Trust-Region Method for Load Flow Solution of Three-Phase Unbalanced Electric Power Distribution System

. At present, the electric power system is getting bigger and more complex, and its loading is also increasing. As a consequence, planning, operation, and control of the power system also become more complicated. It is known that system planning and operation are mostly based on the steady-state condition of the power system, and the system steady-state condition can only be determined from the load ﬂow study. Thus, the development of a reliable and eﬃcient method to solve the load ﬂow problem is necessary so that the system steady-state condition can properly be evaluated. Since the characteristics of the electric distribution system are diﬀerent from those of the transmission system, special treatments are usually required in the distribution system load ﬂow (DSLF) analysis. In this context, several interesting techniques have been proposed in the analysis. In this paper, the application and extension of the trust-region method to solve the three-phase DSLF problem are proposed and investigated. Case studies using 19-node, 25-node, and 123-node distribution systems are also given in this paper. Results of the studies show that the output values obtained by the proposed method are in excellent agreement with those obtained by previously published methods. These results conﬁrm the validity of the proposed method. Case study results also indicate that the proposed method has better computational performances than the forward/backward sweeping (FBS) method.


Introduction
Load flow (or power flow) analysis is basically a solution for the normal operating conditions of a power system.e results of load flow analysis are normally used for system planning, basis data in the operational stage, and electric power system operation and control.Results obtained from load flow studies are also used for system steady-state studies, optimum scheduling of power generation, and system dynamic or stability studies.
e significance of load flow analysis has attracted the attention of many engineers for several decades.Many researchers have spent much of their professional careers looking for solutions to load flow problems.A number of efforts that have been made to solve the load flow problem have produced a number of methods reported in many technical publications [1].
e Gauss-Seidel method was the first method that solves the load flow problem digitally.
e method was invented by Ward and Hale in 1956.However, the method needs a significant amount of computer memory and iteration number as the power system size increases.e disadvantage of the Gauss-Seidel method triggers the development of the Newton-Raphson method, which was first developed by Van Ness and Griffin [1].In contrast to the Gauss-Seidel algorithm, the Newton-Raphson-based methods require only a small number of iterations to obtain the solution.Moreover, the methods do not depend on the size of the system network.
However, the application of the Newton-Raphson-based methods to electric power distribution systems can cause some convergence problems.ese convergence problems arise because the characteristics of the distribution system are different from those of the electric power transmission system.Electric power distribution systems are usually characterized by (i) radial or weakly meshed network structures; (ii) high ratio of line resistance/reactance (R/X); (iii) very large number of branches and nodes; and (iv) unbalanced network and load.e characteristics (i)-(iii) can cause singularity of the Jacobian matrix in the Newton-Raphson iteration schemes, and the solution will be difficult or impossible to obtain.e characteristic (iv), that is, the system unbalance, requires that the system must be modeled in three phases, and consequently, the three-phase load flow study must be used instead of the single-phase load flow study.is will lead to additional computational effort.is computational effort will further increase if the load flow calculation is carried out repeatedly, for example, the case of service restoration, feeder reconfiguration, and optimal placement of capacitor.
us, in addition to being reliable, the load flow solution method of an unbalanced electric power distribution system also needs to be efficient.
Other researchers have also conducted investigations into the load flow problem solution methods [16][17][18][19][20][21][22][23][24][25].ese researchers use a method or technique that utilizes the radial structure of the electric power distribution system. is technique is also known as the FBS technique.In finding a solution to the load flow problem, the forward/backward sweeping technique does not use the Newton-Raphson iterative scheme.erefore, it does not have the convergence issue as in the Newton-Raphson method.e drawback of this method is that it requires some complicated branch numbering and bus ordering techniques.In the method investigated and proposed in this paper, the load flow problem is solved using the trust-region method.e trustregion method is commonly used to solve optimization problems.However, several researchers have carried out studies and applied the method to find a solution to a set of nonlinear equations [26][27][28][29].e results of the studies show that this method is very potential to be used as a technique to solve a set of nonlinear equations.Results of the researchers' studies also show that the trust-region method can overcome the case of a singular Jacobian matrix since the optimization method is used in the solution updating process.
In [30], the trust-region method has successfully been applied to find a solution to the single-phase DSLF problem.However, in [30], the distribution system has been assumed to be balanced, and the single-phase load flow method has been used in the analysis.Since distribution systems are inherently unbalanced and to obtain accurate results, the system unbalance needs to be considered and taken into account in the analysis.As a consequence, the distribution system must be modeled in three phases, and the three-phase load flow method must be used in the analysis.erefore, in this paper, the application and extension of the trust-region method proposed in [30] to three-phase load flow analysis are investigated.e contributions of the present paper can be outlined as follows: (i) e developed algorithm has a better convergence characteristic, and only a small number of iterations are required in the calculation.
(ii) e proposed method is more efficient; that is, the load flow solution can be obtained with a minimum computation time.
To be more systematic, this paper is organized as follows: Section 2 discusses the formulation of the three-phase DSLF problem.Section 3 continues with an explanation of the proposed method for solving the load flow problem.Case study is presented in Section 4, where validation of the proposed method is also given.Finally, Section 5 points out some important conclusions of the paper.

Formulation of Three-Phase DSLF Problem
Load flow problem is usually solved using node analysis where the admittance matrix is frequently used in the analysis.In terms of node quantities, the behavior of a threephase electric power distribution system can be explained using the following relationship: where I abc is the vector of nodal currents, V abc is the vector of nodal voltages, and Y abc is the system admittance matrix.
For distribution system with n nodes, I abc , V abc , and Y abc will have the following forms: where Nodal current in (1) can be expressed in terms of nodal voltage and nodal power as follows: where S abc G is the vector of powers entering the node (generation powers) and S abc G is the vector of powers leaving the node (load powers).
For distribution system with n nodes, formulations for S G abc and S L abc are given by where Substituting ( 4) into (1) results in Equation ( 7) is the formulation of the three-phase DSLF problem.All of the variables (known and unknown) in the formulation are shown in Table 1.It is to be noted that distribution systems are normally fed at one node (substation node).erefore, in DSLF analysis, the substation node is usually considered as a reference node, and the voltage magnitude of this node (|V SS |) is specified at a certain value (e.g., 1.0 pu).Moreover, as the system is only fed at the substation node, power generations at the remaining nodes (load nodes) will be zero.It can be seen that ( 7) is a set of nonlinear equations which has to be solved in load flow analysis to evaluate the steady-state condition of the distribution system.e method of solution to these equations is explained in the next section.

Solution Technique
3.1.Trust-Region Method.Similar to the Newton-Raphson method, the iterative technique is also employed in the trustregion method to find a solution to a set of nonlinear equations [30].Consider a general set of nonlinear equations in terms of vector function F(x) as follows: where n is the number of equations, f i (x) is the i th nonlinear equation, and x � x 1 x 2 . . .x n   T is the vector of unknown variables (to be calculated).
In the iterative method, (8) is solved using where k is the iteration count and d � d 1 d 2 . . .d n   T is the vector of correction factors.
In the Newton-Raphson method, the vector of correction factors is calculated directly.However, as the Jacobian matrix for the distribution system is sometimes singular, the calculation does not always succeed, and the solution cannot be obtained.In the trust-region method, a different technique is used to calculate the vector of correction factors.It is determined through the optimization process, where inverting the Jacobian matrix is not required in the process.In this way, the trust-region method can always produce a valid solution.In the trust-region method, the vector of correction factors is determined using min with constraint : where Δ (k) > 0 is the radius of the trust region.Details on how to choose the radius value in every iteration step can be found in [26][27][28][29][30]. Also, in (10), quantity q(d (k) ) is determined using where and the Jacobian matrix J(x) in ( 13) and ( 14) has the following form: Elements of J(x) can be calculated analytically or numerically.However, calculation using the numerical method is more advantageous because the analytic formulations of the partial derivative are sometimes difficult to obtain.An explanation of the numerical method for the determination of the Jacobian matrix is given in Appendix A.
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Starting Values for Iteration.
To initialize the iteration process in the trust-region method, the following starting values for the unknown variables can be used: Substation power: Load node voltages:

Test Systems and Software Conditions.
To validate the method proposed in Section 3, the following three unbalanced distribution networks areused: (i) 19-bus network [31,32] Single line diagram of this 11 kV distribution network is shown in Figure 1. e system (line and load) data are given in (Tables 2 and 3).
(ii) 25-bus network [31] Single line diagram of this 4.16 kV distribution network is shown in Figure 2. e system (line and load) data are given in (Tables 4-6).
(iii) 123-bus network adopted from [33] Single line diagram of this 4.16 kV distribution network is shown in Figure 3. e system (line and load) data are given in (Tables 7-9).
It is to be noted that MATLAB ® software has been used for all of the computations in the present work (the proposed method algorithm has been implemented as MATLAB ® codes or m-files). .For comparison purposes, output from other methods (i.e., FBS method [16-25, 31, 32]) and results from the OpenDSS tool [34] are also shown in the tables.It can be seen that the output of the proposed method is in excellent agreement with those from other methods.ese results confirm the validity of the proposed method for solving the three-phase DSLF problem.In addition, Table 13 gives computational performances of the proposed method and the FBS method.e computations were run on a PC with Intel Core 2 2.4 GHz processor.It can be seen from Table 13 that the proposed method requires fewer iterations, which indicates that it has a better convergence characteristic.Moreover, the proposed trust-region method is more efficient than the BFS method since the BFS method consumes more computation time than the proposed method.

Conclusions
Load flow analysis is basically a solution for the normal operating conditions of an electric power system.In general, the results of load flow calculation are used for power system planning, basis data in the operational stage, and power system operation and control.In the present paper, the trustregion method has been investigated and proposed to solve the load flow problem of the three-phase unbalanced electric power distribution system.e trust-region method is commonly used to solve the optimization problem.However, this method can be used as a technique to solve nonlinear equation systems arising from the load flow problem formulation.SpecialB treatments that are usually required in the distribution system load flow (DSLF) analysis are not needed in the proposed method.Moreover, the method can always obtain a solution even if the system is illconditioned.Case studies using 19-node, 25-node, and 123node distribution systems have also been given in this paper.Results of the studies show that the output values obtained by the proposed method are in excellent agreement with those obtained by the previously published method.ese results confirm the validity of the proposed method for solving the three-phase unbalanced DSLF problem.Case study results also indicate that the proposed method has better computational performances than the FBS method.In future work, the extension of the method so that it can be implemented in distribution system with distributed energy resources (DERs) can be investigated.is is probably an interesting topic since the penetration of DERs in the distribution network is presently increasing, which complicates the system load flow analysis.
Appendix A e following is the formulation to calculate the Jacobian matrix J(x) of a vector function F(x) at the point x * : Elements of J(X * ) are computed as follows: First row:

Table 1 :
Known and unknown variables in DSLF formulation.

Table 5 :
Type and length of conductor lines of 25-bus network.

Table 8 :
Type and length of conductor lines of 123-bus network.

Table 11 :
Voltage profile of 25-bus network.

Table 10 :
Voltage profile of 19-bus network.