# Power System Voltage Correction Scheme Based on Adaptive Horizon Model Predictive Control
## 3 Adaptive Horizon Parameters
### Voltage Regulation Ability Evaluation  
Under the condition of fulfilling the control variation constraint (10), large control variations are forced to be applied in multi-steps. The limited candidate control actions in one control step have theoretical maximum voltage regulation ability for every bus. The procedure to estimate the ability is summarized as follows:  
1. At \(t_{n}\), the voltage evolution trajectories are predicted within a predetermined prediction horizon [\(t_{n}\), \(t_{n}+t_{pn}\)] based on the trapezoidal numerical simulation. \(t_{pn}\) is the length of prediction horizon applied in the \(n\)-th optimization. If the predicted voltage deviation of load bus \(i\) satisfies the inequality constraint (11) at the initial time instant \(t_{1}\), the voltage magnitude of bus \(i\) is determined as the evaluation target. \[\left|V_{\mathrm{r}}-\hat{V}_{(i)}^{*}(t_{1}+t_{\mathrm{p1}})\right|\geq DB/2\] (11) where \(\hat{V}_{(i)}^{*}(t_{1}+t_{\mathrm{p1}})\) is the voltage magnitude of bus \(i\) without control at the prediction time instant (\(t_{1}+t_{\mathrm{p1}}\)) obtained by numerical simulation. \(DB\) is the control deadband. \(V_{\mathrm{max}}\) and \(V_{\mathrm{min}}\) initially presented in Equation (8) are subject to: \(V_{\mathrm{max}}=V_{\mathrm{r}}+DB/2\), \(V_{\mathrm{min}}=V_{\mathrm{r}}-DB/2\). 2. The trajectory sensitivity matrix of the evaluation targets at time instant \(t_{n}+t_{\mathrm{p}n}\) with respect to the control action of each step can be expressed as: \[ \begin{split} S_{k}=\left[\begin{array}{cccc}S_{k(1,1)}(t_{n}+t_ {\mathrm{p}n})&S_{k(1,2)}(t_{n}+t_{\mathrm{p}n})&\cdots&S_{k(1,N_{j})}(t_{n}+t_ {\mathrm{p}n})\\ S_{k(2,1)}(t_{n}+t_{\mathrm{p}n})&S_{k(2,2)}(t_{n}+t_{\mathrm{p}n})&\cdots&S_{k (2,N_{j})}(t_{n}+t_{\mathrm{p}n})\\ \vdots&\vdots&&\vdots\\ S_{k(N_{i},1)}(t_{n}+t_{\mathrm{p}n})&S_{k(N_{i},2)}(t_{n}+t_{\mathrm{p}n})& \cdots&S_{k(N_{i},N_{j})}(t_{n}+t_{\mathrm{p}n})\end{array}\right]\end{split} \] (12) where \(S_{k(i,j)}\) represents the trajectory sensitivity of the \(i\)-th evaluation target with respect to the \(j\)-th candidate control in the \(k\)-th control step. \(N_{i}\) and \(N_{j}\) are the total number of evaluation targets and available control actions respectively. 3. Based on the voltage prediction and trajectory sensitivity analysis, the maximum control target variation caused by controls applied at sampling point \(k\) is evaluated as follows. When \(S_{k(i,j)}\cdot(V_{\mathrm{r}}-\hat{V}_{(i)}^{*}(t_{n}+t_{\mathrm{p}n}))\geq 0\), the optimization tends to increase the \(j\)-th control in order to raise (when \(S_{k(i,j)}\geq 0\) and \(V_{\mathrm{r}}\geq\hat{V}_{(i)}^{*}(t_{n}+t_{\mathrm{p}n})\)) or reduce (when \(S_{k(i,j)}\leq 0\) and \(V_{\mathrm{r}}\leq\hat{V}_{(i)}^{*}(t_{n}+t_{\mathrm{p}n})\)) the voltage magnitude of bus \(i\) towards its desired value. Under the inequality constraints (9) and (10), the predicted voltage magnitude variation of bus \(i\) reaches its maximum value when the variation of control \(j\) equals its maximum positive value: \[\Delta V_{k(i,j)\mathrm{max}}(t_{n}+t_{\mathrm{p}n})=\left\{ \begin{array}{ ccc}S_{k(i,j)}(t_{n}+t_{\mathrm{p}n})\cdot\Delta u_{(j)\mathrm{max}}&if&u_{k(j)}+ \Delta u_{(j)\mathrm{max}}\leq u_{(j)\mathrm{max}}\\ S_{k(i,j)}(t_{n}+t_{\mathrm{p}n})\cdot(u_{(j)\mathrm{max}}-u_{k(j)})&if&u_{k(j) }+\Delta u_{(j)\mathrm{max}}\geq u_{(j)\mathrm{max}}\end{array} \right.\] (13) where \(\Delta V_{k(i,j)\mathrm{max}}(t_{n}+t_{\mathrm{p}n})\) is the predicted maximum variation of the evaluation target \(i\) at a future time instant \(t_{n}+t_{\mathrm{p}n}\), which is caused by the \(j\)-th control action applied at the sampling point \(k\). \(\Delta u_{(j)\mathrm{max}}\) is the \(j\)-th element in vector \(\Delta u_{max}\), representing the maximum value of the corresponding control variation. \(u_{(j)\mathrm{max}}\) is the \(j\)-th element in vector \(u_{\mathrm{max}}\).