Potential DSSE Tool
In section 3.5, 3.6 and 4.1, five SE solution processes which are classical WLS, CWLS, Orthogonal (QR) Decomposition, Hachtel's Augmented Matrix and WEM methods, are described as candidate DSSE tools. In this section their hypothetical values and performance on test cases are analysed to select one as the most potentially useful method to be used for distribution SE. The selection will be based on the computation time, convergence property, robustness of the process and most importantly, the quality of estimation they can provide.
Classical WLS is a popular and widely used optimizer in power system SE problems especially for transmission systems, due to its excellent performance in removing errors
WLS WEM CWLS QR HACHTEL'S
Max est. error |θ| 4.11 8905.08 4.11 4.11 4.11
Mean est. error |θ| 1.14 117.20 1.14 1.14 1.14
0.00 2000.00 4000.00 6000.00 8000.00 10000.00 % E rr o rs
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generated from low level noises. However, the application of WLS is more challenging at distribution levels where significant numbers of pseudo and virtual measurements may cause deterioration of the gain matrix condition number. One of the major sources of matrix ill-conditioning is the high weighting factors assigned to virtual measurements in the normal equation based SE solution strategy. The matrix in CWLS and Hachtel's Augmented method do not contain such significant values, as virtual measurements are considered as equality constraints, is in a separate equation in these methods. Furthermore, the Hachtel's method does not form any normal equation or gain matrix at all and the equality equation is never squared in the solution equation for CWLS and Hachtel's methods. Hence the CWLS and Hachtel's methods acquire robustness with regard to the ill-conditioning problem due to measurement weights. However any possibility of the presence of bad data in the enforced constraint may leave catastrophic effects on convergence and estimation quality [49], but that is usually an unlikely event to occur.
The Orthogonal Decomposition and the Hachtel's methods can be considered as the most robust methods, since they both do not form a gain matrix or normal equation. Studies under various scenarios give strong evidence in support of the robustness of both methods. In addition to consistency in producing good quality estimated data, the mean iteration values under different scenarios and networks appear not to be affected significantly. However the method benefits from being strongly well conditioned by default. The factorization is mathematically more robust that than factorization as stated in [1]. A few drawbacks of this method include lower sparsity than the gain matrix, memory storage requirement [50] and computation time.
As Hachtel's method tactically avoids calculation, it gives somewhat better conditioning than CWLS method. The Hachtel's method tends to be numerically more stable and is theoretically expected to generate less erroneous solutions than constrained normal equation e.g. CWLS [50]. While Orthogonal Decomposition has been proved to be the most robust but a computationally costly SE solution process, CWLS is more economic in computation time and preserves good conditioning. The Hachtel method can be considered as a compromise between Orthogonal factorization and CWLS, as it provides better robustness than CWLS and faster computation than the QR method [50] [52]. While avoiding formation of complete normal equations, the coefficient matrix of Hachtel's and CWLS remain no longer positive definite, therefore they require more sophisticated ordering and factorization. This can be treated as a trivial problem to be concerned about
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since advanced software technology, such as MATLAB, can execute such complex computation efficiently.
Significant errors in estimated values are obtained from the WEM estimation tool in a few case studies, which indicates that the iteratively reweighted least squares based methods may not be efficiently applied to DSSE where limited real measurements are available. WEM tends to adjust the weight with magnitudes of relevant measurement residual values to attribute greater emphasis to coherent measurement data. The method, therefore, instead of treating all pseudo-measurements equally, prefers a few of them to gain more weight as the solution approaches convergence. The reweighting factors are expected to elect those pseudo-measurements which are closer representation of real states- which is not happening as observed in case studies. The iteratively weight assignment is possibly leading the WEM estimator to trust somewhat more erroneous measurements, deteriorating estimation quality. The error becomes quite large in the cases of impedance data errors and existence of very short branches. One possible reason of such unexpected performance can be attributed to the assumption of Gaussian error distribution. At near convergence point, the WEM method starts behaving more like the WLAV estimator, which is based on the maximum log-likelihood of the Laplace error density function. Therefore, the Gaussian error assumption no longer remains consistent with the objective function as the process approaches to the convergence [17]. Hence, the principle of the WEM method is not performing at the expected level in the case studies.
The quality of estimation is very much similar for the least squares error minimization based algorithms i.e. WLS, CLWS, QR and Hachtel's methods. The calculation time and iteration requirements vary, as the solution processes are different. The convergence characteristics based on required iteration numbers are always better in the case of Orthogonal Decomposition and Hachtel's methods, even in presence of various errors and specific conditions of the network. Their operation time however is longer for the same number of iterations when compared with that of WLS and CWLS. Hachtel's and Orthogonal methods can still be considered to have better convergence properties, as the convergence of WLS and CWLS is delayed at some operational states. The relative convergence characteristics are shown in Table 4. 1 to Table 4. 5.
Considering the pros and cons of five candidate solutions, WEM can be excluded to be considered as a potential DSSE tool, deeming it's lower estimation quality and higher
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execution time. The Orthogonal method is very robust, however its quite high computation time may turn out infeasible to apply to large networks. Completely normal and semi- normal equation based WLS and CLWS methods respectively suffer from ill-conditioning problems and increased iteration requirements when applied to larger (356 node) networks with erroneous DSSE inputs. This may raise strongly the problem of scalability. Hachtel's Augmented Matrix method takes longer estimation time than WLS and CWLS and shorter than WEM or Orthogonal Decomposition methods for the same number of iterations. However, the average number of iterations it requires remain steadily within the range of 3 to 3.5 for various network states and sizes. The quality of estimation is consistent and very similar for WLS, CWLS, QR Decomposition and Hachtel's methods.
The critical analysis confirms that Hachtel's Augmented Matrix method can provide most robust and consistent outcome while maintaining acceptable estimation quality and convergence speed in various scenarios as well as network sizes. Therefore, it is proposed as the most useful potential tool for DSSE.