# Power Load Demand Forecasting Model and Method Based on Multi-Energy Coupling
## 3 Model Construction
### Least Squares Support Vector Machine (LSSVM)  
Support vector machine (SVM) has been widely used in forecasting scenarios and has achieved many achievements [18; 30]. It has excellent processing capacity for small samples and high latitude data [32], which means that it has a good applicability to the situation where the multi-energy coupling quarterly data is relatively small and the system structure is relatively complex. The least square support vector machine (LSSVM) has a significant improvement in the prediction accuracy compared with the traditional support vector machine. Therefore, LSSVM is applied to the main load prediction algorithm under the multi-energy coupling scenario in this paper.  
Nonlinear regression support vector machine (SVM) uses kernel function to change input data in sample space to high-dimensional linear eigenspace with nonlinear transformation. The linear method is used to solve nonlinear problems in the characteristic space and the global optimal solution is obtained. On the basis of standard support vector machine, the least squares support vector regression machine changes inequality constraints into equality constraints to accelerate calculation and improve accuracy.  
The optimal decision function after nonlinear mapping (\(\psi(x)=(\varphi(x_{1}),\varphi(x_{2}),\cdots,\varphi(x_{n}))\)) is as Equation (19).  
\[f(x)=w^{T}\bullet\varphi(x)+b, \tag{19}\]  
where, \(w\in R^{k}(k>d)\) is the weight vector of high-dimensional features, which reflects the way in which each of the key impact factors affects the load and the extent to which it affects the load, \(x_{i}\in R^{d}\) is the input of d-dimensional training samples, \(y_{i}\in R\) is the output of training samples, and \(b\in R\) is the bias. We searched for the optimal \(w\) and \(b\) based on the principle of structural risk minimization, and obtained the solution equation of the optimization problem as Equation (20).  
\[min\frac{1}{2}w^{T}w+r{\sum\limits_{i=1}^{n}{\xi_{i}}^{2}}, \tag{20}\]  
where, \(r>0\) is the penalty parameter and \(\xi_{i}\) is the relaxation variable.  
Compared with the standard support vector machine algorithm, there are differences in constraint conditions.  
\[y\big{[}w^{T}\bullet\varphi(x_{i})+b\big{]}=1-\xi_{i},\ i=1,2,\cdots,n, \tag{21}\]  
apply Lagrange function to solve the optimization problem, then:  
\[L=\tfrac{1}{2}w^{T}\bullet w+r\bullet\tfrac{1}{2}{\sum\limits_{i=1}^{n}{\xi_ {i}^{2}}}-{\sum\limits_{i=1}^{n}{\alpha_{i}\big{[}y_{i}\big{[}w^{T}\bullet \varphi(x_{i})+b\big{]}-1+\xi_{i}\big{]}}}\]  
\[s.t.\ \ \ \left\{\begin{array}{c}w={\sum\limits_{i=1}^{n}{\alpha_{i}y_{i} \varphi(x_{i})}}\\ {\sum\limits_{i=1}^{n}{\alpha_{i}y_{i}}}=0\\ {\alpha_{i}=r\xi_{i}}\\ {y_{i}\big{[}w^{T}\bullet\varphi(x_{i})+b\big{]}-1+\xi_{i}=0}\end{array}\right., \tag{22}\]  
where, \(\alpha_{i}\) is the vector of Lagrange multiplier, \(\alpha_{i}>0\) and \(i=1,2,\cdots,n\). The final forecasting function can be obtained as shown in Equation (23).  
\[f(x)=\sum\limits_{i=1}^{n}{\alpha_{i}K(x,x_{i})+b}, \tag{23}\]  
where, \(K\big{(}x_{i},x_{i}\big{)}=\varphi(x_{i})^{T}\varphi\big{(}x_{i}\big{)}\) is the kernel function satisfying Mercer condition. The kernel function in this paper is the radial basis kernel function, as shown in Formula (24).  
\[K(x,x_{i})=exp\bigg{(}-\frac{\|x-x_{i}\|^{2}}{2g^{2}}\bigg{)}, \tag{24}\]  
where \(g\) is the width coefficient of kernel function.