## 6 Conclusions  
In this paper, we propose a novel resilient cyber-physical demand forecasting system (CDFS) called PRDFS that is resilient to false data injection attacks (FDIAs) since it can distinguish a natural change in the dynamics of demand over time, and a change due to the  
Figure 13: Comparison of demand forecasting performance between PRDFS and BRDFS when no attack is present. Figure 13 shows the demand forecasting accuracy of PRDFS and BRDFS when no attack takes place. The Y-axis of the figure represents the accuracy. The first two bars from the left are for PRDFS and the second two bars are for BRDFS. Blue bars represent the point forecasting accuracy, whereas the orange bars represent the interval forecasting accuracy. The figure shows that PRDFS outperforms BRDFS in both point and interval forecasting accuracy. adversary's attack. To achieve this capability, it uses a signature-based intrusion detection system (SIDS) of malicious nodes and normal nodes. SIDS is trained on labeled training data set using a supervised learning algorithm (GRNN). To automatically generate the training dataset for SIDS with all possible attack paths (defined in the spaces of assessed node trustworthiness and the skewness of residuals of forecasting models for individual sensor time series), an ethical hacking system (EHS) is run. EHS plays repeated attacker-defender games with the defense system (SIDS and TMS) of PRDFS. The training data of SIDS are continuously automatically generated by these games. This is the main novelty of PRDFS. PRDFS uses a trust management system (TMS) to determine the most trustworthy measurement at time step \(t\) from noisy sensor measurements for the aggregate demand time series. It considers the following information while determining the most trustworthy measurement at time step \(t\): spatial data similarity, temporal data similarity, and the minimum expected trustworthiness of the corresponding sensor node at time step \(t\). TMS employs GRNNs to predict the minimum expected trust score of each sensor based on the past trustworthiness scores of its measurements. The demand forecasting system (DFS) of PRDFS uses the lagged values of aggregate demand time series to make out-of-sample forecasts. Our study suggests that among all existing approaches for dealing with the non-stationarity problem, the wavelet multiresolution decomposition approach is the most effective. Our DFS decomposes the time series into multiple component time series using wavelet transform. To avoid the curse of dimensionality, the proposed DFS uses a hierarchical ensemble of convolutional neural networks (CNNs) where each ensemble member is trained with a relatively small number of non-seasonal lag values of the component time series. Each ensemble member is trained sequentially. The first ensemble member is trained on the component time series derived from the original demand time series. Each of the remaining members is trained on the component time series derived from the residual series of the previous ensemble member. The number of ensemble members depends on how many members are required to make the final residual series of the ensemble a white noise series. This is another novelty of the proposed PRDFS. For comparison purposes, we implement a hybrid resilient demand forecasting system that incorporates the best modules of existing demand forecasting systems. We call this resilient demand forecasting system BRDFS. We compare the performance of PRDFS and BRDFS against stealthy FDIAs. The data streams of all sensors are generated synthetically. Both were tested in 11 simulation setups with different percentages of malicious nodes. The performance of our algorithm is highly encouraging. It detects all malicious nodes within the first 30-time steps and it has low false positive rates (FPRs), never exceeding 9% regardless of the percentage of malicious nodes. Its demand forecasting accuracy never drops below 80% in the face of attacks.