Variations of the Gauss Seidel and the gauss implicit z-bus load flow methods for primary-secondary integrated distribution grids

Abstract— The primary and secondary distribution grids are typically designed separately and operated with a radial configuration; therefore, specialized load flow methods only applicable to radial or weakly meshed networks are normally used. However, projections indicate that the distribution grids will be more interconnected in the future, mainly because of the inclusion of distributed generation, voltage and reliability optimization, as well as an efficiency improvement when the primary and secondary networks are considered in an integrated way. For this new meshed grids scenario, the efficient and precise typically used load flow methods for distribution networks are no longer applicable and it becomes necessary using load flow algorithms that are also applicable for meshed configurations, such as the ones classically used for transmission networks like the Newton-Raphson, Gauss-Seidel and Gauss Implicit Z-bus methods, while also procuring to avoid potential singularity problems which may arise when dealing with long radial grids. In this work, variations of the Gauss-Seidel and Gauss Implicit Zbus methods are presented, that are adequate for low and medium voltage grids regardless of the network configuration. Additionally, a linear, direct, and non-iterative load flow variation is presented as well as a comparison between different possible convergence criteria for the classical methods. Index Terms— Distribution, Gauss-Seidel, Ill-conditioned, Linearized, Load flow, Z-bus. I. INTRODUCTION he distribution system has been typically classified in two subsystems: the primary and the secondary electrical distribution networks. The primary distribution grid carries the energy from the power transformers and generators connected to the high voltage transmission grid to the distribution transformers, and the secondary distribution network distributes the electrical power to the final consumers. Under this classical approach, the low and medium voltage grids were classically defined only as power distribution networks, normally designed and operated radially, and fed from a single power source. Nowadays, this classical approach has changed [1]. The technological advancements in renewable generation, power electronics and control capabilities have shifted the paradigm of the low and medium voltage systems, from a solely distribution function to a highly interconnected (meshed) system including distributed generation, advanced control and monitoring capabilities [2]-[4]. In this scenario, the optimal planning of the entire distribution grid is obtained by analyzing both structures (the primary and the secondary distribution grids) as only one system (integrated paradigm) [1], where the complete electrical distribution grid might present a radial or a meshed topology. This new approach introduces the necessity to develop load flow algorithms applicable for distribution grids that can be highly interconnected, for which the existing distribution load flow algorithms presents problems or may not be applicable. The nonlinear characteristic of the power flow equations introduces the possibility of convergence to different solutions. The first load flow methods, such as the NewtonRaphson and Gauss-Seidel, were initially developed for transmission grids, which present a typically meshed configuration and high X/R ratios. For example, the classical Newton-Raphson algorithm is based on the Taylor series where the solution is found by successive approximations or iterations of the linearized power flow expression. On the other hand, the Gauss-Seidel algorithm presents a simpler approach but it requires an elevated number of iterations to calculate the solution. The Gauss Implicit Z-bus is a load flow algorithm based on partitions of the admittance matrix, and it has proven to be adequate when used in distribution networks, mainly because of the reduction in the number of required iterations [5]. However, when these classical load flow algorithms were used in electrical networks with characteristics typically found in electrical distribution grids, efficiency and convergence problems were found, as reported in [6]-[8]. Generally, these problems arise because of the typical radial configuration and the wide range of the grid X/R ratios, which are some of the characteristics that define the electrical distribution networks known as “ill-conditioned” grids, as shown in [9] and [10]. Therefore, a new approach was necessary and new load flow algorithms were created. These load flow algorithms have been specifically designed for radial distribution grids, as the ones used and reported in [11]-[17], and for weakly meshed grids, as reported in [18]-[21], including methods such as the backward-forward sweep, the decoupled Newton-Raphson based methods and the grid reduction algorithms. The backward-forward algorithms are fundamentally based on the radial characteristics of the grid. The algorithm calculates the Roberto Barrenechea, Federico Rypin, Pedro Paiva-Mata, Luis García de Vicuña, and Miguel Castilla Variations of the Gauss Seidel and the Gauss Implicit Z-bus Load Flow Methods for PrimarySecondary Integrated Distribution Grids T line currents starting from the final node towards the source in a “backwards sweep” and then updates the voltages, starting from the source in a “forward sweep”. The decoupled type algorithms combine a “backwards sweep” with a formulation similar to the decoupled Newton-Raphson method. The network reduction algorithms are based on modelling every nonlinear variable by a linear expression, and then using standard circuit calculations taking advantage of the radial characteristic of the grid. All of these methods are strictly based on the classical view, where the medium and low voltage distribution grids are operated radially, and under specific considerations, they may be operated in weakly meshed configurations. Now, considering this new scenario, where the low and medium voltage distribution grids can be highly interconnected and therefore the specialized distribution load flow algorithms can no longer be used, it becomes necessary to develop new algorithms or to review the classical ones from the power transmission systems that are applicable to meshed networks, while addressing the challenges of using them in distribution networks. In this work, the first and main contribution is the derivation of several load flow algorithms for medium and low voltage electrical networks. All the proposed load flow variations are designed to calculate the steady state variables of medium and low voltage grids that may be radial, weakly or heavily meshed, regardless of the voltage and X/R ratio of the primary and secondary networks. The first algorithm is based on the Gauss Implicit Z-bus, where the primary and secondary networks are considered as a whole (integral design), and it is applicable regardless of the configuration of the grid (radial or meshed) and its X/R ratio. The proposed load modelling variation is developed to achieve a load flow formulation that presents the adequate performance of the Z-bus method when used in meshed low voltage networks, while additionally minimizing possible singularity problems due to the sparsity of the admittance matrix when used in long radial and “illconditioned” networks. A variation of the classical Gauss Seidel method is presented, where the loads are also modelled as admittances in an attempt to avoid potential singularity problems when used in distribution long radial and “illconditioned” grids [15]. The high X/R ratio of the high voltage transmission networks allows for an active and reactive power decoupling, and thus for the formulation of approximated linearized load flow algorithms, typically used as a starting vector for another load flow method. However, because of the typically low X/R ratio in medium and low voltage grids, this decoupling of the active and reactive power is not applicable. Therefore, another algorithm, based on a linearized non-iterative load flow variation of the proposed algorithms is developed, and it is presented as a direct and approximate calculation alternative for distribution grids, which can be used as a standalone approximate and fast load flow calculation, or to calculate the starting vector for the initialization of another iterative load flow algorithm. The convergence criterion defines when the algorithm has reached the correct solution. The two classical possibilities are the voltage and the power mismatch criteria. Algorithms like the Gauss-Seidel and the Gauss Implicit Z-bus are classically formulated using a voltage mismatch criterion, which means that only when the voltage calculated in an iteration is equal (or similar) to the one calculated in the previous iteration, it is considered that the correct solution has been reached. Other algorithms, like the Newton-Raphson load flow, use a power mismatch criterion, stating that the convergence to the solution is achieved when the calculated active and reactive power demands are equal (or similar) to the ones initially defined. The final contribution of this work is the proposal for a change of the convergence criteria in the Gauss Seidel and Gauss Zbus algorithms, from a voltage to a power mismatch convergence. The purpose of this change is to optimize the performance of the classical algorithms when used in medium and low voltage grids. A complete performance comparison is presented for the proposed power flow algorithms including the results obtained by using the classical methods. This comparison reveals the main features of the proposals and points out its advantages and limitations. II. REVIEW OF THE CLASSICAL METHODS This section briefly reviews the classical load flow algorithms. A. Gauss-Seidel Load Flow The Gauss-Seidel load flow (abbreviated lfgs from now on) is a method based on representing the loads as voltage dependent current sources at the connection node i [22]: * 1 i inj Inj r i P IV    (1) The voltage is calculated in each iteration using the expression: 1 1 1 1 11 i in r r r r i ii Inj ij j ij j j j i V Y I Y V Y V               (2) where Current injection in node i Inj I i Power injection in node inj P i Voltage at node (in the current iteration) r iiV 1Voltage at node (in the previous iteration) r i V i  Admittance matrix`s self admittance at node ii Y i Mutual admittance between nodes and ij iY j The calculation is repeated until the difference between the voltage values of the current and previous iteration is lower than a given tolerance tol, which is known as the voltage convergence criterion by voltage mismatch. 1; 2,3... rr ii V V tol i N     (3) B. Gauss Implicit Z-bus Load Flow The Gauss Implicit Z-bus load flow (in brief Zimp) is a method based on calculating the nodal voltages using submatrices of the admittance matrix [5]. From the expression: Inj bus nodal I Y V (4) where Current injections vector Inj I Admittances matrix bus Y Nodal voltages vector nodal V Assuming the reference or “slack” node is numbered as N=1, (4) can also be formulated as follows: 11 12 21 22 2, 2, slack slack NN IV Y                    Y YY IV (5) Where the current and voltage vectors are decomposed in the variables for the slack bus Islack and Vslack and the submatrixes for the remaining nodes current injections I2,N and voltages V2,N. The admittance matrix is also decomposed in the slack node’s self-admittance variable Y11, the slack node’s mutual admittances sub-matrixes Y12 and Y21, and the remaining node’s mutual admittances sub-matrix Y22. Solving for the current values: 21 22slack VI Y Y V (6) The voltages are finally calculated as:   1 22 21 slack V  V Y I Y (7) Similar to the Gauss-Seidel algorithm, the calculation is repeated until the voltage mismatch convergence criterion is met. III. MODELLING OF THE DISTRIBUTION GRID COMPONENTS The considerations taken into account for the modelling of the distribution grid components are described below. In this work, the distribution substations were modelled as the system voltage reference or “slack” node, given that these are the main and typically larger power sources and are provided with voltage control capabilities. The transformers were modelled using the standard “pi” model considering only the series resistance and reactance. The typical short distribution lines model was used considering only the series resistance and reactance. The loads are typically modelled either as constant apparent power demands, as impedances or as current demands. In this work, different types of load models were considered and compared. The capacitive compensators were modelled as constant reactive power injections. As expressed in [23] and [24], the power generating units typically used in low and medium voltage grids can be modelled as apparent power specific nodes (PQ), therefore, the PQ model was considered for the power source nodes. IV. PROPOSED METHODS Different variations of the Gauss-Seidel and the Gauss Implicit Z-bus load flow algorithms are presented in this section. One of the main considerations for the proposed load flow variations is that the loads were modelled as voltage dependent admittances connected to the ground instead of as current sources. Thus, the admittances were formulated as the relation between the conjugated apparent power demand and the magnitude of the nodal voltage squared: * 2 i i dem i S yV  (8) where Admittance in node i dem y i Apparent power demand in node i S i Voltage at node i V i By modelling the loads as admittances, opposite to modelling them as current injections like in the Gaus-Seidel and Zimp algorithms, they become a parameter of the system and are included in the admittance matrix Ybus, thus making the formulation naturally less susceptible to potential convergence issues when used in long radial and “illconditioned” distribution grids, where the admittance matrix is quasi-singular, mainly because of their radial configuration, their length, and their typically low X/R ratio. Including the loads in the admittances matrix results in fuller matrixes with a higher determinant, making the algorithm naturally less susceptible to possible singularity problems when inverting the admittance matrix during the iterations. It is also important to consider that when the loads are modelled as current sources, the precision of the value depends on the complex conjugated voltage. Moreover, when the loads are modelled as admittances, as in the proposed variation, it depends on the inverse of the voltage magnitude squared. Therefore, the proposed methods were examined considering different radial and meshed grid examples, in order to evaluate the influence of the proposed modelling variation in the algorithm’s performance (speed and precision). A. Proposed Load Flow Algorithm This algorithm (called lfRBY) is a variation of the Zimp load flow where the loads are modelled as voltage dependent admittances and therefore are included in the system’s admittance matrix, instead of modelling them as currents and including them in the current injections vector as in the Zimp algorithm. From (7) and considering that in this formulation the loads are modelled as admittances, their corresponding current injections vector is fixed at zero I2,N = 0 and the load admittances are included in Y22: 1 * 22 1 2 22 21 * 2 1 0 0 r rslack N r N S V V S V                         V Y Y (9) The loads model is updated according to the new calculated voltage in each iteration, and the process is repeated until the mismatch criterion is met. For reasons explained below, the power mismatch convergence criterion is recommended. B. Proposed Load Flow Algorithm Variation Another variation of the load flow (called lfRBZ) was developed, where the loads modelled as admittances are included in the impedances matrix of the system Z: 1 * 12 1 11 12 21 22 * 2 1 0 0 bus N r N S VZ S V                           Z ZY ZZ (10) The system nodal voltages are defined as follows: 11 12 21 22 2 slack slack N VI Z             Z ZZ V0 (11) Solving for the slack node voltage: 1 11 11slack slack slack slack V Z I I Z V     (12) Solving for the remaining nodes voltages: 2 21N slack IVZ (13) Combining (12) and (13), the nodal voltages are finally calculated by the following formulation: 2 21 11 slack N V Z VZ (14) The calculation process is repeated until the convergence criterion is met. C. Gauss-Seidel Variation A variation of the Gauss-Seidel load flow (called lfgsZ) is proposed, where the loads are modelled as admittances in order to avoid possible singularity problems. Taking the original Gauss-Seidel formulation, but setting the current injections from the loads to zero, and including them as admittances in the admittances matrix, the system voltages are defined and calculated as follows: 11 11 * 2 1 () ii in rr ij j ij j j j i r i load load ii r i Y V Y V VP jQ YV                 (15) where Active power demand in node i load P i Reactive power demand in node i load Q i D. Proposed Linear Non-Iterative Load Flow This algorithm, named lfRBL, allows to calculate precise solutions in only one calculation (without iterations), thus reducing the calculation time to a minimum, and it can be used as an approximated load flow calculation or just to obtain an initial voltage array to be used as a starting point for another load flow algorithm. The proposed direct non-iterative load flow method is based on the consideration that, in low and medium voltage electrical grids, the voltage angle differences are typically close to zero, allowing for the following approximation: * ii VV (16) Considering this approximation, the loads can be modelled as follows: ** * ii i i i SS IV V  (17) Moreover, considering that the voltage magnitudes are required to be maintained at a maximum deviation of typically ±5% from the system’s nominal value, it is possible to use the following linearized expression. 1 i i m V b V (18) By combining (17) and (18), the system’s current injections can be defined as follows: * *2 2* *2 2 2 2 * * * * () () N N N N N N S S V S m V b V S S S m V b V V                              I (19) **2 22 * * 0 0NN N V SS bm SV S                   I (20) Combining (20) with (6) from the Zimp algorithm: ** 22 21 22 * * 0 0slack N N SS b m V S S               V Y Y V (21) The linearized expression for the system voltages calculation is finally defined as follows: * 1 *2 2 22 21 ** 0 0slack NN S S m b V SS                          V Y Y (22) The proposed linearization range is between voltage values of 0.95 and 1 p.u. since it is more frequent for the grid voltages to be below the system rated voltage, and considering that normally the regulations shall enforce that the system voltages will not drop to lower values. In Fig. 1, both the real and the proposed voltage dependant functions f(v) are depicted for comparison. The real inverse voltage function (1/V) is shown in a straight line and the proposed linearized voltage dependant function is shown in the segmented line. It is possible to see that, as well as for the methods presented in [17] and [25], the proposed algorithm is most adequate as long as the nodal voltages are maintained near the nominal system voltage in the considered range (0.95 to 1 p.u.), for which the proposed linearized function f(v) is: ( ) 1.0526 2.0526f v V   (23) E. Proposed Convergence Criteria Variations Given the nonlinear characteristic of the power flow equations, special attention is required towards the convergence criterion to be used, in order to avoid the convergence to a mistaken or a physically illogical solution. Algorithms like the Gauss-Seidel, as well as the majority of the distribution load flow algorithms rely on the criterion of the voltage mismatch for the convergence determination. As defined in (3), the voltage mismatch criterion establishes that the solution is found when the voltage values calculated in the current iteration are similar (given a certain tolerance tol) to the ones calculated in the previous iteration. Depending on its formulation, it is possible for some load flow algorithms, which are less sensitive to voltage variations, to converge to a mistaken or deviated solution when the voltage mismatch criterion is used. Therefore, a variation of the convergence criterion is proposed, where the classically used voltage mismatch criteria of the Gauss-Seidel algorithm is modified to a power mismatch convergence. The power convergence criterion establishes that the solution has been found only when the power demands, calculated in the current iteration are similar (given a certain tolerance tol) to the specified power demands at each node. The power mismatch convergence criterion is defined as follows: 22 2 NN r spec cal r spec N cal PPtol P tol P                     (24) 222 NN r spec cal r spec N cal QQtol Q tol Q                     (25) where Specified active power demand for each node spec P Specified reactive power demand for each node spec Q Calculated active power demand for each node cal P Calculated reactive power demand for each node cal Q If the algorithm is naturally formulated using the voltage mismatch criterion, the change to a power mismatch might imply an additional calculation and, therefore, a longer computation time. This additional calculation is required because of the necessity to calculate the active and reactive power from the calculated voltage values in each iteration. Fig. 1. Inverse voltage and linearized functions. Fig. 2. Electrical network of the case study I. Fig. 3. Electrical network of the case study III. V. CASE STUDIES The previously described load flow algorithms were used to calculate the solution of three different case studies: 1) a meshed primary-secondary grid; 2) an entirely radial grid; and 3) a meshed standard IEEE network. The case studies were selected as a representative sample of the different types of distribution network configurations, in order to obtain an adequate evaluation of the compared load flow algorithms under different scenarios. In order to obtain an adequate comparison, the power mismatch convergence criterion was used for all the evaluated load flow algorithms. A. Case Study I The case study I is a primary-secondary integrated 36 nodes network based on the one reported in [1], where its connections were modified in the referenced paper to obtain an interconnected secondary grid. For the per unit calculations, a power base of 100 kVA, a primary voltage base of 13.8kV and a secondary voltage base of 0.208kV was considered. Fig. 2 shows the overall layout of the grid. B. Case Study II The case study II is an example of a radial grid with 201 nodes based on the distribution grid optimization studies reported in [26] and [27]. For the per unit calculations, a power base of 100 kVA and a voltage base of 10kV was considered. C. Case Study III The case study III is the standard IEEE 5-Bus network used in [28] as an example for calculating and analyzing the effects of variations in the line X/R ratio. Fig. 3 depicts the overall layout of the network. VI. COMPARED LOAD FLOW METHODS A total of 13 load flow algorithms were considered in this study including the classical approaches and the proposed variations. The merit factors for the comparison were the convergence to the solution (defined by the calculated power losses), number of iterations and calculation time. These algorithms are listed below: lfnr: Complete (not decoupled) Newton-Raphson algorithm. lfnrL: lfnr algorithm using lfRBL for a starting vector. lfgs: Gauss-Seidel algorithm. lfgsL: lfgs algorithm using lfRBL for a starting vector. lfgsZ: Proposed method variation. lfgsZL: lfgsZ algorithm using lfRBL for a starting vector. Zimp: Gauss Implicit Z-bus algorithm. ZimpL: Zimp algorithm, using lfRBL for a starting vector. lfRBY: Proposed method variation. lfRBYL: lfRBY algorithm using lfRBL for a starting vector. lfRBZ: Proposed method variation. lfRBZL: lfRBZ algorithm using lfRBL for a starting vector. lfRBL: Proposed linear non-iterative load flow method. VII. RESULTS The calculated results for the case studies I and II are presented in Tables I and II respectively. By reviewing the obtained results, the following observations can be stated regarding the overall performance and the adequacy of the proposed methods. The proposed algorithm lfRBY allows calculating the precise results independently of the grid’s configuration, and with less iterations, hence less calculation time, than the traditional Gauss-Seidel and Newton-Raphson methods, and only comparable in this regard to the Zimp. By analysing the results, it is possible to conclude that using the proposed variation for making and managing fuller admittance matrixes (making the algorithm naturally less susceptible to potential singularity problems), has not deteriorated the performance of the original Zimp algorithm in precision nor speed. The proposed linear non-iterative method has proven adequate for the evaluated study cases, allowing to achieve the results precisely and, being a direct and non-iterative method, was able to achieve the solutions in calculation times significantly lower than any other method. The power losses calculated using the proposed linear non-iterative method lfRBL presented a deviation of less than 1% when compared with the solutions calculated using Zimp. Additionally, in all of the cases, using the proposed lfRBL algorithm as a starting vector has introduced a significant reduction (of more than 70% in some cases) in the required time and number of iterations for all of the evaluated methods, including both the classical algorithms as well as the proposed variations. Simulations were also made changing the convergence criterion of the methods, between the classically used voltage mismatch and the proposed power mismatch criteria. An improvement in the overall precision was achieved by using the proposed variation, specially for the classical Gauss-Seidel algorithm. As presented in Table III, the algorithms where the classical voltage mismatch criterion was established, lfgs(V) and lfgsL(V), present a deviation in power losses of 13% for the meshed study case I, and an even higher deviation for the long radial case study II. These results indicate that the proposed power mismatch convergence criterion has allowed to achieve a more precise solution in the evaluated case studies, and that the use of the proposed linear variation lfRBL as a starting vector for the classical Gauss-Seidel method, also helped correcting the convergence errors introduced by the use of the classical voltage mismatch criterion, by guiding the algorithm towards the correct solution, while also reducing the number of the required iterations. The original line data of the case study III network, presented in Table IV, was modified by reducing the lines X/R ratio in order to evaluate the possibility of convergence problems. The reactance of the line was reduced in steps of 25, 50 and 100 times its initial value and the proposed load flow method lfRBY was used to calculate the solution for each of the resulting networks. The data and results are shown in Tables IV and V, and it is possible to see that the proposed method achieved convergence in all the simulated scenarios. Finally, the radial case study II was also solved using the Oriented Graph “GO” reported in [29], which is a load flow algorithm specifically developed for classical radial distribution networks and it was designed to reduce the computation times by taking advantage of the radial configuration of the grid. The GO reached the solution in 0.09998s, which is comparable to the proposed algorithms. An additional simulation was made for different nodes of the proposed radial case study. The results presented in Fig. 4 indicate a linear behaviour of the GO calculation time with respect to the number of nodes, while the other methods do not, which implies that, for strictly radial systems, when the number of nodes increases, the use of a specific radial load flow algorithm may reduce the overall computation times, however these methods are only applicable for radial distribution grids and not for the new meshed low and medium voltage grids that are expected in the near future. VIII. CONCLUSION Several load flow algorithms have been presented in this paper as variations of the Gauss Implicit Z-bus and the GaussSeidel load flows, including different iterative methods and a linearized non-iterative approach. The proposed load flow algorithms have been proven adequate for its utilization in low and medium voltage grids, reaching precise solutions in less time and number of iterations than the classical load flow methods like the Gauss-Seidel and the Newton-Raphson, and only comparable to the Zimp method. The algorithms have been derived mainly by changing the standard load modelling TABLE I LOAD FLOW RESULTS FOR THE CASE STUDY I Load flow Time (s) Iterations Time/Iteration (s) Power losses (p.u.) lfnr 0.079 3 0.02633 0.0081 lfnrL 0.046 2 0.02300 0.0080 lfgs 1.688 877 0.00192 0.0081 lfgsL 0.437 255 0.00171 0.0081 lfgsZ 1.672 877 0.00191 0.0081 lfgsZL 0.453 255 0.00178 0.0081 Zimp 0.001 2 0.00001 0.0080 ZimpL 0.016 1 0.01600 0.0081 lfRBY 0.001 2 0.00001 0.0080 lfRBYL 0.001 1 0.00001 0.0081 lfRBZ 0.001 2 0.00001 0.0080 lfRBZL 0.016 1 0.01600 0.0081 lfRBL 0.001 N.A. N.A. 0.0081 TABLE II LOAD FLOW RESULTS FOR THE CASE STUDY II Load flow Time (s) Iterations Time/Iteration (s) Power losses (p.u.) lfnr 1.453 4 0.36325 2.2904 lfnrL 0.985 3 0.32833 2.2904 lfgs 753.046 7095 0.10614 2.2904 lfgsL 278.485 3517 0.07918 2.2904 lfgsZ 737.734 7095 0.10398 2.2904 lfgsZL 274.046 3517 0.07792 2.2904 Zimp 0.266 3 0.08867 2.2903 ZimpL 0.188 2 0.09400 2.2904 lfRBY 0.250 4 0.06250 2.2904 lfRBYL 0.203 2 0.10150 2.2904 lfRBZ 0.406 4 0.10150 2.2904 lfRBZL 0.219 2 0.10950 2.2904 lfRBL 0.032 N.A. N.A. 2.2930 as voltage dependent current sources to voltage dependant admittances, providing a formulation that is applicable for both radial and interconnected or meshed electrical systems (for which the specialized distribution load flow algorithms cannot be used), providing good results when used in long radial systems and even grids with ill-conditioned characteristics, while using an admittance matrix formulation which naturally helps avoiding potential convergence problems due to singularity. The presented linear non-iterative load flow method is adequate for its use in medium and low voltage grids, achieving results within less than 1% of error and in considerably less calculation times, while also being adequate regardless of the grid’s configuration. Moreover, the number of required iterations and calculation time of the classical methods such as the Gauss-Seidel algorithms, can be reduced, and even the precision can be improved, by using the proposed linear method as a starting vector. This feature makes the proposed method adequate as a calculation tool for the estimation and deployment of grid’s regulation instructions, for which speed is of great importance. Finally, for the Gauss-Seidel and the Implicit Z-bus algorithms, the proposed convergence criterion variation of a power mismatch instead of the classical voltage mismatch has proven to be adequate in calculating the correct solutions and minimizing possible convergence errors. REFERENCES [1] P. C. Paiva, H. M. Khodr, J. A. Dominguez-Navarro, J. M. Yusta and A. J. Urdaneta, "Integral planning of primary-secondary distribution systems using mixed integer linear programming," in IEEE Transactions on Power Systems, vol. 20, no. 2, pp. 1134-1143, May 2005. [2] D. A. Quijano, J. Wang, M. R. Sarker and A. Padilha-Feltrin, "Stochastic assessment of distributed generation hosting capacity and energy efficiency in active distribution networks," in IET Generation, Transmission & Distribution, vol. 11, no. 18, pp. 4617-4625, 21 12 2017. [3] J. -. Teng, "Modelling distributed generations in three-phase distribution load flow," in IET Generation, Transmission & Distribution, vol. 2, no. 3, pp. 330-340, May 2008. [4] M. S. S. Abad, J. Ma, D. Zhang, A. S. Ahmadyar and H. Marzooghi, "Probabilistic Assessment of Hosting Capacity in Radial Distribution Systems," in IEEE Transactions on Sustainable Energy, vol. 9, no. 4, pp. 1935-1947, Oct. 2018. [5] H. Chiang, T. Zhao, J. Deng and K. Koyanagi, "Convergence/divergence analysis of implicit Z-bus power flow for general distribution networks," 2014 IEEE International Symposium on Circuits and Systems (ISCAS), Melbourne VIC, 2014, pp. 1808-1811. [6] M. Abokrisha, A. Diaa, A. Selim and S. Kamel, "Development of newton-raphson power-flow method based on second order multiplier," 2017 Nineteenth International Middle East Power Systems Conference (MEPCON), Cairo, 2017, pp. 976-980. [7] S. D. Varwandkar, "Realification of Power Flow," in IEEE Transactions on Power Systems, vol. 34, no. 3, pp. 2433-2440, May 2019. [8] M. Baradar and M. R. Hesamzadeh, "AC Power Flow Representation in Conic Format," in IEEE Transactions on Power Systems, vol. 30, no. 1, pp. 546-547, Jan. 2015. [9] R. Pourbagher and S. Y. Derakhshandeh, "Application of high-order Levenberg–Marquardt method for solving the power flow problem in the ill-conditioned systems," in IET Generation, Transmission & Distribution, vol. 10, no. 12, pp. 3017-3022, 2 9 2016. [10] M. Tostado-Véliz, S. Kamel and F. Jurado, "Development of combined Runge–Kutta Broyden's load flow approach for welland ill-conditioned power systems," in IET Generation, Transmission & Distribution, vol. 12, no. 21, pp. 5723-5729, 27 11 2018. [11] Q. Li and V. Vittal, "Convex Hull of the Quadratic Branch AC Power Flow Equations and Its Application in Radial Distribution Networks," in IEEE Transactions on Power Systems, vol. 33, no. 1, pp. 839-850, Jan. 2018. [12] M. Nick, R. Cherkaoui, J. L. Boudec and M. Paolone, "An Exact Convex Formulation of the Optimal Power Flow in Radial Distribution Networks Including Transverse Components," in IEEE Transactions on Automatic Control, vol. 63, no. 3, pp. 682-697, March 2018. [13] R. A. Jabr, I. Džafić and B. C. Pal, "Compensation in Complex Variables for Microgrid Power Flow," in IEEE Transactions on Power Systems, vol. 33, no. 3, pp. 3207-3209, May 2018. [14] N. Yang and H. Chen, "Three-phase power-flow solutions using decomposed quasi-Newton method for unbalanced radial distribution networks," in IET Generation, Transmission & Distribution, vol. 11, no. 14, pp. 3594-3600, 28 9 2017. [15] E. Romero-Ramos, A. L. Tirgo-García and J. A. Romero-Romero, "Improved radial load flow for the smart distribution grid," 2017 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGTEurope), Torino, 2017, pp. 1-6. [16] T. Alinjak, I. Pavić and M. Stojkov, "Improvement of backward/forward sweep power flow method by using modified breadth-first search strategy," in IET Generation, Transmission & Distribution, vol. 11, no. 1, pp. 102-109, 5 1 2017. [17] A. Garces, "A Linear Three-Phase Load Flow for Power Distribution Systems," in IEEE Transactions on Power Systems, vol. 31, no. 1, pp. 827-828, Jan. 2016. [18] K. Saxena and A. R. Abhyankar, "Agent-based decentralised load flow computation for smart management of distribution system," in IET Generation, Transmission & Distribution, vol. 11, no. 3, pp. 605-614, 16 2 2017. [19] X. Wang, M. Shahidehpour, C. Jiang, W. Tian, Z. Li and Y. Yao, "Three-Phase Distribution Power Flow Calculation for Loop-Based Microgrids," in IEEE Transactions on Power Systems, vol. 33, no. 4, pp. 3955-3967, July 2018. [20] I. Džafić and R. A. Jabr, "Real Time Multiphase State Estimation in Weakly Meshed Distribution Networks With Distributed Generation," in IEEE Transactions on Power Systems, vol. 32, no. 6, pp. 4560-4569, Nov. 2017. [21] C. Wang, A. Bernstein, J. Le Boudec and M. Paolone, "Explicit Conditions on Existence and Uniqueness of Load-Flow Solutions in Distribution Networks," in IEEE Transactions on Smart Grid, vol. 9, no. 2, pp. 953-962, March 2018. TABLE III LOAD FLOW RESULTS - VOLTAGE MISMATCH CONVERGENCE Load Flow Study case I Study case II Number of iterations Power losses (p.u.) Number of iterations Power losses (p.u.) lfgs(V) 13 0.0070 7 0.0087 lfgsL(V) 2 0.0081 2 2.2930 TABLE IV LOAD FLOW INITIAL LINES DATA FOR THE CASE STUDY III Line p-q Resistance (p.u.) Reactance (p.u.) 1-2 0.02 0.06 1-3 0.08 0.24 2-3 0.06 0.18 2-4 0.06 0.18 2-5 0.04 0.12 3-4 0.01 0.03 4-5 0.08 0.24 Fig. 4. Calculation time for different load flow algorithms in radial grids. [22] S. Chatterjee and S. Mandal, "A novel comparison of gauss-seidel and newtonraphson methods for load flow analysis," 2017 International Conference on Power and Embedded Drive Control (ICPEDC), Chennai, 2017, pp. 1-7. [23] R. Wang, H. Zhang, C. Li and C. Ma, "Dynamic power flow calculation method of power system with wind power," 2015 5th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (DRPT), Changsha, 2015, pp. 1987-1991. [24] W. Liu, W. Gu, Y. Xu, Y. Wang and K. Zhang, "General distributed secondary control for multi-microgrids with both PQ-controlled and droop-controlled distributed generators," in IET Generation, Transmission & Distribution, vol. 11, no. 3, pp. 707-718, 16 2 2017. [25] P. Chen, Z. Chen and B. Bak-Jensen, "Probabilistic load flow: A review," 2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, Nanjing, 2008, pp. 1586-1591. [26] J. F. Gomez et al., "Ant colony system algorithm for the planning of primary distribution circuits," in IEEE Transactions on Power Systems, vol. 19, no. 2, pp. 996-1004, May 2004. [27] P. M. D. O. Jesus, M. T. P. de Leao and H. M. Khodr, "Remuneration of Distribution Networks using a Fuzzy Multicriteria Planning Algorithm," 2006 International Conference on Probabilistic Methods Applied to Power Systems, Stockholm, 2006, pp. 1-7. [28] H. G. Abood, V. Sreeram, H. Al-Saadi and Y. Mishra, "Solution methods of Ill-conditioned power system state estimation: A comparative study," 2017 IEEE Region 10 Symposium (TENSYMP), Cochin, 2017, pp. 1-6. [29] D. P. Sharma, A. Chaturvedi, G. Purohit, and R. Shivarudraswamy, "Distributed load flow analysis using graph theory", World Acad. Science, Eng. and Tech. Int. J. of Electrical and Computer Eng., 2011, 5, (2), 2011, pp. 146-149. TABLE V LOAD FLOW POWER DATA AND VOLTAGE RESULTS FOR THE CASE STUDY III Node Initial (x/r) (x/r) / 25 (x/r) / 50 (x/r) / 100 Pd (MW) Qd (MVAr) Pg (MW) Qg (MVAr) V(p.u.) Angle V(p.u.) Angle V(p.u.) Angle V(p.u.) Angle 1 0 0 157.657 178.070 1.050 0.000 1.050 0.000 1.050 0.000 1.050 0.000 2 0 0 30.000 -65.100 0.954 -2.031 1.028 1.019 1.029 1.079 1.030 1.109 3 45.000 20.000 0 0 0.914 -5.618 1.008 1.158 1.009 1.283 1.010 1.346 4 80.000 30.000 0 0 0.902 -6.267 1.004 1.250 1.005 1.388 1.006 1.456 5 50.000 25.000 0 0 0.898 -5.662 1.006 1.378 1.008 1.507 1.009 1.571