Weighted Error Modulus (WEM) is a novel iteratively re-weighting approach introduced in this section as a potential DSSE tool. Although the WLS method can be applied successfully to detect and remove outliers, the method is not always efficient in detecting and overcoming the effect of gross measurement errors or bad data. In the presence of gross errors, an alternative optimization criteria known as Weighted Least Absolute Values (WLAV) is more effective. WLAV optimizes the absolute value of the residual vector instead of the quadratic value as in the WLS method [6]. A novel estimator, Weighted Error Modulus (WEM) method that exploits the benefits of both WLS and WLAV methods, is proposed as a candidate DSSE tool. In this approach, the weighting value associated with the measurement is modified iteratively within the WLS method.
In addition to the low level measurement errors, linearization errors from the Taylor series approximation of the optimization equation for and unexpected gross error can also be present in the measurement data. Assuming linearization and other unexpected errors are termed as , the Gauss-Newton solution of WLS optimization function can be expressed as:
Here, – = vector of residuals. According to the Gauss-Newton principle, is negligible provided that the initial guess of the state variable approximates the true value. The measurement is usually within a ±1% error margin with respect to the true value and therefore the WLS method provides good estimation under normal conditions. This implies that the residual vector always has a considerably smaller value. However, this will not remain true if any gross error exists in the measurements. In that case, the assumption H ∆x ≈ r remains no longer valid since in (3.58) would have a significant value. The proposed WEM method utilizes the characteristics of the variation in depending on the accuracy of the measurement to re-weight the measurements. The principle of the algorithm is that
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the measurement data corresponding to a larger residual eventually has reduced weight and the measurement with smaller residual has a gradually increased weighting factor. Hence,
is modified iteratively such that for kth iteration
Here is the measurement re-weighting factor that corresponds to the actual weighting factor, at the first iteration. As the recursion process approaches a convergence point, becomes trivial and . Consequently,
i becomes negligible,
– – and therefore, | |k |
|k+1 at that point. That also implies,
i i
By replacing the value from (3.60) in (3.59), is obtained. The WLS minimization problem can therefore be expressed as follows:
Equation (3.61) represents the final objective function that resembles the WLAV minimization criteria. In such a way, the WEM commences the process with the WLS objective functions while the weighting factor is being recurrently updated. At near convergence point, (3.60) occurs and the minimization function is transformed to WLAV objective function. Essentially, the WEM method attempts to reduce low level measurement errors by the WLS method and gross errors by the WLAV method. Hence, WEM method combines the advantages of the objective functions of both the WLS and the WLAV methods [53]. The entire process is depicted in Fig. 3. 5.
.
There are two nested iteration loops in this algorithm: one outer and one inner iteration as shown in Fig. 3. 5. After accomplishment of each set of inner loop iterations, the elements of the weighting matrix are replaced by the most recent corresponding re-weighting factors. The outer loop controls Gauss-Newton recursion while the inner loop performs the re-weighting, gain matrix and mismatch calculations. The inner loop is assumed to be
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converged when the mismatch vectors of two consecutive inner loop iterations are almost equivalent; while the outer loop's convergence criteria is set as mismatch values approximating zero. The inner loop iteration is capped at a smaller number of iteration (that is five in this thesis) and allowed to terminate even if full convergence is not achieved. The outer loop controls whether the algorithm is satisfying the Gauss-Newton convergence criteria and should terminate.
Fig. 3. 5: WEM method flowchart
The WEM method can be compared with the iteratively re-weighted method known as Schweppe-Huber Generalized-M (SHGM) estimation. The SHGM is developed as a robust estimator to suppress bad data and avoid the influence of the leverage point. It utilizes two different objective functions representing WLS and WLAV criteria. The value of residual remaining above or below certain threshold value determines whether the estimator will behave like a WLAV or WLS estimator respectively. The threshold value is set by the weighting factor and a tuning parameter [1] [17]. The proposed WEM method is similar to the SHGM estimator in a way, since both methods combine the benefits of WLAV and WLS estimators. However, unlike SHGM, WEM does not need to define two different objective functions. The process starts with the WLS objective function, which is gradually transformed to the WLAV equivalent optimization criteria inherently (through the re- weighting procedures). There is also no need to define any tuning parameter for WEM.
start
Calculate state variables for kthiteration, Vk,θk, Pk, Qk
Calculate gain matrix and
mismatches, Δxkjfor jthiteration
Update state variables, Vk+1,θk+1, Pk+1, Qk+1
Converge : Δxkj≈ Δxk(j-1)Or j=5 ? Converge: Δxk≤ 1e-6 ? Calculate Re-weighting factor, ukj = Wk/|HkΔxkj- rk| end k=k+1 j=j+1 NO NO YES YES Wk+1= ukj
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3.7 Concluding Remarks and Discussion
The sensors distributed over the network to read and transmit the operating state information are prone to telemetry failure and erroneous information. Furthermore, it is not practically possible to achieve perfectly simultaneous measurement from all parts of the network. The state estimation tool uses a set measurements to calculate the system state. This estimated data can then be fed into DMS functions [1] [2]. An initial discussion of typical power system state estimation tools has been included in this chapter. The SE building blocks, network components and critical assumptions used for SE have been defined. Important mathematical formulations such as measurement equations, normal equation, and Jacobian matrix are identified and discussed in detail. These formulations are used while implementing DSSE in MATLAB. The chapter excludes current and phase angle measurement equations as the current and phase angle measurements are not considered in this work.
Furthermore, five different SE solution processes are discussed as candidate DSSE tools. In addition to normal equation based classical WLS, alternative solution processes are considered, which include orthogonal factorization, constrained WLS and Hachtel's Augmented Matrix Method. A novel WEM optimization method is introduced as a candidate solution in this chapter.
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CHAPTER 4
SELECTION AND APPLICATION OF HACHTEL'S
AUGMENTED MATRIX METHOD
The performance of five candidate DSSE solutions discussed previously in Chapter 3 is assessed through various case studies in order to select one of these as the most potentially useful DSSE tool in this chapter. The case studies are performed on simulated datasets and model networks that represent general features of UK distribution networks. A detail discussion is provided, leading to the selection of Hachtel's Augmented Matrix method, based on the outcomes of the case studies. One of the important contributions in this chapter is the demonstration of the performance of Hachtel's method on the real networks and authentic datasets in various off line case studies.