## 3 Synthetic Measurement Enhanced Parameter Error Correction
### Overdetermined Parameter FDI Correction Model  
Consider the conjugate of the complex power flow [33]:  
\[ \begin{split} S^{*}_{k-m}&=E^{*}_{k}k_{k-m}\\ &=y_{k-m}V_{k}e^{-j\theta_{k}}(V_{k}e^{j\theta_{k}}-V_{m}e^{j \theta_{m}})+jb^{sh}_{km}V^{2}_{k}.\end{split}  \tag{9}\]  
The expressions for real and reactive power flows can be obtained by identifying the corresponding coefficients of the real and the imaginary parts of (9):  
\[ \begin{split} P_{k-m}=V^{2}_{k}g_{km}&-V_{k}V_{m}g _{km}cos\theta_{km}\\ &-V_{k}V_{m}b_{km}sin\theta_{km},\end{split}  \tag{10}\]  
\[ \begin{split} Q_{k-m}=-V^{2}_{k}(b_{km}+b^{sh}_{km})&+ V_{k}V_{m}b_{km}cos\theta_{km}\\ &-V_{k}V_{m}g_{km}sin\theta_{km}.\end{split}  \tag{11}\]  
Through equation (10), one can derive the real power loss of a line:  
\[ \begin{split} P_{k-m(loss)}&=P_{k-m}+P_{m-k}\\ &=g_{km}(V^{2}_{k}+V^{2}_{m}-2V_{k}V_{m}cos\theta_{km})\\ &=g_{km}|E_{k}-E_{m}|^{2},\end{split}  \tag{12}\]  
Through equation (11), one can derive the reactive power loss of a line:  
\[ \begin{split} Q_{k-m(loss)}&=Q_{k-m}+Q_{m-k}\\ &=-b^{sh}_{km}(V^{2}_{k}+V^{2}_{m})\\ &-b_{km}(V^{2}_{k}+V^{2}_{m}-2V_{k}V_{m}cos\theta_{km})\\ &=-b^{sh}_{km}(V^{2}_{k}+V^{2}_{m})-b_{km}|E_{k}-E_{m}|^{2}.\end{split}  \tag{13}\]  
Equations (10)-(13) provide a model which correlates the real power flow losses, reactive power losses, real power flow, and reactive power flow with system net parameters, as (14) and (15). It is important to note that in this framework, all six of the measurements in (14) are always used in the correction process, but they may not all be real measurements. Any measurements that are not available through a sensor are calculated as SM as described earlier. For example, if the \(P_{k-m}\), and \(Q_{k-m}\) are the only true measurements, \(P_{m-k}\), and \(Q_{m-k}\) will be SM. \(P_{k-m(loss)}\) will be calculated by \(P_{k-m}\) and \(P_{m-k}\), \(Q_{k-m(loss)}\) will be obtained by \(Q_{k-m}\) and \(Q_{m-k}\). \[\tau_{up} \begin{pmatrix}g_{km}\\ b_{km}\\ b^{sh}_{km}\end{pmatrix} = \begin{pmatrix}P_{k-m(loss)}\\ Q_{k-m(loss)}\\ P_{k-m}\\ P_{m-k}\\ Q_{k-m}\\ \end{pmatrix} , \tag{14}\]where \(\tau_{up}\) is defined as:  
\[\tau_{up}=\left( \begin{array}{cccc}|E_{k}-E_{m}|^{2}&0&0\\ 0&-|E_{k}-E_{m}|^{2}&-(V_{k}^{2}+V_{m}^{2})\\ V_{k}^{2}-V_{k}V_{m}cos\theta_{km}&-V_{k}V_{m}sin\theta_{km}&0\\ V_{m}^{2}-V_{k}V_{m}cos\theta_{km}&V_{k}V_{m}sin\theta_{km}&0\\ -V_{k}V_{m}sin\theta_{km}&-V_{k}^{2}+V_{k}V_{m}cos\theta_{km}&-V_{k}^{2}\\ V_{k}V_{m}sin\theta_{km}&-V_{m}^{2}+V_{k}V_{m}cos\theta_{km}&-V_{m}^{2}\\ \end{array} \right). \tag{15}\]  
By linearizing (14) through a Taylor series, considering a Newton-Raphson method at \(n^{th}\) iteration:  
\[\tau_{up_{n}} \begin{pmatrix}\Delta g_{km}\\ \Delta b_{km}\\ \Delta b_{km}^{sk}\end{pmatrix} =A, \tag{16}\]  
where \(A\) is the residual of the set of measurements associated with the correction of line parameters as follows:  
\[A= \begin{pmatrix}Z_{R_{k-m(loss)}}-H_{R_{k-m(loss)}}^{n}\\ Z_{Q_{k-m(loss)}}-H_{Q_{k-m(loss)}}^{n}\\ Z_{R_{k-m}}-H_{R_{k-m}}^{n}\\ Z_{Q_{k-m}}-H_{Q_{k-m}}^{n}\\ Z_{Q_{k-m}}-H_{Q_{k-m}}^{n}\\ Z_{Q_{k-m}}-H_{Q_{k-m}}^{n}\\ \end{pmatrix} . \tag{17}\]  
The expression in (15) shows that augmenting \(\tau_{up}\) and the measurements set, will change the model in (8) into an overdetermined system of nonlinear algebraic equations. Model (16) can be solved considering the minimization problem:  
\[\min_{\Delta g_{km},\Delta b_{km},\Delta b_{km}^{sk}}\left[\tau_{up} \begin{pmatrix} \Delta g_{km}\\ \Delta b_{km}\\ \Delta b_{km}^{sk}\end{pmatrix} -A\right]^{T}W_{p}[\tau_{up} \begin{pmatrix} \Delta g_{km}\\ \Delta b_{km}\\ \Delta b_{km}^{sk}\end{pmatrix} -A]. \tag{18}\]  
One finds the classical WLS SE solution for (18) is:  
\[ \begin{pmatrix}\Delta g_{km}\\ \Delta b_{km}\\ \Delta b_{km}^{sk}\end{pmatrix}