The SCADA system for the power network shown in Figure 5.3 reports the following measurements and variances:

Question

Variance (s²)	Measurement	State	[latex]z_{i} [/latex]
0.010	0.975	[latex]V_{3} [/latex]	1
0.050	0.668	[latex]P_{13} [/latex]	2
0.075	−0.082	[latex]Q_{21} [/latex]	3
0.050	−1.181	[latex]P_{3} [/latex]	4
0.075	−0.086	[latex]Q_{2} [/latex]	5

Estimate the power system states.

Accepted Answer

The first step in the estimation process is to identify and enumerate the unknown states. In this example, the unknowns are [latex]\left[\begin{matrix}x_{1} & x_{2} & x_{3}\end{matrix} ight] ^{T} =\left[\begin{matrix}\delta _{2} & \delta _{3} & V_{3}\end{matrix} ight] ^{T} [/latex] . After the states are identified, the next step in the estimation process is to identify the appropriate functions h(x) that correspond to each of the measurements. The nonlinear function that is being driven to zero to minimize the weighted error is

[latex]F(x)=H^{T}_{x} R^{-1} \left[z-h(x) ight] =0 [/latex] (5.106)

where the set of [latex]z-h(x)[/latex] is given by

[latex]z_{1} -h_{1} (x)=V_{3} -x_{3}[/latex]

[latex]z_{2} -h_{2} (x)=P_{13} -(V_{1}x_{3} Y_{12} \cos (-x_{2}-\phi _{13} )-V^{2}_{1} Y_{13} \cos \phi _{13} )[/latex]

[latex]z_{3} -h_{3} (x)=Q_{21} -(V_{2}V_{1} Y_{21} \sin (x_{1}-\phi _{21} )+V^{2}_{2} Y_{21}\sin \phi _{21} )[/latex]

[latex]z_{4} -h_{4} (x)=P_{3} -(x_{3}V_{1} Y_{31} \cos (x_{2}-\phi _{31} )+x_{3} V_{2} Y_{32}\cos(x_{2}-x_{1}-\phi _{32} ) +x^{2}_{3} Y_{33}\cos \phi _{33} )[/latex]

[latex]z_{4} -h_{5} (x)=Q_{2} -\left[V_{2}V_{1} Y_{21} \sin (x_{1}-\phi _{21} )- V^{2} _{2} Y_{22}\sin \phi _{22} +V_{2}x_{3} Y_{23}\sin (x_{1} -x_{2} - \phi _{23} ) ight] [/latex]

and the matrix of partial derivatives for the set of functions in Equation 5.106 is

[latex]H_{x} =\left[\begin{matrix}{0} \ 0 \V_{1}V_{2} Y_{21}\cos (x_{1} -\phi _{21} )\x_{3}V_{2} Y_{21}\sin (x_{2} -x\phi _{32} ) \V_{1}V_{2} Y_{21}\cos (x_{1} -\phi _{21} )+V_{2}x_{3} Y_{23}\cos (x_{1}-x_{2} -\phi _{23} )\end{matrix} \begin{matrix} 0 \V_{1}x_{3} Y_{32}\sin (-x_{2} -\phi _{13} )\0 \-x_{3}V_{1} Y_{31}\sin (x_{2} -\phi _{31} )-x_{3}V_{2} Y_{32}\sin (x_{2}-x_{1} -\phi _{32} )\ -V_{2}x_{3} Y_{23}\cos (x_{1} -x_{2} -\phi _{23} )\end{matrix} ight] [/latex]

[latex] imes \left[\begin{matrix}{1} \ V_{2} Y_{13}\cos (-x_{2} -\phi _{13} )\0 \V_{1} Y_{31}\cos (x_{2} -\phi _{31} )+V_{2} Y_{32}\cos (x_{2}-x_{1} -\phi _{32} )+2x_{3}Y_{33}\cos \phi _{33} \ V_{2} Y_{23}\sin (x_{1} -x_{2} -\phi _{23} ) \end{matrix} ight][/latex] (5.107)

The covariance matrix of the measurements is

[latex]R=\left[ \begin{matrix} \frac{1}{0.010^{2} } & & \ &\frac{1}{0.050^{2} } & \ & & \frac{1}{0.075^{2} }\&&&\frac{1}{0.050^{2} } \&&&&\frac{1}{0.075^{2} }\end{matrix} ight] [/latex] (5.108)

The Newton–Raphson iteration to solve for the set of states x that minimize the weighted errors is

[latex]\left[H^{T}_{x} (x^{k} )R^{-1}H_{x} (x^{k} ) ight] \left[x^{k-1} -x^{k} ight] =H^{T}_{x} (x^{k} )R^{-1} \left[z-h(x^{k} ) ight] [/latex] (5.109)

Iteration 1
The initial condition for the state estimation solution is the same flat start as for the power flow equations; namely, all angles are set to zero and all unknown voltage magnitudes are set to unity. The measurement functions h(x) evaluated at the initial conditions are

[latex]h(x^{0} )=\left[\begin{matrix}1.0000 \ 0.0202 \ -0.0664\-0.0198\-0.1914 \end{matrix} ight][/latex]

The matrix of partials evaluated at the initial condition yields

[latex]H^{0}_{x} =\left[\begin{matrix}0 &0 & 1.0000\ 0 & -10.0990 & -1.0099 \-0.2257 & 0 & 0\-9.9010&20.0000&1.9604\-1.2158&0.9901&-9.9010 \end{matrix} ight] [/latex]

The nonlinear functions in Equation 5.106 are

[latex]F(x^{0} )=\left[\begin{matrix}0.5655 \-1.4805 \ -0.2250 \end{matrix} ight][/latex]

The incremental updates for the states are

[latex]F(x^{1} )=\left[\begin{matrix}0 0113 \-0.0258 \0 0091 \end{matrix} ight][/latex]

The incremental updates for the states are

[latex]\Delta (x^{1} )=\left[\begin{matrix}-0.0119 \-0.0625 \ -0.0154 \end{matrix} ight][/latex]

leading to the updated states

[latex]\left[\begin{matrix}\delta ^{1 }_{2} \\delta ^{1 }_{3} \V ^{1 }_{3} \end{matrix} ight] =\left[\begin{matrix}-0.0119 \-0.0625 \ -0.9846 \end{matrix} ight][/latex]

where [latex]\delta _{2}[/latex] and [latex]\delta _{3}[/latex] are in radians. The error at iteration 0 is

[latex]\varepsilon ^{0}=1.4805[/latex]

Iteration 2
The updated values are used to recalculate the Newton–Raphson iterations:

[latex]h(x^{1} )=\left[\begin{matrix}0.9846 \ 0.6585 \ -0.0634\-1.1599\-0.724 \end{matrix} ight][/latex]

The matrix of partials is

[latex]H^{1}_{x} =\left[\begin{matrix}0 &0 & 1.0000\ 0 & -9.9858 & -0.3774 \-0.2660 & 0 & 0\-9.6864&19.5480&0.7715\-0.7468&0.4809&-9.9384 \end{matrix} ight][/latex]

The nonlinear function evaluated at the updated values yields

[latex]F (x^{1} )=\left[\begin{matrix}0. 0113 \-0.0258\0. 0091 \end{matrix} ight][/latex]

he incremental updates for the states are

[latex]\Delta (x^{2} )=\left[\begin{matrix}0. 0007 \-0.0008 \0. 0013 \end{matrix} ight][/latex]

leading to the updated states

[latex]\left[\begin{matrix}\delta ^{2 }_{2} \\delta ^{2}_{3} \V ^{2 }_{3} \end{matrix} ight] =\left[\begin{matrix}-0.0113\-0.0633 \ 0.9858 \end{matrix} ight][/latex]

The error at Iteration 1 is

[latex]\varepsilon ^{1}=0.0258[/latex]

The iterations are obviously converging. At convergence, the states that minimize the weighted measurement errors are

[latex]x=\left[\begin{matrix}-0.0113 \-0.0633 \0. 9858 \end{matrix} ight][/latex]

Question 5.9: The SCADA system for the power network shown in Figure 5.3 r...

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