# Power System Voltage Correction Scheme Based on Adaptive Horizon Model Predictive Control
## 5 Simulations
### Case A: Nine-Bus Test System  
Simulation on a simple test system is presented in this section to demonstrate the basic idea and implementation procedure of the proposed scheme. The topology of the test system is shown in Figure 6. The available candidate control actions include the shunt capacitors at bus6, the AVR set points of generator at bus2 and bus3, the OLTCs of the transformer between bus9 and bus3. The constraint of control variation in one control step for the shunt capacitors, AVR set points and the OLTCs are 0.1 p.u., 0.04 p.u. and 1.67% respectively.  
Line between bus 4 and bus 6 are tripped at \(t=10\) s. The evolution of voltages at load bus 5, 6 and 8 are illustrated in Figure 7. Due to the loss of the transmission system, the voltage magnitudes drops below the lower bound if no control actions are applied.  
The voltage evolution with the proposed control scheme is shown in Figure 8. The step-like voltage variations are caused by the discrete control feature of the reactive power compensation and voltage regulation devices, which is accordant with the voltage response feature in real-life applications. At the initial time instant \(t_{1}=10\) s, the proposed controller predicts the system trajectory over a predefined prediction horizon \(t_{\mathrm{p1}}=60\) s. The voltage magnitude of load bus 6 and bus 8 fall out of the lower limit 0.95 p.u at the end of the prediction horizon \(t_{1}+t_{\mathrm{p1}}=70\) s, thus the voltage magnitudes of these two buses are determined as the evaluation targets. The horizon parameters are updated at the initial time of each optimization according to the applicability judgment index \(I_{sm}\). The value of \(I_{sm}\) and the applied horizon parameters in this case is listed in Table 1, where \(n\) represents the \(n\)-th optimization.  
Figure 6: Nine-bus test system.  
When \(n=1\) and \(m=1\), the applicability judgment index \(I_{s1}\) is obtain according to Equation (17). As is shown in Table 1, the value of the element \(I_{s(1)1}=-0.00195<0\), thus the proposed controller increases \(m\) and calculate \(I_{s2}\). Since all the elements of \(I_{s2}\) are positive, the control horizon of the current optimization \(t_{c1}\) and the prediction horizon of the next optimization \(t_{p2}\) are both determined as 20 s. The control model of the current optimization is then formulated and solved. Considering the time consuming of communication and calculation, the above procedure is started at 10 s, and the first step of the so-computed control sequence is applied a sampling interval later. When \(n=2\) and \(m=1\), all the elements of \(I_{s1}\) are positive, thus \(t_{c2}=t_{p3}=10\) s. According to the system trajectory prediction implemented during the third optimization, the voltage magnitudes of all the system buses has restored to the desired range and stops declining, the receding-horizon optimization is then completed. The adaptive horizons applied in the simulation is graphically illustrated in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c} \hline \(n\) & \(I_{s1}\) (p.u.) & \(I_{s2}\) (p.u.) & **Control Horizon (s)** & **Prediction Horizon (s)** \\ \hline
1 & (\(-0.0195\), \(0.0128\)) & (0.0298, 0.0483) & 20 & 60 \\
2 & (0.0285, 0.0429) & & 10 & 20 \\ \hline \end{tabular}
\end{table}
Table 1: The applicability judgment index \(I_{sm}\) in case A.  
Figure 8: Voltage evolution using AH-MPC in case A.  
Figure 7: Voltage evolution without control in case A.  
The control actions are listed in Table 2, where \(U_{Inf,Gi}\) represents the AVR set point of generator \(i\), \(C_{i}\) is the applied shunt capacitor at bus \(i\), \(n_{t(i-j)}\) is the ratio of the transformer connecting bus \(i\) and \(j\).