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Generalised‐fast decoupled state estimator

2018; Institution of Engineering and Technology; Volume: 12; Issue: 22 Linguagem: Inglês

10.1049/iet-gtd.2018.5076

1751-8695

Yanbo Chen, Zhi Zhang, Hengfu Fang, Yansheng Lang, Jin Ma, Shunlin Zheng,

Power System Reliability and Maintenance

IET Generation, Transmission & DistributionVolume 12, Issue 22 p. 5928-5938 Research ArticleFree Access Generalised-fast decoupled state estimator Yanbo Chen, Corresponding Author Yanbo Chen yanbochen2008@sina.com School of Electrical & Electronic Engineering, North China Electric Power University, The Main Teaching Building, No. 2, Beinong Road, Zhu Xin Zhuang, Beijing, People's Republic of ChinaSearch for more papers by this authorZhi Zhang, Zhi Zhang School of Electrical & Electronic Engineering, North China Electric Power University, The Main Teaching Building, No. 2, Beinong Road, Zhu Xin Zhuang, Beijing, People's Republic of ChinaSearch for more papers by this authorHengfu Fang, Hengfu Fang China Electric Power Research Institute, Road No. 15, Qinghe Xiaoying, Beijing, People's Republic of ChinaSearch for more papers by this authorYansheng Lang, Yansheng Lang China Electric Power Research Institute, Road No. 15, Qinghe Xiaoying, Beijing, People's Republic of ChinaSearch for more papers by this authorJin Ma, Jin Ma School of Electrical and Information Engineering, University of Sydney, Sydney, NSW, 2006 AustraliaSearch for more papers by this authorShunlin Zheng, Shunlin Zheng School of Electrical & Electronic Engineering, North China Electric Power University, The Main Teaching Building, No. 2, Beinong Road, Zhu Xin Zhuang, Beijing, People's Republic of ChinaSearch for more papers by this author Yanbo Chen, Corresponding Author Yanbo Chen yanbochen2008@sina.com School of Electrical & Electronic Engineering, North China Electric Power University, The Main Teaching Building, No. 2, Beinong Road, Zhu Xin Zhuang, Beijing, People's Republic of ChinaSearch for more papers by this authorZhi Zhang, Zhi Zhang School of Electrical & Electronic Engineering, North China Electric Power University, The Main Teaching Building, No. 2, Beinong Road, Zhu Xin Zhuang, Beijing, People's Republic of ChinaSearch for more papers by this authorHengfu Fang, Hengfu Fang China Electric Power Research Institute, Road No. 15, Qinghe Xiaoying, Beijing, People's Republic of ChinaSearch for more papers by this authorYansheng Lang, Yansheng Lang China Electric Power Research Institute, Road No. 15, Qinghe Xiaoying, Beijing, People's Republic of ChinaSearch for more papers by this authorJin Ma, Jin Ma School of Electrical and Information Engineering, University of Sydney, Sydney, NSW, 2006 AustraliaSearch for more papers by this authorShunlin Zheng, Shunlin Zheng School of Electrical & Electronic Engineering, North China Electric Power University, The Main Teaching Building, No. 2, Beinong Road, Zhu Xin Zhuang, Beijing, People's Republic of ChinaSearch for more papers by this author First published: 27 September 2018 https://doi.org/10.1049/iet-gtd.2018.5076Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Nowadays the fast-decoupled state estimation (FDSE) is widely used in almost every power system control centre. FDSE is effective and efficient for most transmission systems but it may not converge for systems with a large ratio of branch resistance to reactance (R/X); meanwhile the branch current magnitude measurements (BCMMs) cannot be reliably used in FDSE, thereby limiting its applications especially for the distribution systems where BCMMs abound. In this study, the above two problems have been addressed by transforming all measurements so that they can be classified as quasi-real power measurements and quasi-reactive power measurements, leading to a generalised FDSE (GFDSE) with a solid theoretical foundation. The formulation of GFDSE is based on only the assumption, rather than three assumptions used in FDSE. As a result, GFDSE has good adaptability to transmission systems as well as distribution systems; additionally, BCMMs can be reliably used in GFDSE. Case studies based on IEEE benchmark systems and a real grid of China demonstrate that the proposed GFDSE has very good convergence properties for transmission systems and distribution systems; and at the same time, the proposed GFDSE is also superior to FDSE in terms of computational efficiency under almost all cases. 1 Introduction In recent years, in order to cope with the dual pressures of the energy crisis and the climate change, the smart grid has become a common choice of the world. In order to achieve the comprehensive, real-time and accurate perception of smart grids, advanced measurement instruments, phase measurement units together with the conventional supervisory control and data acquisition (SCADA) systems have been more and more widely used in smart grids, bringing huge amounts of data and information to be processed for the smart grid control centres. The reliability of data and information is critical to the smart grid. How to quickly filter the data in order to obtain the authentic information is a key issue to be addressed. As a data filter, the traditional state estimation (SE) technology needs to be improved and will play a more and more important role in smart grids [1]. The transmission system SE (TSSE) has been studied and applied for many years, while the research and application of distribution system SE (DSSE) is relatively insufficient. DSSE is indispensable for active distribution network modelling and integrated operation with distributed energy resources [2]. The difference between DSSE and TSSE mainly includes the following aspects: (i) in distribution systems, real-time measurements are limited and the observability is sometimes not satisfied; furthermore, there are often a lot of branch current magnitude measurements (BCMMs) in distribution systems; (ii) distribution systems under radial and weakly-meshed operations have numerous unbalanced three phase branches with high R/X ratios and unbalanced loads separated by short distances [2]. The above two key features determine that TSSE cannot be used directly in the distribution systems. On the other hand, as the different application fields of SE, there are some similarities between TSSE and DSSE. The following is the comparison of TSSE and DSSE from three aspects of measurement data, core algorithm and robustness, also a brief discussion about the correlation between active power and reactive power is made. 1.1 Measurement data Real-time measurements from SCADA system generally guarantee the observability of TSSE; whereas real-time measurements, pseudo measurements together with virtual measurements are often used together in DSSE to guarantee the observability. The measurements in the distribution systems often come from different equipment and have different sampling periods and precision. How to use different types of measurements is an important issue for DSSE. Some research has been dedicated to solving these problems [3–7]. The use of pseudo measurements and virtual measurements is a crucial characteristic of DSSE, the authors of [8–13] have already studied this problem. 1.2 Core algorithm The earliest proposed SE uses the weighted least square (WLS) method [14–17], which is the optimal estimate under the maximum likelihood estimation with the assumption that the measured noise follows the Gauss distribution. WLS has good convergence properties, so it is widely applied in both TSSE and DSSE. The main differences among WLS-based DSSE methods are basically the choices of the state variables and the techniques to incorporate heterogeneous measurements [2, 6, 18–21]. To improve the efficiency of WLS, fast-decoupled SE (FDSE) was proposed [22, 23]. Compared with WLS, FDSE has the following advantages: (i) it requires less computer memory for its implementation; (ii) its computational efficiency is higher than WLS. The above advantages make FDSE widely accepted in the industry and used for TSSE in almost every power system control centre all over the world. On the other hand, FDSE has two limitations which should be carefully considered before using it: (i) there may be cases when the network parameters or operating conditions violate the decoupling assumptions (especially for the distribution systems where R/X is high), thereby leading FDSE do not converge; (ii) FDSE cannot be reliably used in the presence of BCMMs, thereby affecting the estimation accuracy and more importantly limiting its application especially for the DSSE where BCMMs abound. The above two limitations of FDSE may be attributed to the three prerequisites of FDSE which will be elaborated further in Section 2. Some research has been dedicated to the above two issues of FDSE. In [24], two different decoupling techniques using node voltages in rectangular co-ordinates are proposed: the exact decoupled technique (EDT) and the modified fast-decoupled technique (MFDT). EDT is suitable for high R/X networks, but the coordinate transformation in EDT requires extra computation in the iterative process; MFDT uses the method of retaining nonlinear term to improve the accuracy, but the premise is that the R/X ratio is not high, and BCMMs are not taken into account. In [25], the technique of measurement transformation is presented in FDSE for a distribution system, and the convergence of this method is proved better when the R/X ratio is closer to 1, but the measurement transformation requires pairing transformed measurements, which reduce the accuracy of the calculation. In [26], a new FDSE is presented by the rotation transformation, this method is suitable for networks with a high ratio of R/X, but the rotation transformation will greatly reduce the accuracy and efficiency of calculation, and rotated measurements do not include BCMMs still. Recently, a new FDSE for distribution systems is proposed in [27], the advantage of this method is that BCMMs can be used in FDSE by approximately reformulating BCMMs as active and reactive branch loss measurements; the drawback is that the computational efficiency has been affected by additional state variables, and the applicability of this approach to distribution systems with high ratio of R/X has not been discussed. 1.3 Robustness As conventional WLS and FDSE are inherently sensitive to bad data, much attention has been devoted to the issue of bad data suppression, such as the largest normal residual (LNR) test [14, 17] and other approaches. Despite the wide applications of either the WLS or the FDSE with LNR tests, they are difficult to handle multiple conforming gross errors. For remaining unbiased estimation despite the existence of different types of gross errors, various robust state estimators (RSEs) have been proposed. Recently, some interesting RSEs have been proposed for TSSE by [28, 29] and [1], respectively. The robustness of DSSE has been studied in [11, 12] and so on. However, the robustness of SE is beyond the scope of this paper. 1.4 Correlation between active and reactive power It is considered that active power and reactive power are typically correlated in [30–32]. In order to simplify the analysis, it is assumed that the active power and reactive power are irrelevant in the derivation process in this paper, we will leave this topic for the next research work. In this paper, the two limitations of conventional FDSE have been comprehensively addressed. The major contributions of this paper lie in threefold: (i) the measurements are transformed so that they can be classified as quasi-real power measurements and quasi-reactive power measurements, which have better decoupling performances, thereby leading to better convergence characteristics of the GFDSE proposed on normal systems as well as systems with a large ratio of R/X, i.e. making GFDSE suitable for DSSE; (ii) BCMMs are specially processed so that they can be used in GFDSE, thereby improving the accuracy of SE and more important making the proposed GFDSE have good adaptability to distribution systems as well as transmission systems; and (iii) the formulation of GFDSE is based on only one assumption, rather than three assumptions used in conventional FDSE, as a result, GFDSE has good adaptability to distribution systems as well as transmission systems. The remaining of this paper is organised as follows: conventional FDSE are briefly reviewed in Section 2. The measurements transformation method is proposed in Section 3, Section 3 also gives the method of handling BCMMs so as to use them directly in GFDSE. The formulation method and calculation procedures of the proposed GFDSE are also presented in Section 3. Case studies on standard IEEE test systems and a real power grid of China are given in Section 4. Finally, conclusions are drawn in Section 5. 2 Brief reviews on conventional FDSE 2.1 Measurement equations of SE For SE, the relationship between state variables and measurements can be described by the following equation: (1)where z is an m-dimensional measurement vector, usually including bus voltage magnitude measurements, real and reactive power injection measurements, real and reactive power flow measurements, BCMMs and so on; is the n-dimensional state vector (voltage magnitudes and phase angles in polar coordinates) with n = 2N − 1; N is the number of buses; denotes the phase angles of all buses except for the reference bus, denotes the voltage magnitude of all buses; is the non-linear vector function mapping the state vector to the measurement vector; e is an m-dimensional measurement error vector with variance R (an m × m diagonal matrix). The details about the measurements equation can be found in [17]. 2.2 Formulation of WLS The classical WLS model is given by (2)where R −1 is an m × m weighted diagonal matrix. Optimisation problem (2) can be solved via the Gauss–Newton method with the iteration Normal Equation (NE) as follows: (3)where k is the iteration index; x k is the solution vector obtained in the k th iteration; H (x) = ∂h (x)/∂x is the Jacobian matrix of the measurement equation; G (x k) = H T (x k)R −1 H (x k) is the gain matrix in the k th iteration. 2.3 Formulation of conventional FDSE The main computational burden associated with the WLS solution algorithm (3) is caused by the calculation and triangular decomposition of the gain matrix. To improve the computational efficiency, conventional FDSE is proposed by using two techniques: (i) maintaining a constant but an approximate gain matrix, based on the fact that the elements of the gain matrix do not significantly change between flat start initialisation and the converged solution; (ii) only using real (reactive) power measurements to estimate the angles (voltage magnitude) of all buses, based on the fact that the sensitivities of the real (reactive) power equations to changes in the magnitude (phase angle) of bus voltages are very low, especially for high-voltage (HV) transmission systems. In this formulation, the measurements and their related arrays can be partitioned as follows: (4)where denotes real power measurements, including the active power bus injections and active power flows in branches; denotes reactive power measurements, including bus voltage magnitudes, the reactive power bus injections and reactive power flows in branches; note that in conventional FDSE, BCMMs are not included in either of these groups of measurements; denotes the non-linear measurement function corresponding to , denotes the non-linear measurement function corresponding to ; , , , ; denotes the active/reactive weighted diagonal matrix. The following three assumptions are used in the formulation of FDSE [17]: (i) assume flat start operating conditions, i.e. all bus voltage magnitudes being at a nominal magnitude of 1.0 p.u. and all angles being at zero; (ii) the ratio of branch resistance to reactance, i.e. R/X, is very small; (iii) ignore the off-diagonal blocks and in H and compute the gain matrix using this approximation. This will also remove the off-diagonal blocks in the Jacobian matrix and the gain matrix, yielding a constant and decoupled gain matrix evaluated at the flat start (5)where and are the corresponding parts in the nodal admittance matrix; ignoring the branch series resistances in forming and will lead to the so-called XB or BX formulation of FDSE, respectively. The right hand side of (3) is also divided into two blocks (6)Thus, the NE (3) is replaced by (7)Note that the dimensions of two gain sub-matrices and are half of the size of the fully coupled gain matrix , and they are computed and decomposed into their triangular factors one-off at the beginning of the iteration, thereby making FDSE much more efficient than WLS. However, it should be pointed here that: (i) for transmission systems (especially for ultra-HV and HV transmission systems), the assumptions of FDSE often hold, as a result, FDSE has good adaptability to transmission systems (especially for HV transmission systems); whereas when the network parameters or operating conditions violate the decoupling assumptions (especially for distribution systems, where R/X is large), FDSE may not converge; (ii) BCMMs cannot be reliably used by FDSE, thereby affecting the estimation accuracy and more importantly limiting its application especially for the DSSE where BCMMs abound. 3 Proposed generalised-fast decoupled state estimator The analysis above indicates that the difficulties of the conventional FDSE stem from its three prerequisites. Therefore, if the number of prerequisites of FDSE can be reduced and BCMMs can be used, then FDSE will have good adaptabilities to different systems. This motivates us to propose a generalised FDSE (GFDSE). 3.1 Network model and measurements Firstly, the appropriate network model needs to be established. By modelling a three-winding transformer as three two-winding transformers, all transformers and transmission lines can be modelled by two-port π-models (as shown in Fig. 1a). Fig. 1Open in figure viewerPowerPoint Unified model of transmission lines and transformers(a) Two-port π-model of a network branch; (b) Equivalent circuit In Fig. 1a, ys = gs + jbs = 1/(rij + jxij) is the series susceptance; rij + jxij is the series impedance; bc is the line charging susceptance (specifically, for transformer branch bc = 0); k is the tap ratio (ignoring phase shifting transformer). For a transmission line, k = 1. The equivalent circuit of Fig. 1a is given in Fig. 1b, where , , , , , and . There are usually four types of measurements from SCADA system: the voltage magnitude at bus i, vim; real and reactive power injections at bus i, Pi and Qi; real and reactive power flow from bus i to bus j, Pij and Qij; and the branch current magnitude measurement from bus i to bus j, Iij. The errors of the measurement equations are not shown here for simplification and clearness (8) (9) (10) (11) (12) (13)where vi and θi are the state variables to be estimated; gsi, bsi, gij and bij are the same as shown in Fig. 1b; Gij + Bij are the ij th elements of the complex bus admittance matrix; ; ; ; . 3.2 Jacobian matrix of GFDSE based on only one assumption Only one assumption (i.e. the flat start operating conditions) is used in the formulation of GFDSE, rather than three ones used in FDSE. The exact expression of the Jacobian matrix of GFDSE is as follows. Note that the square of BCMMs is used instead of BCMMs. 3.2.1 Exact elements corresponding to z a, i.e. H aa and H ar 3.2.2 Exact elements corresponding to z r, i.e. H ra and H rr 3.2.3 Exact elements corresponding to BCMMs 3.3 Measurement transformation of GFDSE The following analysis is performed with the assumption that real and reactive measurements, including both real and reactive power injections as well as real and reactive power flows, are available in pairs, which is basically satisfied in the actual systems. In addition, the usage method of voltage magnitude measurements in GFDSE is the same as that in conventional FDSE. 3.3.1 Transformation method of Pi and Qi in GFDSE In conventional FDSE, the modified equations for i th real and reactive injection powers are as follows: (14)where , , and their elements can be obtained from Section 3.2. Introduce the matrix (where is to be solved), then multiply (14) with it from the left, then we have (15)In order to guarantee the good decoupling property of active and reactive injection powers, the values of non-diagonal elements should be close to 0, thus the value for should satisfy and . See Appendix 1 for the specific proof. Substituting and into (15) yields the modified equations in GFDSE for i th real and reactive injection powers as follows: (16)From (16), it can be seen that can be regarded as quasi active power measurements, can be regarded as quasi reactive power measurements, and they have very good decoupling properties for different systems. 3.3.2 Transformation method of Pij and Qij in GFDSE In conventional FDSE, the modified equations for i th real and reactive power flows are as follows: (17)where, , , their specific constant values can be obtained from the subsection B. Introduce two parameters and (where and are to be solved), and use them to construct the modified equations in GFDSE for real and reactive power flows as follows: (18)In order to guarantee the good decoupling property of real and reactive power flows, the values of non-diagonal elements should be close to 0, thus the value for and should satisfy and . See Appendix 2 for the specific proof. Substituting and into (18) yields the modified equations in GFDSE for i th real and reactive power flows ( and ) as follows: (19)Equation (19) shows that can be regarded as quasi active power measurements and can be regarded as quasi reactive power measurements, and they have very good decoupling properties for different systems. 3.3.3 Transformation method of BCMMs in GFDSE In conventional WLS, the modified equations for BCMM are as follows: (20)where, , , their specific constant values can be obtained from the subsection B. From (20), it can be seen that both the sensitivity of to changes in the magnitudes of bus voltages and the sensitivity of to changes in the phase angles of bus voltages are large, as a result, BCMMs are neither real power measurements nor reactive power measurements. This is the reason why BCMMs cannot be used in conventional FDSE. However, (20) can be rewritten as the following form: (21)From (21), it can be seen that is similar to reactive measurement residuals, therefore can be regarded as a quasi-reactive measurement in GFDSE, which solves the problem that BCMMs cannot be used in conventional FDSE. Note that before the iterative process converges, the correct value of is unknown, but it can be replaced by the estimated values that have been obtained in the current iteration. 3.4 Modified equations and NEs of GFDSE After the above analysis, all measurements can be transformed and partitioned into two parts in GFDSE: Quasi real power measurements, including and . Quasi reactive power measurements, including , , and . The corresponding quasi active and quasi reactive modified equations are (22) and (23), respectively. (22) (23)where, , all elements of are equal to zero except that the element corresponding to is equal to one; It can be seen that (22) and (23) are decoupled, in each iteration, need to be updated, and then (22) and (23) can be solved separately. Equations (22) and (23) can be written into more compact forms: (24) (25)where, , , and can be obtained from (22) and (23) directly. The above overdetermined correction (24) and (25) can be solved by WLS, leading to the NEs of GFDSE: (26) (27)where and are the gain matrices of GFDSE; and are the weighted diagonal matrices, see Appendix 3 for their values. Remark.(i) The formulation of GFDSE is based on only one assumption (i.e. the flat start operating conditions), rather than three ones used in conventional FDSE, as a result, GFDSE has good adaptabilities to different systems (including normal running distribution systems with large R/X); (ii) BCMMs can be reliably used by GFDSE, thereby improving the estimation accuracy and more importantly making GFDSE applicable for the DSSE where BCMMs abound; (iii) based on only one assumption, quasi-real power measurements and quasi-reactive power measurements used in GFDSE have better decoupling properties than real and reactive power measurements used in conventional FDSE, thereby making GFDSE have better convergence properties and higher computational efficiency. 3.5 Flowchart of GFDSE The flowchart of the proposed GFDSE can be summarised in Fig. 2, where and are the convergence tolerances. Here we set . Motivated by the Gauss–Seidel algorithm for solving linear equations, in each iteration, the latest angles obtained by solving (26) can be used in solving (27), thereby improving the computational efficiency furtherly. Fig. 2Open in figure viewerPowerPoint Flowchart of the proposed GFDSE 4 Case studies To evaluate the performance of the proposed GFDSE, the IEEE benchmark systems as well as practical power systems, are used here for case studies. For the tests on IEEE systems, the branch parameters are set up firstly (if necessary); then the power flow is calculated; finally, the noise is superimposed on the results of the power flow calculation to simulate the measured value. While for the tests on real power systems, measurements are obtained directly from the SCADA system. The algorithm is coded in JAVA and performed on an Intel(R) Core(TM) i3 PC, 2.40 GHz processor with 2 GB RAM. 4.1 Tests on the IEEE-4 bus system The correctness of the approximations of Jacobian elements in Section 3.2 is important for the formulation of GFDSE. The approximations of Jacobian elements in Section 3.2 of the IEEE-4 bus system are given in Table 1 (denoted by J1). For comparison, the elements of the corresponding Jacobi matrix in the first iteration of WLS are given in Table 2 (denoted by J2). The error rate between J1 and J2 is defined as Table 1. Approximations of Jacobian elements in Section 3.2 of the IEEE-4 bus system Meas. State v1 v2 v3 v4 P1 10.9 −7.1 −3.8 0 −36.2 −10.5 0 P12 7.1 −7.1 0 0 −36.2 0 0 Q1 46.7 −36.2 −10.5 0 7.1 3.8 0 Q12 36.2 −36.2 0 0 7.1 0 0 Table 2. Elements in the corresponding Jacobi matrix in the first iteration of WLS on IEEE-4 bus system Meas. State v1 v2 v3 v4 P1 11.2 −7.3 −3.9 0 −36.6 −10.6 0 P12 7.3 −7.3 0 0 −36.6 0 0 Q1 48.1 −36.7 −10.6 0 7.2 3.9 0 Q12 37.6 −36.7 0 0 7.2 0 0 Test results show that the maximum error rate between J1 and J2 is 3.9%, corresponding to . Such error rates are acceptable, thereby illustrating the correctness of the proposed method. 4.2 Tests on other systems with normal R/X The performance of GFDSE is further tested on IEEE benchmark systems with normal R/X and a China-1170 real power system. Among them, IEEE 43 and IEEE 69 are distribution systems, others are transmission systems. In all tests on the IEEE benchmark systems, measurements including power flow measurements, injection power flow measurements, voltage magnitude measurements and BCMMs are used. The real measurements are created using load flow results with additional small Gaussian noises. The standard deviations of noise are set to be 0.0025 and 0.01 p.u., respectively. Measurements set allocations are listed in Table 3. Test results under different standard deviations of noise are given in Table 3. Table 3. Test systems and measurement sets Systems Numbers Buses/lines vi Pi/Qi Pij/Qij Iij IEEE 9 9/9 9 9/9 12/12 12 IEEE 14 14/20 14 14/14 34/34 34 IEEE 30 30/41 30 30/30 74/74 74 IEEE 39 39/46 39 39/39 68/68 68 IEEE 43 43/42 43 43/43 84/84 84 IEEE 57 57/80 57 57/57 160/160 160 EEE 69 69/68 69 69/69 136/136 136 IEEE 118 118/186 118 118/118 354/354 354 IEEE 300 300/411 300 300/300 608/608 608 China 1170 1170/1751 767 739/739 2854/2854 2854 Bold values indicate distribution systems, non-bold values indicate transmission systems. 4.2.1 Standard deviations of noise are set to be 0.0025 p.u. Firstly, test results are given when the standard deviations of measurement noise are set to be 0.0025 p.u. To judge the convergence property and the optimality of obtained solutions, it needs to observe the variations of the voltage magnitudes and the phase angles, which reflects directly the characteristic of the algorithm. Fig. 3 shows the changes in the variations of voltage magnitudes (maxV-118/300) and the variations of phase angles (maxA-118/300) with iterations for the IEEE 118/300 bus systems. The general requirement for an excellent SE algorithm is that the variations decrease to zero monotonically and rapidly. Fig. 3 certainly verifies that the proposed GFDSE substantiates this claim. Fig. 3Open in figure viewerPowerPoint Variations of voltage magnitudes and phase angles with iterations for IEEE 118/300 bus systems The estimation accuracy and computational efficiency of WLS, FDSE(XB) and the proposed GFDSE without using BCMMs are tested, and then WLS and GFDS