## 3 Our Proposed Work
### Forecast Module
\begin{split} Minimize\ \ \ e(k)=y(k)-\hat{y}(k),\\ \forall k\in\{1,2,3,\ldots,m\}\end{split}  \tag{13}\]  
We use the method of least squares to determine the parameter matrices, which is given as follows,  
\[ \begin{split} Minimize\ \ \ J=\sum_{k=1}^{m}e^{T}(k)e(k),\\ \forall k\in\{1,2,3,\ldots,m\}\end{split}  \tag{14}\]  
Subject to the most feasible solution of Equation (14), we use the multi-variate auto-regressive model presented in [17], because it solves the objective function in relatively less time with reasonable accuracy, as compared to the typically used learning rules, like gradient descent, Widrow-Hoff and delta [18]. According to [17], the parameter matrices are given as follows,  
\[\sum_{i=1}^{n}W(i)R(j-i)=0,\ \ \ j=\{2,3,\ldots,n\} \tag{15}\]  
\[\sum_{i=1}^{n}\overline{W}(i)R(i-j)=0,\ \ \ j=\{2,3,\ldots,n\} \tag{16}\]  
where \(W(1)=I_{D}\) (\(I_{D}\) is the identity matrix), \(\overline{W}(1)=I_{D}\) and \(R\) is the cross co-relation given as:  
\[R(i)=\frac{1}{n}\sum_{k=i}^{n-1-i}[x(k)-m][x(k-i)-m]^{T} \tag{17}\]  
Figure 4: An artificial neuron. In Equation (11), \(m\) is the mean vector of the observed data,  
\[m=\frac{1}{n}\sum_{k=i}^{n}x(k) \tag{18}\]  
Based on these equations, [17] defines the following prediction error co-variance matrices. \[\left. \begin{array}{l}V_{t}=\sum_{k=1}^{n}W_{t}(k)R(-k)\\ \overline{V}_{t}=\sum_{k=1}^{n}\overline{W}_{t}(k)R(-k)\\ \Delta_{t}=\sum_{k=1}^{n}W_{t}(-k)R(t-k+1)\\ \overline{\Delta}_{t}=\sum_{k=1}^{n}\overline{W}_{t}(k)R(-t+k-1)\end{array} \right\} \tag{19}\]  
The recursive equations are as follows:  
\[\left. \begin{array}{l}W_{t+1}(k)=W_{t}(k)W_{t+1}(t+1)\overline{W}_{t}(t-k+1) \\ \overline{W}_{t+1}(k)=\overline{W}_{t}(k)\overline{W}_{t+1}(t+1)\overline{W}_ {t}(t-k+1)\end{array} \right\} \tag{20}\]  
\[\left. \begin{array}{l}W_{t+1}(t+1)=-\Delta_{t}\overline{V}_{t}^{-1}\\ \overline{W}_{t+1}(t+1)=-\overline{\Delta}_{t}V_{t}^{-1}\end{array} \right\} \tag{21}\]  
In order to find the weights, Equations (20) and (21) are solved recursively. For further details about the weight update mechanism, Equations (15)-(21), readers are suggested to read [17]. Figure 5 is a pictorial representation of the steps involved in the data forecast module. Once the weights in Equations (20) and (21) are recursively adjusted as per the objective function in Equation (13), the output matrix is then binary decoded and de-normalized to get the desired load time series. The stepwise algorithm of the proposed methodology is shown in Algorithm 1. Figure 5: Data forecast module. ```
1:Pre-conditions:\(i=\)number of days, and \(j=\) number of hours per day
2:\(P\ \leftarrow\) historical load data
3:Compute \(P_{max}^{c_{i}}\ \ \forall i\in\{1,2,3,\ldots,n\}\)
4:Compute \(P_{nrm}\)
5:Compute \(Med_{i}\ \ \forall i\in\{1,2,3,\ldots,n\}\)
6:for all\((i\in\{1,2,3,\ldots,n\})\)do
7:for all\((j\in\{1,2,3,\ldots,m\})\)do
8:if\((P_{nrm}^{(i,j)}\leq Med_{i})\)then
9:\(P_{b}^{i,j}\ \gets 0\)
10:elseifthen
11:\(P_{b}^{i,j}\ \gets 1\)
12:endif
13:endfor
14:endfor
15:Remove redundant and irrelevant features using Equation (4)
16:Compute \(TS\) and \(VS\) using Equations (8) and (9), respectively
17:Compute \(y(1)\) by letting \(W(1)=I\) and \(\overline{W}(1)=I\)
18:while Maximum number of iterations not reached do
19:if\(J(k+1)\leq J(k)\)then
20:\(y(k)\gets y(k+1)\)
21:elseifthen
22: Train ANN as per Equations (20) and (21)
23: Compute \(y(k+1)\) and go back to Step (18)
24:endif
25:endwhile
26:Perform decoding
27:Perform de-normalization
```  
**Algorithm 1** Day-ahead load forecast. Note: our proposed prediction model predicts tomorrow's load on the basis of historical load till today. Thus, the prediction model never fails, _i.e._, for every next day, the model needs information till the current day. However, the proposed model is unable to predict the load for more than tomorrow provided the historical load information till today.