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10-01-2011, 03:40 PM

Ulaş Eminoğlu, M. Hakan Hocaoğlu

In this paper, the convergence ability of Distribution Systems Power Flow Algorithms, which are widely used for distribution systems analysis, is compared with different voltage-dependent load models. The convergence ability of methods are also evaluated for different tolerance values, voltage levels, loading conditions and R/X ratios, under the wide range exponents of loads. Results show that Ratio-Flow method is preferable from other methods.


In the literature, there are a number of efficient and reliable load flow solution techniques, such as; Gauss- Seidel, Newton-Raphson and Fast Decoupled Load Flow . Hitherto they are successfully and widely used for power system operation, control and planning. However, it has repeatedly been shown that these methods may become inefficient in the analysis of distribution systems with high R/X ratios or special network structures. Accordingly, a number of methods proposed in the Literature specially designed for the solution of power flow problem in radial distribution networks. The methods developed for the solution of ill-conditioned radial distribution systems may be divided into two categories. The first type of methods is utilised by proper modification of existing methods such as, Newton- Raphson. On the other hand, the second group of methods is based on forward-backward sweep processes using Kirchooff’s Laws or making use of the well-known bi-quadratic equation which, for every branch, relates the voltage magnitude at the receiving end to the voltage at the sending end and the branch power flow for solution of ladder networks.

Shirmohammadi et al., have presented a compensation-based power flow method for radial distribution networks and/or for weakly meshed structure using a multi-port compensation technique and basic formulations of Kirchhoff’s Laws. The radial part is solved by a straightforward two step procedure in which the branch currents are first computed (backward sweep) and then the bus voltages are updated (forward sweep). In the improved version, branch power flow was used instead of branch complex currents for weakly meshed transmission and distribution systems by Luo. In , Baran and Wu propose a methodology for solving the radial load flow for analysing the optimal capacitor sizing problem. In this method, for each branch of the network three non-linear equations are written in terms of the branch power flows and bus voltages. The number of equations is subsequently reduced by using terminal conditions associated with the main feeder and its laterals, and the Newton-Raphson method is applied to this reduced set. The computational efficiency is improved by making some simplifications in the jacobian. Consequently, numerical properties and convergence rate of this algorithm have been studied using the iterative solution of three fundamental equations representing real power, reactive power and voltage magnitude by Chiang in . In , G. Renato Cespedes makes use of wellknown bi-quadratic equation which, for every branch, relates the voltage magnitude at the receiving end to the voltage at the sending end and branch power flow. In , only voltage magnitudes are computed, bus phase angles do not appear in the formulation which was also used by Das et al., in. Jasmon, in , have proposed a load flow technique which, for every branch, leads to a pair of quadratic equations relating power flows at both ends with the voltage magnitude at the sending end for the voltage stability analysis of radial networks. Haque, in [19], have formulated the load flow problem of the distribution system in terms of three sets of recursive equations and analysed load flow results for various voltage dependent load models. The effects of various load models on the convergence pattern of the method are also studied. The effect of voltage-dependency of load on the results and convergence characteristics of power flow solution are also analysed in , where the proposed method is also based on Kirchoff’s Laws. In authors have proposed Ratio-Flow method which is based on forward-backward ladder equation for complex distribution system by using voltage ratio for convergence control. This method were applied with standard Newton-Raphson method for complex distribution systems, which have multiple sources or relatively strong connected loops with extended long radial feeders including laterals, to solve the load flow problem. In , R. Ranjan and D. Das have proposed a new method to solve radial distribution networks. They have used simple algebraic recursive expression of voltage magnitude and the proposed algorithm used the basic principle of circuit theory. D. Zimmerman and H. D. Chiang in [23], have formulated load flow problem as a function of the bus voltages and equations are solved by Newton’s method. The method has been compared with classical Newton-Raphson and Forward-Backward sweep methods by using a number of test cases. Although required iteration number considerable favoured from classical methods for small tolerances, no results has been provided on the accuracy of the solution in terms of bus voltage magnitudes or angles. The results provided in suggest that undertaken comparisons only cover network structures which are inherently convergent ie. solutions can also be obtained using classical Newton-Raphson method. In ref., the authors have proposed forward-backward substitution method which is based on the Kirchhoff’s Laws. In backward substitution, each branch current is calculated by Kirchhoff’s Current Law (KCL). Using these currents, the node voltages are calculated by Kirchhoff’s Voltage Law in forward substitution at each iteration. The voltage magnitudes at each bus in an iteration are compared with their values in the previous iteration. If the error is within the tolerance limits, the procedure is stopped. Ladder Network Theory given in ref. is similar to the Forward-Backward Substitution method. In Ladder Network Theory, the currents in each branch are computed by KCL. In addition to the branch currents, the node voltages are also computed by KVL in each iteration. Thus magnitude of the swing bus voltage is also determined. The calculated value of swing bus is compared with its specified value. If the error is within the limit, the procedure is stopped. Otherwise, the forward and backward calculations are repeated as in forward-backward substitution method. The aim of this paper is to compare the convergence ability of distribution system load flow methods which are widely used for distribution systems analysis. The method, analysed in this paper, are classical Newton- Raphson method, Ratio-Flow, Forward- Backward Substitution method and Ladder Network Theory , The convergence ability of methods are also evaluated for different tolerance values, different voltage levels, different loading conditions and different R/X ratios, under the wide range exponents of loads. Algorithms are implemented with Matlab codes.

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