## 3 Principles of the Scheme
### Adaptive Secondary Voltage Control (ASVC) Function  
First, consider that the approximate model of the small disturbance voltage-var control is represented by  
\[\left[ \begin{array}{c}\mathbf{\Delta Q_{G}}\\ \mathbf{\Delta Q_{L}}\end{array} \right]=\left[ \begin{array}{cc}B_{GG}&B_{ GL}\\ B_{LG}&B_{LL}\end{array} \right]\left[ \begin{array}{c}\mathbf{\Delta}\left| \mathbf{V_{G}}\right|\\ \mathbf{\Delta}\left|\mathbf{V_{L}}\right|\end{array} \right] \tag{1}\]  
where \(\mathbf{\Delta}\)\(Q\) and \(\mathbf{\Delta}\left|\mathbf{V}\right|\) are the reactive power and voltage magnitude change vectors; \(B\) stands for the system susceptance matrix; and the subscripts \(L\) and \(G\) denote the load and the voltage-controlled buses, respectively. In the above matrix equation, load voltage changes \(\mathbf{\Delta}\left|\mathbf{V}\right|\) can be expressed as  
\[\mathbf{\Delta}\left|\mathbf{V_{L}}\right|=J_{1}\mathbf{q}-J_{2}\mathbf{u} \tag{2}\]  
where  
\[J_{1}=B_{LL}^{-1} \tag{3}\] \[J_{2}=J_{1}B_{LG}\]  
\[\mathbf{q}=\mathbf{\Delta Q_{L}} \tag{4}\] \[\mathbf{u}=\mathbf{\Delta}\left|\mathbf{V_{G}}\right|.\]  
Notice that \(\mathbf{q}\) and \(\mathbf{u}\) are considered as reactive power load disturbances and control variables, respectively. In this study, the feedback control law is used to develop the secondary voltage controller. The control vector is determined by using the worst-case design, the technique which addresses the problem by applying the minimization of the maximum load voltage deviation as an objective function for the optimal control model. Mathematically, the problem is to search an optimal set of control solutions such that the \(l_{\infty}\) norm of load voltage changes is minimized. Suppose that PMUs are placed at the chosen pilot nodes; the voltage deviations at the observed buses can be elaborated by  
\[\mathbf{\Delta}\left|\mathbf{V_{p}}\right|=\left|\mathbf{V_{p}^{*}}\right|- \left|\mathbf{V_{p}}\right| \tag{5}\]  
where \(\left|\mathbf{V_{p}^{*}}\right|\) and \(\left|\mathbf{V_{p}}\right|\) denote the pilot-node set-point voltages determined by tertiary level and pilot-node measured voltages obtained from installed PMUs, respectively. In the proposed secondary voltage control, both voltage profile and static voltage stability margin are adopted to serve as trigger signals, as shown in Figure 3. It utilizes pilot-bus voltage changes \(\Delta\left|\mathbf{V_{p}}\right|\) as input signals, and generates control actions \(\mathbf{u}\) as output signals. Based on different numbers of pilot buses being used, the following conditions are considered. (1) All Load Buses as Pilot Buses: This means that voltage measurements are available at all load buses. Under this circumstance, the first term \(J_{1}\mathbf{q}\) in Equation (2) can be replaced by \(\Delta\left|\mathbf{V_{p}}\right|\); therefore, the problem formulation of the voltage control strategy is stated as  
\[ \begin{split}\underset{\mathbf{u}}{\min}\left\|\,\Delta\left| \mathbf{V_{p}}\right|-J_{2}\mathbf{u}\,\right\|_{\infty}\\ \text{\emph{subject to}}\,\left|\mathbf{V_{G}^{\min}}\right| \leqslant\left|\mathbf{V_{G}}\right|\leqslant\left|\mathbf{V_{G}^{\max}}\right| \end{split}  \tag{6}\]  
where \(\left|\mathbf{V_{G}^{\min}}\right|\) and \(\left|\mathbf{V_{G}^{\max}}\right|\) are the lower and upper bounds of generator voltage magnitudes, respectively. (2) Partial Load Buses as Pilot Buses: This means that voltage measurements are only available at the given pilot buses, and thus \(\Delta\left|\mathbf{V_{p}}\right|\) only has some elements of the vector \(J_{1}\mathbf{q}\) under this situation. Indeed, \(\Delta\left|\mathbf{V_{p}}\right|\) can be related to \(\mathbf{q}\) by the equation \(J_{p}\mathbf{q}=\Delta\left|\mathbf{V_{p}}\right|\), in which \(J_{p}\) is the matrix with the rows of \(J_{1}\) corresponding to the pilot points selected. Since there are fewer measurements than variables to be estimated, the equation \(J_{p}\mathbf{q}=\Delta\left|\mathbf{V_{p}}\right|\) is underdetermined.