APPLICATION OF FORWARD BACKWARD PROPAGATION ALGORITHM FOR THREE PHASE POWER FLOW ANALYSIS IN RADIAL DISTRIBUTION SYSTEM

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1 Simposium Nasional RAP V 2009 SSN : APPLCATON OF FORWARD BACKWARD PROPAGATON ALGORTHM FOR THREE PHASE POWER FLOW ANALYSS N RADAL DSTRBUTON SYSTEM Agus Ulinuha Department of Electrical Engineering, Muhammadiyah University of Surakarta Jl. A. Yani Paelan, PO BOX 1 Kartasura Surakarta, Postcode Phone: Fax ulinagus1970@gmail.com Astrak Power flow analysis is ackone of power system analysis and design. t is essential for power system analysis in operational as well as in planning stages. Power flow calculation is normally performed y simply considering that the system under study is completely alance. Actual distriution system is inherently unalance and therefore three-phase power flow is necessary to accurately simulate the unalance system. However, inclusion of unalance increases the dimension of prolem and, as distriution system is commonly constructed as radial system with high R/X ratio, it may also cause the sophisticated power flow algorithms fail to converge. The roust algorithm for three-phase power flow is therefore needed. n this paper, three-phase power flow for unalance distriution system is carried out using Forward-Backward Propagation Technique. The equivalent injection current method is employed to represent the loads and shunt admittances. The algorithm starts with mapping the distriution network to determine the forward and ackward propagation paths. The ackward propagation is used to calculate ranch currents using the us injection currents. The forward propagation is employed to calculate us voltages using the otained ranch currents and line impedances. The algorithm offers roust and good convergence characteristics for radial distriution system. The algorithm is presented for the EEE 34-us system with satisfied generated results. Keyword: Convergence; equivalent injection current; forward-ackward propagation; unalance system ntroduction Power flow calculation is ackone of power system analysis and design. t is essential for analysis of any power system in the operational as well as planning stages. The calculation is initially carried out y formulating the network equation. Node-voltage method, which is the most suitale form for many power system analyses, is commonly used. Mathematically, power flow prolem requires solution of simultaneous nonlinear equations and normally employs an iterative method, such as Gauss-Seidel and Newton-Raphson. Power flow calculation is normally performed y simply considering that the system under study is alance. Hence, the calculations are carried out for single phase assuming that the other two phases are exactly the same ut with the 120 degrees phase difference. The asymmetry in lines and loads produces a certain level of unalance in real power system and this is considered as disturance whose level should e controlled to maintain the electromagnetic compatiility of the system (Mayordomo, zzeddine et al. 2002). For distriution system, in particular, the system is inherently unalanced, due to factors such as the unalance of customer loads, the presence of unsymmetrical line spacing, and the comination of single, doule and three-phase line sections. Therefore, three-phase power flow is necessary to accurately simulate the unalance system. nclusion of unalance increases the dimension of prolem as all the three phases need to e considered instead of single phase alanced representation. On the other hand, distriution system commonly constructed as radial system with high R/X ratio may cause the sophisticated power flow algorithms fail to converge. The roust algorithm for three-phase power flow is therefore needed. Decomposition of the coupled unalance system into positive, negative and zero symmetrical components is the most popular approach used in three-phase power flow algorithms (Lo and Zhang 1993; Zhang and Chen 1994). This eliminates the mutual coupling etween phases so a three-phase power flow can e run three times, once for each phase. However, if the coupling etween sequences occurs, then there is no real advantage in decomposing the system into the symmetrical components. Furthermore, this may result in significant error in calculation. Another approach is decoupling the three-phase system into individual phases y introducing compensation current injections (Chen, Chen et al. 1991; Cheng and Shirmohammadi 1995; Lin and Teng 2000; Vieira, Freitas et al. 2004). Therefore, a three-phase power flow can e solved independently for every phase without utilization of E-66

2 Simposium Nasional RAP V 2009 SSN : symmetrical components. All components are modeled y phase voltages, admittances and independent current sources. This approach will work well as long as every component can e modeled in the admittance matrix or can e converted into equivalent injection current. The methods for three-phase distriution networks can e asically divided into two classes, Gauss-Seidel (Teng 2002; Vieira, Freitas et al. 2004) and Newton-Raphson (Garcia, Pereira et al. 2000; Lin and Teng 2000; da Costa, de Oliveira et al. 2007). Gauss-Seidel method needs much iteration and is known to e slow. Newton- Raphson has good convergence characteristic, ut the Jacoian that needs to e partially or totally calculated in every iteration makes this approach unattractive. This paper presents three-phase power flow for unalance distriution system using forward-ackward propagation technique. The algorithm works directly on the system without any modification. Therefore, there is no need to decompose the system into symmetrical components as well as to decouple the system into individual phases. However, the conversion of load and shunt element into their equivalent injection current is necessary. Distriution line charging is usually too small to e included (Lin and Teng 2000). The algorithm is implemented on EEE 34-us system including asymmetrical lines and unalance loads. System Modeling Line Modeling The accuracy of three-phase power flow results greatly depends on the line impedance model used. Therefore, an exact model of a three-phase line section needs to e firstly developed. The model of distriution line feeder as in (Kersting and Phillips 1995) will e developed and used in this paper. An equivalent circuit for a threephase line section is shown in Fig. 1. Fig. 1. Three-phase line model The modeling of three-phase lines starts with determination of self and mutual impedances of a line section which are functions of the conductors and the spacing etween conductors on the pole or underground. The modified form of Carson s equations are used to determine the self and mutual impedances of this model and are given y the following equations: z ii = ri j [ ln(1/ GMRi ) ] Ω/mi (1) [ ln(1/ D ) 7.934] zij = j ij + Ω/mi (2) Where r i is the conductor resistance (Ω/mile), GMR i is the conductor geometric mean radius (ft), and D ij is the spacing etween conductors i and j (ft). Application of (1) and (2) to the three-phase line indicated in Fig. 1. results in a 4 4 primitive impedance matrix : zaa za zac zan [ ] = za z zc zn Zprim (3) zca zc zcc zcn zna zn znc znn By applying Kron reduction, this matrix may e reduced into 3 3 phase impedance matrix whose elements are determined y the following equation: Zij = zij zin znj znn (4) And the resulted phase impedance matrix is: Zaa Za Zac [ Z ] = ac Za Z Zc (5) Zca Zc Zcc E-67

3 Simposium Nasional RAP V 2009 SSN : Load Modeling The load (alance and unalance) is represented y its equivalent injection current. The load modeling as in (Vieira, Freitas et al. 2004) is adopted in this paper. For three-phase loads connected in wye or single-phase loads connected line to neutral, the equivalent injection currents at k th us are determined y the following equation: k k k Pm m m = ; m [a,, c] : phase (6) k Vm * Where P m, Q m, V * m denote real power, reactive power, and complex conjugate of the voltage phasor for each phase, respectively. For three-phase loads connected in delta or single-phase loads connected line to line, the equivalent injection currents at the k th us are determined y the following equations: k P a a Pca ca a = Va * V * Vc * Va * k P c c P a a = V * Vc * Va * V * (7) k Pca ca P c c c = Vc * Va * V * Vc * Where P n, Q n, n [a, c, ca], represents the real and reactive load connected etween the respective phases, and V * m, m [a,, c] denotes the complex conjugate of the voltage phasor for each phase, respectively. Shunt Admittance Modeling Three phase shunt capacitors can also e represented y equivalent injection currents (Vieira, Freitas et al. 2004). Assuming that the capacitor ank has an ungrounded wye connection, the current injection in each phase can e expressed y: sh sh y ( k k k a = 2Va + V + Vc ) 3 sh sh y ( k k k = Va 2V + Vc ) (8) 3 sh sh y ( k k k c = Va + V 2Vc ) 3 Where y sh = jq 0 / V 0 2 ; Q 0 is the nominal reactive power per phase and V 0 is the magnitude of the nominal voltage each phase. f the capacitor ank has a grounded wye connection, then the injection current is: sh sh k a = y Va sh sh k = y V (9) sh sh k c = y Vc Forward-Backward Algorithm Power flow is calculated y initially mapping the distriution network to determine forward and ackward propagation paths followed y ranch current calculation and us voltages calculation. Branch Current Calculation Using ackward propagation path, ranch current can e calculated using the equivalent us injection currents. The ranch current is successively calculated for the network ends toward the source (swing). The voltage at each us therefore needs to e firstly determined. For the first iteration, the voltage at each us is set to 1.0 pu with the angle of 0, -120 and 120 degrees for phase a,, and c, respectively. These voltages are updated during the iteration and therefore the injection currents will also change. E-68

4 Simposium Nasional RAP V 2009 SSN : Fig. 2. Part of a distriution system Fig. 2 indicates a part of distriution system and us injection currents. The relationships etween ranch currents and injection currents are: jk = k jl = l (10) ij = jk + jl k Where jk is the ranch current etween us j and us k, and j injection currents at us, respectively. Bus Voltage Calculation Using forward propagation path, the voltage at each us is calculated using the otained ranch currents and line impedance. The voltages are consecutively calculated from the source (swing) toward the network ends. The voltage of swing us is kept constant at pu, pu, and pu for phase a,, and c, respectively. For the part of system indicated in Fig. 2, the us voltages can e calculated as follows: V j = Vi Zij ij Vk = V j Z jk jk (11) Vl = V j Z jl jl Where V j is the voltage at us j and Z jk is the impedance of line section etween us j and k. The updated us voltages are then used to calculate the us injection currents. t should e noted that all calculations are carried in the three-phase frame. Convergence Criteria The outlined steps for ranch currents and us voltages calculations are invoked during the power flow iteration. The iteration converges if the different of us voltages for the consecutive iterations is equal to or less than the prescried tolerance. The voltage mismatch of us j at the n th iteration is given y: n n n 1 ΔV j = V j V j for phase a,, c n Re (Δ V j ) < ε ; j all uses (12) n m(δ V j ) < ε ; j all uses Where ε is the prescried tolerance. f these equations are satisfied, then the iteration stops. Otherwise, iteration process is repeated. Once the load flow iteration converges all the ranch currents and voltage at each us are known. The real and reactive power loss can therefore e calculated. The three-phase power flow algorithm using Forward-Backward Propagation technique is given in the flowchart of Fig. 3. E-69

5 Simposium Nasional RAP V 2009 SSN : Fig. 3. The flowchart for three-phase power flow calculation using Forward-Backward Propagation technique Simulation System Data The simulation is carried out for the EEE 34-us system (Kersting 1991) indicated in Fig. 4. The system includes alance as well as unalance loads. A minor modification is performed for the system to only include three phase asymmetrical lines. However, the load data and the capacitor data are remain the same. The system load and ranch data are respectively given in Tale 1 (A and B) and Tale 2. Tale 1(a) Balance Loads of the EEE 34-us System Phase A Phase B Phase C Note 1. Swing: us MVA ase: 2.5 MVA Bus# Υ/Δ kw kvar kw kvar kw kvar 860 Υ Υ Δ Υ Δ Tale 1() Unalance Loads of the EEE 34-us System Bus# Υ/Δ Phase A Phase B Phase C kw kvar kw kvar kw kvar 806 Υ Δ Δ Υ Υ Υ Δ Δ Υ Δ Δ Υ Υ Δ Υ Υ Υ Δ Υ Υ Δ E-70

6 Simposium Nasional RAP V 2009 SSN : Fig. 4. The EEE 34-us system used for simulation Tale 2. Branch Data of The EEE 34-Bus System Bus length Line Condctr Neutral Condctr Config. Fr To (ft) R GMR R GMR BACN BACN BACN BCAN BACN BACN BACN ABCN BACN ABCN ABCN BCAN BACN BACN BACN BACN BACN BACN BACN BACN BACN BACN BACN BACN BACN BACN BCAN BACN ABCN BACN BACN ACBN BACN ' 3' 3' 4' 24' Fig. 5. Overhead line spacing for the EEE 34-us system Fig. 5 shows the spacing distances etween the phase conductors and the neutral conductor. The line configuration is given in Tale 2 (3 rd column) indicating that the phase conductors are sequentially placed starting from the left side position. E-71

7 Simposium Nasional RAP V 2009 SSN : Simulation Results The system is run using the algorithm presented in Fig. 3. The simulation is coded using Matla version 7.0 (R14). For the system of Fig.4, the three-phase power flow calculation takes 5 iterations to converge. This iteration numer is fairly small indicating that the algorithm is roust for unalance power flow calculation. The results of simulation including magnitude and angle of voltage at every phase are given in Tale 3. The simulation also indicates the real and reactive power losses of 8.2 kw and 3.4 kvar, respectively. Tale 3. Simulation Results of The EEE 34-Bus Unalance System us Volt at phase c Volt at phase c Volt at phase c no mag Angle mag Angle mag Angle Conclusion Three-phase power flow for an unalance distriution system is carried out using Forward-Backward Propagation Technique. The EEE 34-us unalance system is used for simulation. The main conclusions are: The implemented algorithm works directly on the simulated system and, therefore, there is no need to transform the system into the symmetrical components as well as to decouple the three-phase into individual phases; The algorithm is roust for the radial unalance distriution system with good convergence characteristic. Reference: Chen, T. H., M. S. Chen, et al, (1991), "Distriution system power flow analysis-a rigid approach", EEE Transactions on Power Delivery, 6(3): Cheng, C. S. and D. Shirmohammadi (1995), "A three-phase power flow method for real-time distriution system analysis", EEE Transactions on Power Systems, 10(2): da Costa, V. M., M. L. de Oliveira, et al., (2007), "Developments in the analysis of unalanced three-phase power flow solutions", nternational Journal of Electrical Power & Energy Systems, 29(2): E-72

8 Simposium Nasional RAP V 2009 SSN : Garcia, P. A. N., J. L. R. Pereira, et al., (2000), "Three-phase power flow calculations using the current injection method", EEE Transactions on Power Systems, 15(2): Kersting, W. H., (1991), "Radial distriution test feeders", EEE Transactions on Power Systems, 6(3): Kersting, W. H. and W. H. Phillips, (1995), "Distriution feeder line models", EEE Transactions on ndustry Applications, 31(4): Lin, W.-M. and J.-H. Teng, (2000), "Three-phase distriution network fast-decoupled power flow solutions", nternational Journal of Electrical Power & Energy Systems, 22(5): Lo, K. L. and C. Zhang, (1993), "Decomposed three-phase power flow solution using the sequence component frame", EE Proceedings-Generation, Transmission and Distriution, 140(3): Mayordomo, J. G., M. zzeddine, et al., (2002), "Compact and flexile three-phase power flow ased on a full Newton formulation", EE Proceedings-Generation, Transmission and Distriution, 149(2): Teng, J.-H., (2002), "A modified Gauss-Seidel algorithm of three-phase power flow analysis in distriution networks", nternational Journal of Electrical Power & Energy Systems, 24(2): Vieira, J. C. M., Jr., W. Freitas, et al., (2004), "Phase-decoupled method for three-phase power-flow analysis of unalanced distriution systems", EE Proceedings-Generation, Transmission and Distriution, 151(5): Zhang, X. P. and H. Chen, (1994), "Asymmetrical three-phase load-flow study ased on symmetrical component theory", EE Proceedings-Generation, Transmission and Distriution, 141(3): E-73

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