Short term electric load forecasting using hybrid algorithm for smart cities

Abstract

Many day-to-day operation decisions in a smart city need short term load forecasting (STLF) of its customers. STLF is a challenging task because the forecasting accuracy is affected by external factors whose relationships are usually complex and nonlinear. In this paper, a novel hybrid forecasting algorithm is proposed. The proposed hybrid forecasting method is based on locally weighted support vector regression (LWSVR) and the modified grasshopper optimization algorithm (MGOA). Obtaining the appropriate values of LWSVR parameters is vital to achieving satisfactory forecasting accuracy. Therefore, the MGOA is proposed in this paper to optimally select the LWSVR’s parameters. The proposed MGOA can be derived by presenting two modifications on the conventional GOA in which the chaotic initialization and the sigmoid decreasing criterion are employed to treat the drawbacks of the conventional GOA. Then the hybrid LWSVR-MGOA method is used to solve the STLF problem. The performance of the proposed LWSVR-MGOA method is assessed using six different real-world datasets. The results reveal that the proposed forecasting method gives a much better forecasting performance in comparison with some published forecasting methods in all cases.

Appendices

Appendix A: Phase space reconstruction based on KPCA

Due to the complication of the historical datasets, the use of time series reconstruction technique has become necessary in STLF. So, the KPCA method is utilized in this paper to reconstruct the non-linear time series used in STLF which leads to overcoming the drawbacks of the conventional reconstruction methods. KPCA is an unsupervised method that executes the principal component analysis in the feature space of a kernel which is non-linearly related to the input space [43]. The merit of KPCA over other non-linear methods comes from making the time series structure more linear by mapping the original inputs into a high dimensional feature space via a kernel map.

In this paper, the generally used Gaussian kernel is utilized and can be defined as follows:

$$ Q({x}_{i},{x})=e^{\left( -\frac{\|{x}_{i}-{x}\|^{2}}{2\sigma^{2}}\right)} $$

(A.1)

The details of the basic KPCA can be found in [15, 43].

Appendix B: Support vector regression (SVR)

The quadratic programming and kernel functions are the main elements used to implement the SVR algorithm. Where the SVR’s parameters can be calculated by solving the quadratic programming problem with different constraints. Also, the flexibility of kernel functions helps the algorithm to enhance its searching ability [9].

The main idea of SVR is to transfer the input data x into a high dimensional feature space through a nonlinear mapping and execute a linear regression in that feature space [9] as:

$$ f(x)=(w\cdot\varphi(x))+b $$

(B.1)

To get the weight coefficients and the bias, the following optimization problem is solved [44]:

$$ \min\limits_{w,b,\zeta_{i},\zeta_{i}^{*}}~\frac{1}{2}\|w\|^{2}+C \sum\limits_{i=1}^{N_{t}}(\zeta_{i}+\zeta_{i}^{*}) $$

(B.2)

$$ \text{subject~to} \left\{ \begin{array}{cc} y_{i}-w\varphi(x_{i}) +b\quad\quad & \leq\varepsilon+\zeta_{i}^{*}\\ w\varphi(x_{i})+b-y_{i}\quad\quad & \leq\varepsilon+\zeta_{i}\\ \zeta_{i},\zeta_{i}^{*} & \geq0 \end{array}\right. $$

where x_i is transferred to higher dimensional space via φ. $\zeta _{i}, \zeta _{i}^{*}$ depend on the ε-insensitive tube |y − (w^Tφ(x) + b)|≤ ε.

Based on the Lagrangian and the Karush-Kuhn-Tucker conditions, the solution of the model can be found in dual representation instead of high dimension as follows [9]:

$$ \min\limits_{\eta,\eta^{*}}\left\{ \begin{array}{c} \frac{1}{2}\sum\limits_{i,j=1}^{N_{t}}(\eta_{i}-\eta_{i}^{*})(\eta_{j}-\eta_{j}^{*})Q(x_{i},x_{j}) \\ +\varepsilon\sum\limits_{i=1}^{N_{t}}(\eta_{i}+\eta_{i}^{*})-\sum\limits_{i=1}^{N_{t}}y_{i}(\eta_{i}-\eta_{i}^{*}) \end{array}\right. $$

(B.3)

$$ \text{subject~to} \left\{ \begin{array}{c} \sum\limits_{i=1}^{N_{t}}(\eta_{i}-\eta_{i}^{*})=0\\ 0\leq \eta_{i},\eta_{i}^{*}\leq C \end{array}\right. $$

By solving this dual representation problem, the regression output can be written by [9]:

$$ \widehat{y}=\widehat{f}(x)=\sum\limits_{i=1}^{N_{t}}(\eta_{i}-\eta_{i}^{*})Q (x_{i},x) +b $$

(B.4)

The most commonly employed kernel function is the Gaussian kernel defined in (A.1).