7]*J[7]); v[9]=v[4]-(z[8]*J[8]); v[10]=v[5]-(z[9]*J[9]); v[11]=v[8]-(z[10]*J[10]); v[12]=v[8]-(z[11]*J[11]); v[13]=v[8]-(z[12]*J[12]); v[14]=v[9]-(z[13]*J[13]); v[15]=v[14]-(z[14]*J[14]); Steps 1, 2 and 3 are repeated until convergence is achieved. VI. CONVERGENCE CRITERION We used the maximum real and reactive power mismatches at the network nodes as our convergence criterion. As described in the solution method, the nodal current injections, at iteration k, are calculated using the scheduled nodal power injections and node voltages from the previous iteration. The node voltages at the same iteration are then calculated using these nodal current injections. Here, the power injection for node i at kth iteration, Si(k) is calculated as Si(k) = Vi(k)(Ii(k))* - Yi │Vi(k)│2 --------(6.1) The real and reactive power mismatches at bus i are then calculated as ∆Pi(k) = Re [ Si(k) – Si ] ∆Qi(k) = Im [Si(k) – Si ] VIII. ALGORITHM FOR BACKWARD/FORWARD SWEEP BASED DISTRIBUTION LOAD FLOW METHOD Step 1: Read power system data, i.e., no. of buses, no. of lines, slack bus, base kV, base kVA, bus data, line data. Step 2: Starting from the root node, number the nodes and branches in the network by using Breadth First Search method and Branch Numbering Scheme respectively. Step 3: Calculate the injected powers, i.e., Pinj(i) = Pgen(i) – Pload(i) Qinj(i) = Qgen(i) – Qload(i) , i=1,2,……,n i=1,2,………,n Step 4: Set iteration count, k=1. Step 5:Set convergence є=0.001, ∆Pmax=0.0and Qmax=0.0. Step 6: Calculate nodal current injection at network node ‘i’ as Ii(k) = ( Si / Vi(k-1) )* - YiVi(k-1) i=1,2,…….,n 1542 S. Sunisith et. al., Backward/Forward Sweep Based Distribution Load Flow Method International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.9, pp. 1539-1544 ISSN 2078-2365 http://www.ieejournal.com/ Step 7: Backward sweep: Calculate current in the branch L, JL as JL(k)= - IL2(k) + ∑ (currents in branches emanating from node L2) L=b, b-1 ,…….,1 Step 8: Forward sweep: Calculate the voltage at node L2 as VL2 = VL1(k) – ZL JL(k) L=1,2,….,b Step 9: Calculate the power injection at node ‘i’ as Si(k) = Vi(k)(Ii(k))* - Yi │Vi(k)│2 Step 10: Calculate real and reactive power mismatches as ∆Pi(k) = Re [ Si(k) – Si ] ∆Qi(k) = Im [Si(k) – Si ] Fig. 9.1 single line diagram of IEEE 15 bus system using BFS and Branch numbering scheme i=1,2,………,n Step 11: Check ∆Pi(k) > ∆Pmax, then set ∆Pmax=∆Pi(k) Test results of backward/forward sweep based distribution load flow method 1. The no. of iterations taken for convergence = 3 2. Convergence criteria = 0.001 (ii) IEEE 31-BUS DISTRIBUTION SYSTEM ∆Qi(k) > ∆Qmax , then set ∆Qmax=∆Qi(k) Step 12: If ∆Pmax <= є and ∆Qmax <= є , then go to step 14 Else go to step 13 Step 13: Set k=k+1 and go to step 4. Step 14: Print that problem is converged in ‘k’ iterations. Step 15: Stop IX. TEST RESULTS No. of buses = 31 No. of lines = 30 Substation bus = 1 Base kV = 23 Base kVA = 100 (i) IEEE 15- BUS DISTRIBUTION SYSTEM No. of buses = 15 No. of lines = 14 Sub station bus = 1 Base kV = 11 Base kVA = 100 Fig. 9.2 single line diagram of IEEE 31 bus system using BFS and Branch numbering scheme 1543 S. Sunisith et. al., Backward/Forward Sweep Based Distribution Load Flow Method International Electrical Engineering Journal (IEEJ) Vol. 5 (2014) No.9, pp. 1539-1544 ISSN 2078-2365 http://www.ieejournal.com/ Test results of backward/forward sweep based distribution load flow method 1. The no. of iterations taken for convergence = 5 2. Convergence criteria = 0.001 (iii) IEEE 69-BUS DISTRIBUTION SYSTEM No. of buses = 69 No. of lines = 68 Sub station bus = 1 Base kV = 12.66 Base kVA = 1000 Test results of backward/forward sweep based distribution load flow method 1. The no. of iterations taken for convergence = 3 2. Convergence criteria = 0.001 (iv) IEEE 85-BUS DISTRIBUTION SYSTEM No. of buses = 85 No. of lines = 84 Sub station bus = 1 Base kV = 11 Base kVA = 100 Test results of backward/forward sweep based distribution load flow method 1. The no. of iterations taken for convergence = 3 2. Convergence criteria = 0.001 [1] Jatin Singh Saini, M.P.Sharma, S.N.Singh, “Voltage profile improvement of rural distribution network by conductor replacement”, International Electrical Engineering Journal (IEEJ), Vol. 5 (2014) No.7, pp. 1490-1494, ISSN 2078-2365. [2] Shirmohammadi D, Hong HW, Semlyen A, et al. “A compensation based power flow method for weakly meshed distribution and transmission networks”. IEEE Trans Power Syst 1988;3 (2):753-61. [3] Zhang Fang, Cheng Carol S. A modified Newton method for radial distribution system power flow analysis. IEEE Trans Power Syst 1997;12 (1): 389-97. [4] Van Amerongen RAM. “A general purpose version of the fast decoupled load flow”. IEEE Trans Power Syst 1989;4 (2):760-70. [5] Zhang BM, Wu WC. “A three phase power flow algorithm for distribution system power flow based on loop analysis method”. Electrical Power and Energy Systems 30 (2007) 8-15. [6] Berg R. Hawkins ES, Pleines WW. Mechanized calculation of unbalanced load flow on radial distribution circuits. IEEE Trans Power Appar Syst 1967;86 (4):415-21. [7] “Computer Methods in Power System Analysis” by Glenn W. Stagg and Ahmed H. EL-Abiad. [8] “Principles of Power System” by V.K.MEHTA and ROHIT MEHTA. [9] “Electric Power Distribution” by A S PABLA. X. CONCLUSION Firstly, a mass of methods to solve the radial distribution power flow problem are introduced in this paper. Subsequently, the reason, why the convergence of these widely used methods deteriorates when the network is radial, is well analyzed. In this case, an improved solution is needed to deal with radial network. So, a theoretical formulation of backward/forward sweep distribution load flow method is developed in this paper. Furthermore, the implementation of this method in distribution management system remains very straightforward. This method takes full advantage of the radial structure of distribution systems, to achieve high speed, robust convergence and low memory requirements. The numerical tests proved that, this method is very efficient for radial distribution network. This method is also applicable to weakly meshed networks with compensation based power flow method. REFERENCES 1544 S. Sunisith et. al., Backward/Forward Sweep Based Distribution Load Flow Method