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Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 BRANCH CURRENT BASED STATE ESTIMATION: EQUALITY-CONSTRAINED WLS AND AUGMENTED MATRIX APPROACHES WESLEY PERES, EDIMAR J. OLIVEIRA, JOÃO A. P. FILHO, JOSÉ L. R. PEREIRA,GUILHERME O. ALVES Department of Electrical Engineering, Federal University of Juiz de Fora (UFJF), Juiz de Fora, MG, Brazil E-mails: [email protected] Abstract This paper presents a comparison study among three methods for solving the Branch Current Based Distribution State Estimation (BCSE) problem: normal equations method, normal equations with constraints and augmented matrix approach. Different types of measurements are considered. The methods are evaluated taking into account their numerical stability, computational requirements and implementation issues. For this purpose, three systems have been used to perform the simulations. Keywords Distribution system state estimation, smart distribution system, branch current state estimation, phasor measurements units. ments (Baran, et al., 2009) and Advanced Metering Infrastructure (Baran & McDermott, 2009). In (Baran, et al., 2009) is proposed a methodology for topology errors processing and bad data identification using the BCSE method. In (Wang & Schulz, 2004) the BCSE method was modified to consider the polar representation of branch currents as state variables. The performance of SE methods can be strongly improved by using the new measurements technologies that are currently available. As an example, the Phasor Measurements Units (PMUs) are becoming increasingly widespread in transmission systems (Phadke, et al., 2009). PMUs can directly measure the bus voltage and the phase angles, because they are synchronized by the GPS universal clock. PMUs can also measure the current phasors through all incident branches. At the distribution level, the use of PMUs in the SE process looks very promising and it has been subject of recent papers. In (Pau, et al., 2013) the BCSE method has been modified to include phasor measurements. The solution of the WLS SE problem using Normal Equations (NE) can almost always be successfully carried out (Abur & Expósito, 2004). However, it is well known that under certain circumstances that are likely to occur in actual systems, the NE will become prone to numerical instabilities. As a result, the SE process may diverge. In order to compare some solutions for the numerical ill-conditioning problem associated with the rectangular nodal voltage based state estimators, (Holten, et al., 1988) have conducted a study with different methods: (i) normal equations method; (ii) orthogonal transformation method; (iii) hybrid method; (iv) normal equations with constraints and (v) augmented matrix approach (so called Hatchel's formulation). According to the authors, the methods (iii) and (v) have presented a good compromise between numerical stability and computational efficiency. It should be stressed that this comparison study was performed taking into account the single phase state estimators, that are used at the transmission level. Despite the study presented in (Holten, et al., 1988), investigations concerning the numerical sta- 1 Introduction The operation of smart distribution systems is a recognized challenge to be faced by distribution operators in the next few years. The high penetration level of Distributed Generators (DG) will call for new tools for real time monitoring. In this context, Distribution System State Estimation (DSSE) together with the availability of new measurement technologies will play an important role in the Distribution Management System (DMS). State Estimation (SE) for real time monitoring of transmission systems has been well established over the past four decades (Abur & Expósito, 2004). However, the research on DSSE dates back to the early 1990s (Baran & Kelley, 1994). Nonetheless, a very limited number of practical implementations have been performed, probably due the lack of proper infrastructure (Huang, et al., 2012). Distribution and transmission systems are very different from each other. The major differences can be summarized taking into account the distribution systems characteristics: (i) the number of measurements is very limited, (ii) in order to achieve the observability, the historical load data (that have limited accuracy) is used as pseudomeasurements; (iii) high R/X ratio; (iv) radial and weakly meshed operation; (v) unbalanced operation, etc. Therefore, the state estimators available for monitoring transmission systems are not suitable for distribution systems. In order to take into account the characteristics aforementioned, a new algorithm for DSSE was proposed in (Baran & Kelley, 1995). In this formulation, the real and imaginary parts of branch currents are used as state variables. The methodology is based on the Weighted Least Squares (WLS) and the normal equations. The voltage measurements have not been considered and simplifications have been done to obtain a constant gain matrix. This method is referred in the literature as Branch Current based State Estimation (BCSE). The BCSE method has been found to be faster than the nodal voltage based state estimators. Because of this, many papers have proposed improvements on BCSE method to include voltage measure- 3198 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 bility of the BCSE method for distribution systems should be conducted. This analysis has not been conducted before and it is a contribution of this paper. As previously mentioned, the future distribution systems will call for new tools for a reliable monitoring. In this context, it is important to investigate the performance of different approaches for solving the BCSE method under ill-conditioning problems. This paper presents a comparison study among three methods for solving the BCSE method: (i) normal equations method (NE); (ii) normal equations with constraints (NE/C) and (iii) augmented matrix approach (AM). This paper is a preliminary study conducted by the authors. Future works will investigate the performance of other methods for solving the BCSE method, such as those presented in (Holten, et al., 1988). Different types of measurements are considered. The methods are evaluated taking into account their numerical stability, computational requirements and implementation issues. It should be stressed that if any PMU is not installed in the system, the substation voltage (which is the slack node) cannot be estimated, due to a lack of a reference angle for the estimation process. In this case, the state variable vector will be composed only by branch currents, as defined in equation (3). (3) In order to obtain the estimated state , a WLS problem is performed to minimize the weighted sum of the squares of the residuals: (4) The term in equation (4) corresponds to the inverse of the weighting matrix associated with measurements. Generally, the elements of correspond to the variances of each measurement. The solution of the optimization problem presented in equation (4) can be obtained through an iterative process. Thus, if is the value of the estimated state at the iteration , the value at the next iteration will be: 2 Branch Current based Distribution System State Estimation including PMUs In this section, the methodology presented in (Pau, et al., 2013) is revised. The normal equation method is used to solve the SE problem. + where is the Jacobian matrix and Gain matrix: 2.1 Mathematical Formulation (5) is the (6) Consider a vector composed of a set of measurements, such as: real and reactive power flow, real and reactive power injection, current magnitude, voltage magnitude, voltage phasor and current phasor. The measurements are a nonlinear function of the state variables : (7) It is important to emphasize that the phasor measurements (voltage and currents) are included in the Jacobian matrix through their rectangular representation (real and imaginary parts). However, PMUs provide these types of measurements in terms of magnitude and phase angles. Therefore, it is necessary to perform a coordinate transformation (polar to rectangular) as proposed in (Singh, et al., 2009) and (Korres & Manousakis, 2011). The mathematical expressions associated with the Jacobian matrix can be obtained in (Baran & Kelley, 1995) and (Pau, et al., 2013). (1) where is the noise vector assumed to be composed by independent zero mean Gaussian variables, with diagonal covariance matrix . When at least one PMU is installed in the system, the substation voltage can be estimated. Therefore, the state variable vector will be composed by the real and imaginary parts of branch currents ( branches) together with the real and imaginary parts of substation voltage: 2.2 Algorithm The solution of the BCSE method can be separated in two steps that should be executed at every iteration: estimate the branch current by using the equation (5); calculate the nodal voltages through a forward step starting from the substation bus. Because of this, the separation of the layers in the radial system is required. For additional information on layers separa- (2) The state variable vector defined in equation (2), is composed by ( ) elements, where is the number of branches. In this case, ( ) is associated with the number of current variables (real and imaginary parts for each branch). The other two elements correspond to the real and imaginary parts of substation voltage. 3199 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 Telemetered and phasor measurements have low standard deviation , while the pseudomeasurements are assigned with a higher to highlight the lower confidence given to such quantities. The standard deviations (squared) appear in the covariance matrix , as shown in equation (4). In order to emphasize the accuracy of virtual measurements, high weighting factors are used (inverse of square of the standard deviation) (Holten, et al., 1988). According to (Abur & Expósito, 2004), the following specific sources of ill-conditioning have been described in the literature: very large weighting factors used to enforce virtual measurements; short and long lines simultaneously present at the same bus; a large proportion of injection measurements. It should be stressed that in the distribution systems, these sources of ill-conditioning can appear in a stronger way. Additionally, it is important to remember that the pseudomeasurements are modeled as power injection measurements. However, since the accuracy of pseudomeasurements is lower than telemetered measurements, low weights are assigned to pseudomeasurements (Lu, et al., 1995). In order to solve the ill-conditioning problems, different approaches have been proposed in the literature (Holten, et al., 1988) and (Abur & Expósito, 2004). In this paper, only two methods are analyzed to solve the ill-conditioning problem: (i) normal equations with equality constraints and (ii) augmented matrix approach. tion, the reader is referred to (de Araujo, et al., 2010). In this paper a backward approach is carried out to get the initial value of current. The initial value of voltage at every node is set to be a per unit value. Thereafter, using the injected power at every node, the value of branch currents is obtained. The following steps are performed: Step 1) Initialize the values of voltages and currents; Step 2) Calculate the updates of the state variables by using (5); Step 3) Calculate the nodal voltages through a forward step starting from the substation node; Step 4) If the increment is smaller than a convergence tolerance, stop. Otherwise: 1. if the number of iteration is smaller than a pre-specified maximum iteration number, go to step 2. 2. if it is not, the iterative process has not achieved the convergence, stop. 2.3 Weakly meshed distribution feeders Distribution systems operate with radial configuration due to protection issues (Blackburn & Domin, 2014). However, under some circumstances, distribution systems operate with loops created by closing normally open tie-switches. In order to deal with weakly meshed distribution feeders, a strategy has been taken into account in (Baran & Kelley, 1995). For each loop, an equality constraint is included, representing the Kirchhoff Voltage Law (KVL). The KVL says that the sum of voltage drops around the loop is equal to zero. Weakly meshed distribution feeders will not be considered in this paper. For additional information, the reader is referred to (Baran & Kelley, 1995). 3.1 Normal Equations with Constraints (NE/C) The use of large weights for modeling very accurate virtual measurements leads to ill-conditioning of the gain matrix . One way to avoid the use of large weights, is to model these measurements as explicit constraints in the WLS BCSE (Abur & Expósito, 2004): (8) st 3 Alternative Formulations of the BCSE method This optimization problem can be solved by the Lagrangian method and leads to the following linear system to be solved at every iteration: Beyond the telemetered and phasor measurements, there are two additional types of measurements (Abur & Expósito, 2004) and (Holten, et al., 1988) : Pseudomeasurements: historical load data used to improve the observability in the DSSE problem; Virtual measurements: are kind of information that does not require metering, such as zero injection buses. (9) (10) 3200 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 4 Performance Indexes where is a parameter used to control the numerical stability of the problem and are the Lagrange multipliers associated with the equality constraints. In order to compare the three methods under analysis (normal equation method, normal equation with equality constraints and augmented matrix approach), two performance indexes are used: (i) condition number of the coefficient matrix and (ii) sparsity degree. This section describes these indexes. 3.2 Augmented Matrix Approach (AM) Similar to the virtual measurements, regular measurements (telemetered and phasor) and pseudomeasurements equations can be written as equality constraints if the associated residuals are retained as explicit variables (Abur & Expósito, 2004): st 4.1 Condition Number (11) According to (Holten, et al., 1988), the condition number of the coefficient matrix can be used to measure the ill-conditioning of a problem. In accordance with (Abur & Expósito, 2004), the condition number can be calculated as shown in equation (13). (13) This optimization problem can also be solved by the Lagrangian method and leads to the following linear system to be solved at every iteration: (12) where is a parameter used to control the numerical stability of the problem, are the Lagrange multipliers associated with the first set of equality constraints and are the Lagrange multipliers associated with the second set of equality constraints. 3.3 Implementations Issues The first comparison among the methodologies under analysis is about the method used for factorization. According to the literature (Holten, et al., 1988), the coefficient matrices of the linear systems associated with NE/C and AM methods (equations (9) and (12)) are no longer positive definite. For a symmetric positive definite matrix, numerical stability is guaranteed when the pivots are taken from the diagonal in any order. Therefore, optimal ordering can be performed using only sparsity criterion. It is true for the Gain matrix of the NE method. For the NE/C and AM methods, row pivoting to preserve numerical stability must be combined with sparsity-oriented techniques during LU factorization (Abur & Expósito, 2004). In this paper, the solution of the linear systems was carried out by using the routines available in the MATLAB environment. These routines were found to be very robust. Regarding to the size of the linear system, it should be stressed that the solution of the NE/C and AM methods is not particularly expensive, since the coefficient matrices are very sparse. where represents a given matrix norm. One approximation which yields a good estimation of the condition number is the ratio , where and are the largest and the smallest absolute eigenvalues of the coefficient matrix. In solving the equation , because of the errors introduced in and during the solution process, it is needed to know by how much does the results or differ from the true solution , where the matrix and the vector represent errors. If they differ greatly we say the matrix is ill-conditioned. The condition number is equal to unity for identity matrices and tends to infinity for matrices approaching singularity. The more singular the coefficient matrix is, the more ill-conditioned the linear system will be (Abur & Expósito, 2004). Therefore, lower values for condition numbers are desirable. The condition number of the Gain matrix in the normal equation method is the square of the condition number of the Jacobian matrix. 4.2 Sparsity Degree As previously mentioned, for the normal equation with constraints and augmented matrix methods, the dimension of the linear system is increased. In order to evaluate the sparsity of the coefficient matrices, the sparsity degree is defined in equation (14). This index yields the percentage of null elements with respect to the total of elements. (14) where is total of elements of coefficient matrix (square of the number of state variables) and is the number of nonzero elements. The sparser a matrix is, the lower the number of elements stored will be. 3201 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 Other parameters are used to compare the methods, such as: the number of iteration and the computational time required by each one. Table 1 summarizes the measurement planning adopted for the simulations. It should be emphasized that no PMU is installed in the system. Therefore, the substation voltage is not included in the state variable vector and the total of state variables is equal to the double of the number of branches. 5 Results Table 1. Measurements Set of the 18-bus Test System. In this section the results obtained by using the three methods under analysis are presented: normal equations (NE); normal equations with constraints (NE/C); augmented matrix approach (AM). The MATLAB environment was used to perform the simulations. The simulations were carried out using an Intel Core i7 2.93 GHz computer with 8 GB of RAM and Windows 7 64-bit operating system. The study has been conducted by using three distribution test systems from the literature: 18-bus, 69-bus and 83-bus. The measurement set for each system was obtained by adding a Gaussian error to the power flow solution. The following standard deviations were considered for telemetered and phasor measurements: voltage magnitude: 8e-3; current magnitude: 4e-3; phasor measurements: 4e-5; power injection: 1e-2. Pseudomeasurements were supposed to contain a maximum error of 50% with respect to the nominal values (Gaussian distribution). The weights of these measurements were set to be 1. Since the pseudomeasurements are not accurate, their weights will be kept constant. In order to ensure the observability, all load buses are treated as pseudomeasurement buses. Virtual measurements (zero injection buses): the weighting factors were varied in a range of values. This procedure was adopted in (Holten, et al., 1988). A convergence tolerance was set to 1e-4. The scaling factor used to control the numerical stability of NE/C and AM methods was set to 1. Number of Measurements Regular Virtual Pseudo 4 14 20 Total Number States Measu. 34 38 Redundancy Level 1.11 Regular Measurements , , , In order to evaluate the performance of the methodologies, the weighting factors associated with the virtual measurements were varied. The condition numbers of the coefficient matrix are presented in Table 2. Table 2. Condition Number of the Coefficient Matrix. Method NE NE/C AM Weighting factors (Virtual Measurements) 1e6 1e8 1e10 1e12 4.960e7 4.959e9 4.959e11 4.983e13 9.603e5 4.179e4 From the results of Table 2 it is possible to see that the AM method presented better numerical stability than the other methods. In addition, the condition number of coefficient matrix remained constant for NE/C and AM methods. On the other hand, the condition number of the NE method increases with the weighting factors. An additional simulation was executed (weighting factors equal to 1e14 and 1e16) and the NE method has not converged. Although the condition numbers of the coefficient matrix was found to be sensitive to the method used, the number of iterations is the same and the average execution times are close, as shown in Table 3. Table 3. Number of Iterations and Average Execution Time (sec). Method NE NE/C AM Iterations 7 7 7 Execution time 0.25 0.29 0.28 Finally, Table 4 presents the sparsity degree of each method. As previously mentioned, the solution of the NE/C and AM methods is not particularly expensive, due to their higher sparsity degree. 5.1 18-bus test system This system has a radial configuration and its data can be obtained in (Ahmadi & Green, 2009). The original version without distributed generation was used and the on line diagram is shown in Figure 1. Table 4. Sparsity Degree. Method NE NE/C AM Figure 1. On line diagram of the 18-bus test system 3202 Number of elements (34)2 (48)2 (72)2 Nonzero elements 224 270 316 Sparsity degree (%) 80.6 88.3 93.9 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 of NE/C method (Holten, et al., 1988). This procedure aims at obtaining better solutions. 5.2 69-bus test system It is a radial system and its data can be obtained in (da Silva, et al., 2008). This system was modified through the inclusion of a distributed generator as proposed in (Dias, et al., 2012). A DG of 1870 kW and 1159 kVar was installed at bus 61. Figure 2 shows the on line diagram. Table 6. Condition Number of the Coefficient Matrix. NE NE/C AM Table 7 shows that the iteration number was the same for both methods. In addition, the average execution time is very close. 1 2 3 4 36 29 47 5 37 30 48 6 38 31 49 7 39 32 50 8 28 33 34 53 35 54 11 55 12 Table 7. Number of Iterations and Average Execution Time (sec). Method NE NE/C AM 40 9 51 41 10 52 42 13 57 14 58 15 66 44 67 45 Iterations 3 3 3 Execution time 0.30 0.34 0.33 Finally, Table 8 presents the sparsity degree of each method. As previously mentioned, the solution of the NE/C and AM methods is not particularly expensive due to their high sparsity degree. 43 56 Weighting factors (Virtual Measurements) 1e6 1e8 1e10 1e12 3.846e12 3.857e12 5.724e12 5.709e14 3.8623e12 1.731e5 Method 46 68 Table 8. Sparsity Degree. 69 59 16 60 17 61 18 62 19 63 20 64 21 65 22 Number of elements (138)2 (178)2 (290)2 Method NE NE/C AM Nonzero elements 4269 4536 1492 Sparsity degree (%) 77.6 85.7 98.2 5.3 83-bus test system 23 24 The 83-bus system of the Taiwan Power Corporation (TPC) has been used for optimal reconfiguration and capacitor allocation studies (de Oliveira, et al., 2010). In this paper, this system is used for evaluating the state estimation methodologies under analysis. Its data can be obtained in (Chiou, et al., 2005). Table 9 summarizes the measurement planning adopted for the simulations. A PMU is installed at the substation node (node 84). In this case the substation voltage is directly measured. 25 26 27 Figure 2. Diagram of the 69-bus test system Table 5 summarizes the measurement planning adopted for the simulations. It is possible to notice that one PMU was installed at bus 61. Therefore, the voltage substation is considered to be a state variable in the SE problem. Table 5. Measurements Set of the 69-bus Test System. Number of Measurements Regular Virtual Pseudo 16 40 96 , , , , Total Number States Measu. 138 152 Regular Measurements , , , , , , , , Table 9. Measurements Set of the 83-bus Test System. Redundancy Level Number of Measurements Regular Virtual Pseudo 33 34 132 1.10 , , , , , The condition numbers associated with the coefficient matrix of each method are presented in Table 6. It is interesting to notice that the condition number of NE/C method is larger than those associated with NE method in some cases. It has already been noticed in the literature and a solution would be optimizing the scaling factor in the coefficient matrix , , , , Total Number States Measu. 168 199 Regular Measurements , , , , , , , , , , , , , , , , Redundancy Level 1.18 , , , , , , The condition numbers associated with the coefficient matrix of each method are presented in Table 10. 3203 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 Table 10. Condition Number of the Coefficient Matrix. Method NE NE/C AM Regarding to the numerical stability, the NE method was found to be prone to convergence problems depending on the weighting factors assigned to the virtual measurements. Finally, the AM method presented the best sparsity degree. In general, the NE/C and AM methods are not expensive to solve with respect to the effort required by the NE method. Concluding, although the NE was found to be numerically unstable when high weighting factors are used, it is possible to conclude that all methods can present a satisfactory performance in practical applications. Concerning future studies, the analysis of other systems will be evaluated as well as other solution techniques, such as the orthogonal transformation. Moreover, other issues will be evaluated, such as observability analysis and bad data processing techniques. Weighting factors (Virtual Measurements) 1e6 1e8 1e10 1e12 2.5516e10 2.6860e10 4.3899e11 4.3692e13 2.5869e10 4.9024e5 Table 11 shows that the iteration number was the same for both methods. In addition, the average execution time is very similar. Table 11. Number of Iterations and Average Execution Time (sec). Method NE NE/C AM Iterations 3 3 3 Execution time 0.33 0.38 0.38 Finally, Table 12 presents the sparsity degree of each method. Table 12. Sparsity Degree. Method NE NE/C AM Number of elements (168)2 (202)2 (367)2 Nonzero elements 1794 1940 1649 Sparsity degree (%) 93.64 95.25 98.78 Acknowledgment The authors wish to thank CAPES, CNPq/INERGE, FAPEMIG and CEPEL for supporting this research. The first author thanks CAPES by the CAPES-REUNI scholarship. For the 83-bus system, the same previous conclusions arise: the AM method is the most stable among the methods considered. All methods considered present a high sparsity degree, providing an efficient performance. However the AM approach yields the largest sparsity degree. Concerning computational effort, both methods presented similar performance. References Abur, A. & Expósito, A. G., 2004. Power System State Estimation: Theory and Implementation (Power Engineering (Willis)). s.l.:CRC Press. Ahmadi, A. & Green, T., 2009. Optimal power flow for autonomous regional active network management system. s.l., s.n., pp. 1-7. Baran, M., Jung, J. & McDermott, T., 2009. Including voltage measurements in branch current state estimation for distribution systems. s.l., s.n., pp. 1-5. Baran, M., Jung, J. & McDermott, T., 2009. Topology error identification using branch current state estimation for distribution systems. s.l., s.n., pp. 1-4. Baran, M. & Kelley, A., 1994. State estimation for real-time monitoring of distribution systems. Power Systems, IEEE Transactions on, Aug, 9(3), pp. 1601-1609. Baran, M. & Kelley, A., 1995. A branch-currentbased state estimation method for distribution systems. Power Systems, IEEE Transactions on, Feb, 10(1), pp. 483-491. Baran, M. & McDermott, T., 2009. Distribution system state estimation using AMI data. s.l., s.n., pp. 1-3. Blackburn, J. L. & Domin, T. J., 2014. Protective Relaying: Principles and Applications, Fourth Edition. s.l.:CRC Press. Chiou, J.-P., Chang, C.-F. & Su, C.-T., 2005. Variable scaling hybrid differential evolution for solving network reconfiguration of distribution 5 Conclusions This paper presented a comparison study among three different methods for solving the branch current based distribution system state estimation: normal equation (NE); normal equation with constraints (NE/C); matrix augmented approach (AM). The comparison study has been conducted in order to evaluate the impact of virtual measurement weighting factors on the convergence behavior of the branch current based estimation method. For this class of state estimators, this study had not been conducted before and it is a contribution of this paper. Three distribution systems were used to perform the simulations. With respect to implementation issues, the NE/C and AM methods require more sophisticated factorization and ordering techniques. In this paper, the MATLAB routines were used and were found to be very effective. Both methodologies spent the same number of iterations and average execution time was very close. 3204 Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014 systems. Power Systems, IEEE Transactions on, May, 20(2), pp. 668-674. da Silva, I. et al., 2008. A Heuristic Constructive Algorithm for Capacitor Placement on Distribution Systems. Power Systems, IEEE Transactions on, Nov, 23(4), pp. 1619-1626. de Araujo, L. R. et al., 2010. Comparisons between the three-phase current injection method and the forward/backward sweep method. International Journal of Electrical Power \& Energy Systems , 32(7), pp. 825-833. de Oliveira, L. W. et al., 2010. Optimal reconfiguration and capacitor allocation in radial distribution systems for energy losses minimization. International Journal of Electrical Power & Energy Systems , 32(8), pp. 840-848. Dias, B. H. et al., 2012. Hybrid heuristic optimization approach for optimal Distributed Generation placement and sizing. USA, Power and Energy Society General Meeting, 2012 IEEE, pp. 1-6. Holten, L. et al., 1988. Comparison of different methods for state estimation. Power Systems, IEEE Transactions on, Nov, 3(4), pp. 17981806. Huang, Y.-F.et al., 2012. State Estimation in Electric Power Grids: Meeting New Challenges Presented by the Requirements of the Future Grid. Signal Processing Magazine, IEEE, Sept, 29(5), pp. 33-43. Korres, G. N. & Manousakis, N. M., 2011. State estimation and bad data processing for systems including PMU and SCADA measurements. Electric Power Systems Research , 81(7), pp. 1514-1524. Lu, C., Teng, J. & Liu, W.-H., 1995. Distribution system state estimation. Power Systems, IEEE Transactions on, Feb, 10(1), pp. 229-240. Pau, M., Pegoraro, P. & Sulis, S., 2013. Efficient Branch-Current-Based Distribution System State Estimation Including Synchronized Measurements. Instrumentation and Measurement, IEEE Transactions on, Sept, 62(9), pp. 2419-2429. Pau, M., Pegoraro, P. & Sulis, S., 2013. WLS distribution system state estimator based on voltages or branch-currents: Accuracy and performance comparison. s.l., s.n., pp. 493-498. Phadke, A., Thorp, J., Nuqui, R. & Zhou, M., 2009. Recent developments in state estimation with phasor measurements. s.l., s.n., pp. 1-7. Singh, R., Pal, B. & Jabr, R., 2009. Choice of estimator for distribution system state estimation. Generation, Transmission Distribution, IET, July, 3(7), pp. 666-678. Wang, H. & Schulz, N., 2004. A revised branch current-based distribution system state estimation algorithm and meter placement impact. Power Systems, IEEE Transactions on, Feb, 19(1), pp. 207-213. 3205