## 4 Results
### Proposed Supplementary Adaptive Droop Controller  
#### 4.2.1 Linear Droop Using ADP  
The APC method with adaptive droop using ADP was implemented in each PV inverter of the twelve-house benchmark. The simulation time step was taken as 100 us, so the time delay for one time and two time-delayed feedback signal of frequency were 100 us and 200 us, respectively. The ADP was implemented using the s-function block of MATLAB/Simulink. The initial weights of the action and critic network of ADP were randomly initialized between \(-0.1\) and \(0.1\). The voltage at the POC and power curtailment were given as inputs in per unit form so that it can be used with any network parameters. The neural networks were then trained offline based on the procedure described in [29].  
As the voltage at the POC is same for the symmetrical houses in every 20 m distance, a single ADP controller was used for each house pair to update the droop values during the training of the ADP. The network is assumed to have a simple communication link between each house pair for exchanging voltage, power curtailment, and droop coefficient values. The ADP controllers are used in odd numbered houses, and the average value of POC voltage and power curtailment of the houses in the house pair are used for training purpose. The adjustment for the droop values is sent to the even numbered houses by its house pair through the same communication link. The training was continued until reduced power curtailment was achieved with the voltage at POC being under allowable limits for all the houses. The proposed method was tested using the final trained weights. The testing phase for the ADP constitutes online learning, set up with a learning rate of  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline \multirow{2}{*}{**Droop Function**} & \multicolumn{2}{c}{**Maximum Power Curtailment (kW)**} & \multicolumn{2}{c}{**Total Energy Curtailed**} \\  & **H7/H8** & **H9/H10** & **H11/H12** & **(kWh)** \\ \hline Linear & 0.44 & 1.66 & 2.17 & 22.72 \\ Quadratic & 0.17 & 1.20 & 1.95 & 15.27 \\ Exponential & 0.11 & 0.95 & 1.92 & 12.40 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Maximum power curtailment and energy curtailed for different droop functions.  
Figure 10: Voltage profile of twelve houses (House 1 to House 12) and curtailed power in APC method with exponential droop.  
0.001. This procedure of reinforcement based learning aids in updating the control action, which leads to improved performance of the controller designed.  
Figure 11 presents the voltage profile of the twelve houses and the power curtailment after using APC method using ADP. The maximum voltage at houses 9 to 12 is 1.057 p.u., which is less than \(V_{cr12}\), and higher than in APC method with linear droop. The maximum power curtailment is 1.25 kW for houses 11 and 12. The maximum curtailments from house 7 to 10 are 1.11 kW. The adjusted droop after the action of supplementary control ADP is shown in Figure 12. The first six houses have constant droop function as calculated in Equation (7). This constant droop is adjusted for houses 9 to 12 by ADP, which leads to lower curtailment while allowing the voltage towards the upper bound limits. The energy output from PV inverters has increased to 470.83 kWh compared to 466.87 kWh in linear droop-based APC and energy curtailment has decreased to 18.77 kWh.  
Figure 11: Voltage profile of twelve houses (House 1 to House 12) and curtailed power in linear droop-based APC method using ADP.  
Figure 12: Droop values of twelve houses (House 1 to House 12) in linear droop-based APC method using ADP. Total droop (\(m_{T}\)) for each house is the sum of constant droop (\(m=2.5\) kW/V) and droop adjustment from ADP (\(m_{ADP}\)).