Short-term photovoltaic power forecasting based on human body amenity and least squares support vector machine with fruit fly optimization algorithm

Along with our country paying more attention to environment and energy sources, photovoltaic (PV) power generation becomes more important in theory and using because of little pollution and high energy utilization. In recent years, with the development of photovoltaic in China, the scale of the PV power station is increasing. However, influenced by external uncertain factors, the output characteristic of PV is present in volatility and intermittency, also has a strong impact on the economical efficiency, stability and security of electric power supply. PV power forecasting is an important part of management modernization of PV utilization, which has attracted more and more attentions from the academic and the practice. Accurate PV power forecasting can relieve the conflict between electricity supply and demand. The short-term forecasting will contribute to arrange the output of the conventional electric power supply by the electric power dispatching system (Damousis et al., 2004). In the meantime, according to the forecasting results can be achieve slide control and reduced the impact of grid by the randomness of PV power generation. Furthermore, these results would be able to assist the operation control of micro grid with variety types of distributed generation (Ai-Hamadi and Soliman, 2005; Almonacid et al., 2009).

However, because the forecasting of PV source has complex and non-linear relationships with several factors such as solar irradiance, temperature, humidity and wind speed and other non-linear factors, it is fairly difficult for forecasting PV output precisely. Lu et al. (2011) applied metrical information including temperature, pressure, humidity and solar irradiance as inputs to predict the day type, and then calculated the power output of PV power generation under that day type. However, the cycle of this forecasting model is very shorter, only in per hour. Benghanem et al. (2009) used the historical weather data to estimate the solar irradiance of the forecasting day and worked out the power output of PV. Although this model can be extended to forecast the 24 hours, the forecasting results may be failure with the weather variations. Yorukoglu and Celik (2006) considered the PV power generation based on time series method, but the unified fitting and generalization was adopted in different weather types. Rehman and Mohandes (2008) took the various meteorological factors into consideration, yet increased the inputs of the network, slowed down the training speed of the network and reduced the accuracy of the forecasting results. Among many factors that affect PV power generation, the solar irradiance has strong randomicity, so that it is rather difficult for forecasting, especially the short-term forecasting (Wang et al., 2012). In addition, few PV power stations are equipped with the real-time monitoring system of solar irradiance.

Moreover, owing to the rising of intelligence techniques, many new intelligence forecasting algorithms were used for PV power generation forecasting. Keerthi and Lin (2003) proposed a hybrid model combining support vector regression and differential evolution algorithm to forecast the output power of PV, and this method was proved to outperform the SVR model with default parameters, regression forecasting model and back propagation artificial neural network. Hsu and Chen (2003) implemented a knowledge-based expert system to support the choice of the most suitable forecasting model, and the usefulness of this method was demonstrated by a practical application. Mellit and Pavan (2010) proposed the artificial neural network technique for short-term PV power forecasting, and it was applied on an actual PV power station of Italy dependent on its historical data. Zhu and Tian (2011) developed a short and medium term PV power forecasting model by using radial basis function neural networks, and the result indicated that the proposed model has a high accuracy and stability.

In conclusion, every forecasting algorithm is almost based on the PV changing regularity of one or two factors, while ignoring the other factors. Those methods have defects and shortcomings of themselves. Reasonably combining with each other, can effectively improve the accuracy of the forecasting results.

Human body amenity, which refers to take no effective measures under the premise of protection, can describe the degree of comfort by the person in the natural environment. This index can effectively use the various meteorological factors, reduce the input of network and improve the precision of forecasting.

The least squares support vector machine (LSSVM) which was extended by support vector machine (SVM) is a powerful regression tool with a dynamic network structure (Cortes and Vapnik, 1995). It can translate a quadratic programming problem with inequality constraint into the solution of linear matrix. Because of the strong nonlinear mapping capability, the LSSVM can effectively solve the nonlinear problems, and it has been widely applied to a variety of fields including short-term load forecasting, sales forecasting and wind speed forecasting, and so on. The forecasting performance of LSSVM model largely depends on the parameters of kernel function and penalty factor. These parameters which chose by the experience and experiment are not only time consuming, but also low forecasting precision. Therefore, some scholars studied how to select proper parameters of the LSSVM model, such as using simulated annealing algorithm (SA) to optimize the parameters (Pai and Hong, 2005). However, these mete-heuristic algorithms are usually program complexity and beyond understanding.

Fruit fly optimization algorithm (FOA) proposed by the scholar Pan (2012) is a novel evolutionary computation and optimization technique. This new optimization algorithm has the advantages of being easy to understand and to be written into program code which is not too long compared with other algorithms. Therefore, this paper attempted to use the FOA to automatically select the parameters value of the LSSVM for improving the LSSVM’s forecasting accuracy in the short-term PV power forecasting.

The rest of this paper is organized as follows: Section 2 sets up the model of human body amenity. Section 3 introduces the LSSVM and FOA methods, then proposes a hybrid forecasting model combined LSSVM with FOA into solving the short-term PV power forecasting in detail. Section 4 introduces the process of the sample data used in this paper and further computations, comparisons and discussion. Section 5 concludes this paper.

3.1. Least Squares Support Vector Machine (LSSVM)

The support vector machine (SVM) is a trainable and machine learning method, which is used to solve the problem of pattern recognition and has better generalization abilities. The least squares support vector machine (LSSVM) is the further improvement of the SVM (van Gestel et al., 2004; Du et al., 2008).

Set $\left\{ x _ { i } , y _ { j } \right\} ( i = 1,2 \ldots , l )$ as the training sample set, among of them, $x _ { i } \in R ^ { n }$ as the n-dimensional input vector, y_i as the output value of one-dimensional and l as the sample number.

Taken the sample x_i to map from Rⁿ into characteristic space $\varphi \left( x _ { i } \right)$, then, the fitting function can be represented as:

$y(X)=\omega \varphi (X)+b$      (3)

where $\omega$ is the weighted vector, b is the constant.

According to the minimization principle of the structural risk, the estimates of \omega and b can be solved. Then,

$\min g(\omega ,\zeta )=\frac{1}{2}\omega \omega +\gamma \sum\limits_{i=1}^{l}{\zeta _{i}^{2}}$     (4)

The constraint conditions are changed from the inequality constraints to the equality in SVM:

${{y}_{i}}=\omega \varphi ({{x}_{i}})+b+{{\zeta }_{i}}$        (5)

where $\gamma$ is the penalty coefficient, which control the degree of punishment beyond the sample error, $\xi_i$ is the error of estimation.

In order to solve this problem, the Lagrangian multiplier method is introduced. The model can be calculated as follows:

$L(\omega ,b,\zeta ,\lambda )=g(\omega ,\zeta )-\sum\limits_{i=1}^{l}{{{\lambda }_{i}}[\varphi ({{x}_{i}})\omega +b+{{\zeta }_{i}}-{{y}_{i}}]}$      (6)

where $\lambda _ { i } ( i = 1,2 , \dots , l )$ is the lagrangian multiplier.

According to the optimal condition of Karush-Kuhn-Tucker (KKT), the partial derivatives of equation (6) are expressed as:

$\left\{ \begin{matrix}   \frac{\partial L}{\partial \omega }=0  \\   \frac{\partial L}{\partial b}=0  \\   \frac{\partial L}{\partial \zeta }=0  \\   \frac{\partial L}{\partial \lambda }=0  \\\end{matrix} \right.$        (7)

$\left\{ \begin{matrix}   \omega =\sum\limits_{i=1}^{l}{{{\lambda }_{i}}\varphi ({{x}_{i}})}  \\   \sum\limits_{i=1}^{l}{{{\lambda }_{i}}=0}  \\   {{\lambda }_{i}}=\gamma {{\zeta }_{i}}  \\   {{y}_{i}}=\omega \varphi ({{x}_{i}})+b+{{\zeta }_{i}}  \\\end{matrix} \right.$      (8)

The parameters of $\omega$ and $\zeta$ are eliminated, the linear equations can be represented as:

${{y}_{i}}=\varphi ({{x}_{i}})\sum\limits_{i=1}^{l}{{{\lambda }_{i}}}\varphi ({{x}_{i}})+\frac{{{\lambda }_{i}}}{\gamma }+b$     (9)

Based on the principle of Hilbert-Schmidt, the kernel function which meets the Mercer condition can be introduced, such as:

$K({{x}_{i}},{{x}_{j}})=\varphi {{({{x}_{i}})}^{T}}\varphi ({{x}_{j}})$       (10)

It is used to convert the inner product calculation of transformation space into a particular function of the original space, thereby indirectly solve the mapping of high-dimensional input space. By solving the equation (9), the regression model of LSSVM can be calculated as follows:

$y(x)=\sum\limits_{i=1}^{l}{{{\lambda }_{i}}K(x,{{x}_{i}})}+b$         (11)

wherein $K \left( x , x _ { i } \right) = \exp \left\{ - \left\| x - x _ { i } \right\| ^ { 2 } / 2 \sigma ^ { 2 } \right\}$, the radial basis kernel function is often adopted.

Therefore, the LSSVM model has two parameters such as kernel function $\sigma$ and penalty coefficient $\gamma$ that need to be determined, which are very important in using LSSVM model for forecasting. The parameters of $\sigma$ and $\gamma$ determine the squared bandwidth of the Gaussian RBF kernel and the trade-off between the training error minimization and smoothness, respectively. Many researchers selected the parameters by priori knowledge or individual experience, which may be un-efficient for forecasting. Therefore, we should develop an automatically efficiently method for selecting the appropriate parameters in the LSSVM model.

In order to achieve this goal, this paper used the fruit fly optimization algorithm (FOA) to automatically determine the parameter values of the LSSVM model.

3.2. Fruit fly optimization algorithm

Fruit fly optimization algorithm (FOA) is a new swarm intelligence method, which was proposed by Pan (2012). The FOA is a new method for finding global optimization based on the food finding behavior of the fruit fly. The fruit fly is superior to other species in vision and osphresis. The food finding process of fruit fly is as follows: firstly, it smells the food source by osphresis organ, and flies towards that location; then, after it gets close to the food location, the sensitive vision is also used for finding food and other fruit flies’ flocking location, and it flies towards that direction. Figure 1 shows the food finding iterative process of fruit fly swarm.

Figure 1. Food finding iterative process of fruit fly swarm

In this paper, based on the food finding characteristics of the fruit fly, it is divided into several necessary steps, just as followings:

Step 1: The main parameters of the FOA are the maximum iteration number maxgen, the population sizepop and the initial fruit fly swarm location as bellows:

$Init\begin{matrix}   {}  \\\end{matrix}X\_axis$      (12)

$Init\begin{matrix}   {}  \\\end{matrix}Y\_axis$      (13)

Step 2: Give the random direction and distance for the search of food using osphresis by an individual fruit fly. 

${{X}_{i}}=X\_axis+RandomValue$       (14)

${{Y}_{i}}=Y\_axis+RandomValue$      (15)

Step 3: Since the food location cannot be known, the distance to the origin is thus estimated first (Dist), then the smell concentration judgment value (S) is calculated, and this value is the reciprocal of distance.

$Dis{{t}_{i}}=\sqrt{X_{i}^{2}+Y_{i}^{2}}$      (16)

${{S}_{i}}=\frac{1}{Dis{{t}_{i}}}$   (17)

Step 4: Substitute smell concentration judgment value (S) into smell concentration judgment function (or called Fitness function) so as to find the smell concentration (Smell_i) of the individual location of the fruit fly.

$Smel{{l}_{i}}=Function({{S}_{i}})$      (18)

Step 5: Find out the fruit fly with maximal smell concentration (finding the maximal value) among the fruit fly swarm.

$[bestSmell\begin{matrix}   {}  \\\end{matrix}bestIndex]=\max (Smell)$     (19)

Step 6: Keep the best smell concentration value and x, y coordinate. At this moment, the fruit fly swarm will use vision to fly towards that location.

$Smellbest=bestSmell$    (20)

$X\_axis=X(bestIndex)$      (21)

$Y\_axis=Y(bestIndex)$     (22)

Step 7: Enter iterative optimization to repeat the implementation of Steps 2-5, then, judge if the smell concentration is superior to the previous iterative smell concentration, if so, implement the Step 6.

FOA belongs to a kind of interactive evolutionary computation, and also is a part of the artificial intelligence. Therefore, it has been widely applied to solve the problems in military, engineering, management and financial affairs.

3.3. Fruit fly optimization algorithm for the parameters selection of the LSSVM model

Figure 2. The flowchart of the FOALSSVM model

Selecting appropriate values of the kernel function

 and penalty coefficient  are particularly important. In this paper, the FOA was used for selecting the suitable parameter values of the LSSVM model in order to effectively improve the short-term photovoltaic power forecasting accuracy.

The flowchart of the FOA for the parameters of the LSSVM model (abbreviated as FOALSSVM) is shown in Figure 2.

The details of the FOALSSVM are shown as follows:

Step 1: Initialization parameters.

The maximum iteration number maxgen, the population size sizepop, the initial fruit fly swarm location (X_axis, Y_axis), and the random flight distance range FR should be determined at first. In this paper, suppose maxgen=100, sizepop=10,

 and .

Step 2: Evolution starting.

Set gen=0, and give the random flight direction and the distance for food finding of an individual fruit fly.

Step 3: Preliminary calculation.

Calculate the flight distance Dist_i of the fruit fly i, and then calculate the smell concentration judgment value S_i. Input S_i into the LSSVM model for the short-term photovoltaic power forecasting. According to the result of photovoltaic power forecasting, calculate the smell concentration Smell_i (also called the fitness function value). The Smell_i is employed the root-mean-square-error (RMSE) which measures the deviation between the forecasting value and the actual value.

$\left\{ \begin{matrix}   \gamma =20*\max ({{S}_{i}})  \\   \sigma =\min ({{S}_{i}})  \\\end{matrix} \right.$    (23)

$RMSE=\sqrt{\frac{\sum\limits_{i=1}^{n}{{{(P_{i}^{r}-P_{i}^{f})}^{2}}}}{n}}$      (24)

where

 and  are the actual and training output power of PV power station, respectively; n is the number of training sample.

Step 4: Offspring generation.

The offspring generation is generated according equation (12) to equation (22). Then input the offspring into the LSSVM model and calculate the smell concentration value again. Set gen=gen+1.

Step 5: Circulation stops.

When gen reaches the max iterative number, the stop criterion satisfies, and the best parameter values of the LSSVM model can be obtained. Otherwise, go back to Step 2.

The LSSVM algorithm has been widely used in variety of fields, but it rarely finds that the LSSVM have been applied to the PV power forecasting. In this paper, a hybrid model based on human body amenity and LSSVM with FOA were proposed for the short-term PV power forecasting. The human body amenity can effectively use the various meteorological factors, reduce the input of network and improve the precision of forecasting. Meanwhile, the FOALSSVM model uses the FOA to automatically select the appropriate parameter values of LSSVM in order to improve the forecasting accuracy. For comparison, another two models such as SVM and LSSVM were selected. The simulating results show: the FOA can select the appropriate parameter values of the LSSVM model, which could effectually improve the forecasting accuracy of PV output power. Compared with the SVM and LSSVM algorithm, the values of APE and MAPE are obviously smaller than that of other two algorithms.