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Photovoltaic (PV) output power has regularity, volatility, and randomness. First of all, this paper carried on a metrological analysis to PV system data. Then, this paper analyzed the relationship between PV historical data, PV power forecasting model, and forecast error. By spectrum analysis of PV power, the PV power is decomposed into periodic components, low frequency residual components, and high frequency residual components. Making a specific analysis of these three components determines the minimum modeling error value, which reflects the unpredictable part of the PV power. Determining the minimum modeling error for PV forecasting not only objectively evaluates the quality of the PV forecasting model but also can determine the prediction accuracy standard according to different PV power generation targets. The examples given in this paper illustrate the effectiveness of the method.

Renewable energy power is an important solution to global warming. Solar energy generated from PV systems is one of the fastest and the most promising growing renewable energy types [

The prediction of PV power is to use a certain modeling method based on historical data [

The above methods only focus on finding effective forecasting methods and do not take into account the predictability and unpredictability of the PV power time series itself. In this paper, the regularity of the PV power is fully excavated, as well as its physical explanation, so as to achieve the greatest modeling accuracy. Firstly, the periodic characteristics of PV power are analyzed. Then, the PV power is decomposed by Fourier decomposition to extract the corresponding periodic component including daily cycle components, low frequency components, and high frequency components which are analyzed and explained physically. Finally, the minimum modeling error is determined from the high frequency components. In order to verify the effectiveness of the method, the minimum error of PV power is analyzed under different forecast horizons and different locations. The minimum modeling error determined is then compared with the standard deviation of the prediction error obtained from the three PV power prediction modeling methods, that is, continuous method, artificial neural networks, and generalized regression audit network of PV power generation combined forecast, respectively. The prediction of PV power is similar to the load forecasting, and the research on the evaluation of the load regularity has existed. The necessity of the evaluation of the load regularity has been expounded in [

This paper is organized as follows. Section

When sunlight shines on the surface of a solar cell, its semiconductor interface converts light energy into electrical energy due to the PV effect. Solar cell output power by the solar radiation intensity, temperature, humidity and wind speed, and other factors, the equivalent circuit shown in Figure

Equivalent circuit of PV cell.

In order to facilitate the analysis of PV cell characteristics and avoid complicated calculations, the practical equivalent of PV cells can be approximated to meet the required accuracy in practical engineering applications, and an equivalent model of PV cells can be established [

^{2};

The influence of solar radiation intensity on the output characteristics of PVcells was analyzed based on (^{2}, ^{2}, ^{2}, and ^{2}. The IV characteristic and PV characteristic curve of a certain type of PV cell were obtained by using MATLAB/Simulink simulation software, as shown in Figure

Effect of radiation intensity on PV output.

As the PV power by a variety of complex factors, such as temperature, humidity, wind speed, and component status. Therefore, it is impossible to predict the PV power 100% accurately. How to determine the unpredictable degree of PV power sequence is the goal of this paper.

The PV power can be predicted due to its regularity. This rule can be represented by modeling PV power based on historical data within a training window of a specified length. The modeling error is

The total error of the PV power prediction can be expressed as

This paper focuses on the analysis of the influence of modeling error on the prediction accuracy of PV power and its composition [

If the PV power curve for each day of the historical data (total

The special case of the above analysis is only to illustrate the relationship between the errors, and it can not be directly used for the actual PV power prediction error analysis. In fact, according to any one or more of the PV power modeling methods, it is always possible to decompose a set of PV sequences into

Comparing (

From the above analysis, the size of the modeling error is related to the modeling method and the regularity of

In the practice of PV power prediction, the relative error of the prediction error is the percentage of the PV power value at the corresponding time. As an important index, the relative modeling error can be defined as follows:

The statistical characteristics of the relative modeling error is closely related to the prediction accuracy of PV power. For the actual PV power data, despite the effort to improve the PV power modeling method, it can not make the modeling error infinite to 0. Because the PV power has a certain degree of randomness, in the process of improving the PV power prediction method, the error of modeling generally has a nonzero lower limit, which mainly reflects the inherent nonregularity of the PV power. For PV power data, it is important to estimate the relative error and determine the upper limit of the prediction accuracy.

PV power sequence significantly satisfies the Dirichlet condition [

The finite element Fourier decomposition of the PV power time series in a given modeling time domain is decomposed as follows [

For the intuitive and effective representation of the signal contained in the components of the energy, it can make the spectrum [

Most of data used in this paper come from the National Key Laboratory of Northeast China Electric Power University, Jilin Province, China. The sampling interval is 15 minutes. Details of the PV test platform are shown in Table

PV power plant information.

Item | Data |
---|---|

Longitude | 126.5098 |

Latitude | 43.8294 |

Altitude | 80 m |

Azimuth | 0° |

Tilt | 37° |

Single capacity | 25 W |

Sampling interval | 15 min |

Sampling time | 2015–2017 |

System storage | 64 G |

Mounting disposition | Flat roof |

Field type | Fixed tilted plane |

Installed capacity | 10 kWp |

Material | Polycrystalline silicon |

PV module | JKM245P |

The type of sensor | ZZ-S-COMB-B |

Accuracy of humidity | 3% RH (25°C) |

Accuracy of temperature | 0.5°C (25°C) |

Accuracy of radioactivity | 7% W/m^{2} (25°C) |

Distributed PV panels.

PV power data acquisition platform.

Immediate data collection system

PV power data storage system

The first part of Figure

PV power curve and spectrum from 2015/01/01 to 2015/01/14.

The data are collected from 6 am to 5 pm. Through the observation of the spectrum, from which the maximum energy of the three corresponding to the frequency of analysis, the extraction results are as shown in Table

High energy frequency extraction.

Item | First high energy | Second high energy | Third high energy |
---|---|---|---|

Frequency [Hz] | | | |

Cycle [h] | 12 | 56 | 15.25 |

The first high energy corresponding frequency is converted to a period of about 12 hours. The time of the collected data per day is 12 hours. Through extensive analysis of different PV power data, the daily cycle characteristics of all PV power data are obvious because the daily periodicity of the light determines the daily cycle of the PV power. This also confirms the daily cycle of PV power output from a mathematical point of view.

The power data are decomposed by spectral analysis. The respective components in equation (

The finite element Fourier decomposition of the PV power time series in a given modeling time domain is decomposed as shown in Figure

Decomposition results of PV power in a region from 2015/01/01 to 2015/01/14.

Actual output time from 2015/01/01 to 2015/01/14 [min]

Day cycle component time from 2015/01/01 to 2015/01/14 [min]

Low frequency residual component time from 2015/01/01 to 2015/01/14 [min]

High frequency residual component time from 2015/01/01 to 2015/01/14 [min]

Figure

It can be seen from the observation that the amplitude of the low frequency residual fraction is relatively large, and the amplitude of the low frequency residual component can not directly reflect the influence of the variable-related factors such as weather on the PV output.

The low frequency residual component contains some related factors to affect the PV power, and the cycle of low frequency is greater than 12 h. If the relevant factors

Practice has proved that the low frequency residual component of the cycle is greater than 12 h; low frequency residual component modeling will usually have a certain improvement in PV power prediction.

Because the cycle of high frequency is less than 12 h, it is difficult to predict it. According to the time series analysis point of view, we can use the time series analysis method to model the high frequency residual components. By analyzing the variation characteristics of PV power in different periods at different time periods, the high frequency residual component autocorrelation function and the partial correlation function are as shown in Figure

Canonical autocorrelation function and partial correlation function of high frequency residual component.

It can be seen from Figure

The time-series modeling of the high frequency residual components in Figure

The modeling results are shown in Figure

Comparison of actual power and AR prediction power of high frequency residual component.

In fact, since the PV power prediction is a multistep prediction, it is required to perform extrapolated prediction of multistep (at least 48 steps) without supplementing the new sample observations. When the high frequency residual component is predicted by the time series model of the high frequency residual component, the number of steps is predicted. According to the literature [

Figure

Standard deviation curve of multistep prediction error.

Figure

The model methods used in this paper can be summarized in the flow chart shown in Figure

Model flow chart.

Define

If the PV power prediction method can only effectively predict the daily cycle component, then

The PV power data for 14 days in three different areas are divided into

The standard deviation of the relative modeling error of the different PV power components with no real contribution to the prediction accuracy is defined.

Item | | |
---|---|---|

Area A [%] | 4.43 | 5.23 |

Area B [%] | 4.56 | 5.78 |

Area C [%] | 5.20 | 5.63 |

As can be seen from Table

Since the high frequency residual component is unpredictable in the actual prediction, the relative modeling error obtained by

Taking the PV power sequence

For the PV power sequence

The regularity of PV power generation is different in time and space. Table

Standard deviation of relative modeling error of regional B PV power in different periods.

Item | | |
---|---|---|

Time 1 | 3.65 | 4.21 |

Time 2 | 4.47 | 5.75 |

Time 3 | 3.74 | 4.57 |

Full time period | 3.95 | 4.84 |

In order to show that the proposed method of PV power regularity evaluation is effective, the standard deviation of the actual prediction error of three different PV power prediction methods is compared with the standard deviation of the minimum modeling error in this paper. The historical data for the three regions are collected from October 1, 2015, to October 30, 2015, and the PV power for the day of October 31, 2015, is predicted. The forecast results are shown in Table

Comparison between the standard deviation of the actual prediction error and the standard deviation of the minimum modeling error.

Item | The smallest modeling error [%] | Standard deviation of actual prediction error [%] | ||
---|---|---|---|---|

Method 1 [ | Method 2 [ | Method 3 [ | ||

Area A | 3.55 | 4.42 | 5.22 | 5.85 |

Area B | 4.72 | 5.24 | 6.13 | 5.01 |

Area C | 5.19 | 5.58 | 6.25 | 6.34 |

Table

Comparison of standard error and standard deviation of the standard deviation of the actual prediction error for each period of the A PV power.

Item | The smallest modeling error [%] | Standard deviation of actual prediction error [%] | ||
---|---|---|---|---|

Method 1 [ | Method 2 [ | Method 3 [ | ||

2015-01 | 4.123 | 4.454 | 5.156 | 4.825 |

2015-02 | 3.930 | 4.359 | 4.645 | 5.698 |

2015-03 | 4.516 | 4.989 | 4.958 | 4.684 |

2015-04 | 4.587 | 5.648 | 5.605 | 5.869 |

2015-05 | 4.432 | 4.685 | 4.788 | 6.698 |

2015-06 | 5.123 | 5.658 | 5.687 | 6.685 |

Average | 4.785 | 4.967 | 5.140 | 5.743 |

Method 1 is the continuous method, using the previous day’s PV power output data as the next day PV power prediction data. Method 2 is wavelet decomposition and artificial networks, that is, with the ability of ANN to address nonlinear relationships, theoretical solar irradiance and meteorological variables are chosen as the input of the hybrid model based on WD and ANN. The output power of the PV plant is decomposed using WD to separate useful information from disturbances. The ANNs are used to build the models of the decomposed PV output power. Method 3 is combined with forecasting of PV Power Generation based on firefly algorithm-generalized regression auditing network. Firstly, to simplify model input dimensions, multiple linear factors influencing PV output are compressed and extracted with principal component analysis (PCA) method. Then the first principal component extracted from PCA combined with grey correlation degree is used to filter similar historical days. Next, the chosen days are, respectively, brought into two models, least square support vector machine (LS-SVN), and modified BP network (MBP), and the two predictions are repeated: the first is to forecast for similar day and then firefly algorithm for generalized regression neural network (FFA-GRNN) is applied to train weight coefficients; the second is to ultimate forecast for test sets.

As can be seen from Table

This paper presents A evaluation method of the PV power prediction quality. The variation of PV power varies with time and area, which is affected by light intensity, temperature, and other factors. There are inherent unpredictable factors in actual PV power generation, which are determined by the influence of some complicated factors and the characteristics of multistep prediction. Ignoring the inherent differences in PV power, it is impractical to unify the accuracy requirements of PV power forecasting. This supports a large number of PV power forecasting practices supported by the analytical methods in this paper. By using the concept of minimum modeling error presented here, the minimum modeling error can be determined by analyzing historical data of PV power, and the upper limit of the prediction accuracy of PV power can be estimated.

The authors declare that there are no conflicts of interest.

This work has been supported by Project 51307017 funded by the National Natural Science Foundation of China.