Solar Irradiation Forecasting using RBF Networks for ...

Solar Irradiation Forecasting using RBF Networks for PV Systems with Storage Lucio Ciabattoni, Gianluca Ippoliti and Sauro Longhi

Matteo Cavalletti and Marco Rocchetti

Dipartimento di Ingegneria dell’Informazione Universit`a Politecnica delle Marche Via Brecce Bianche, 60131 Ancona, Italy Email:{l.ciabattoni, gianluca.ippoliti, sauro.longhi}@univpm.it

Energy Resources SPA Via Ignazio Silone, 60035 Jesi (AN) , Italy Email: {marco.rocchetti, matteo.cavalletti}@energyresources.it

Abstract—In this paper a Radial Basis Function (RBF) neural network is proposed to obtain the 24-hr forecast of the solar irradiation on the horizontal plane in the city of Ancona, Italy. This information is used to estimate the production of a PhotoVoltaic (PV) plant in order to provide the ’Gestore dei Servizi Energetici’ (the main italian provider of energy services) with the power production profile of the next day. The company Energy Resources SPA has experimentally tested the proposed solution by a 14 KWp PV plant and a lithium battery pack. The battery pack is used to store the exceding power produced or to supply the lack of power compared with the reference.

I. INTRODUCTION Due to the strong increase of renewable energies, the significance of the prediction of meteorological quantities, such as wind speed and solar irradiance, is rising. Today, wind power prediction systems have already shown their strong economic impact and improve the integration of wind energy into the electricity grid (see eg. [17]). Accordingly, the prediction of solar yields will become more and more important for utilities, which have to integrate increasing amounts of solar power, especially for countries where legislation encourages the deployment of solar power plants [25]. In Italy, where the installed photovoltaic (PV) power amounts is more than 5 GW, a proposal of law of 2009 included 20% more incentives for the correct hourly prediction of the power production of the next day (with an error of less than 10%) for at least 300 days/year. Forecast information on the expected solar power production is necessary for the management of electricity grids, for scheduling of conventional power plants and also for decision making on the energy market, as described in [18]. Depending on the application and the corresponding time scale different approaches for modelling and forecasting may be appropriate. There have been a lot of researchers engaged in the modeling of solar irradiance. For example, the existing models include those called clear-day solar radiation, half-sine, Multi Layer Perceptron Network (MLP), Wavelet Decomposition Network (WD), Seasonal Auto Regressive Integrated Moving Average (SARIMA), WD-SARIMA, Support Vector Regression (see [19], [22], [23] and reference therein). However, the methods based on time series have not been efficient in all cases (most of them are efficient only for forecast up to 5-10 minutes [2], [20]) and contrarily they may yield to noised results. Indeed, neural Networks provide a generic black-box representation

for implementing mappings. Our forecasting scheme is based on RBF Neural Network (that requires less time for model development than others Neural Networks see eg. [21]) and is used to derive predictions of solar irradiation based on forecasts up to 48 hours ahead provided by meteo websites and Energy Resources’ database of measured solar irradiation. The irradiation forecast is used as input for a PV’s model to obtain the forecast of the PV production, in a customizable way with the type of panel and inverter used. The irradiation is forecasted on the horizontal plane for panels south oriented. Experimental tests are performed on a 14 KWp PV plant, using SMA inverters and Renergies 220P panels (see [28], [29]), in order to provide the power net with the exact hourly amount of predicted energy. In order to have an ”energy buffer” we use a battery pack with a battery charger and another solar inverter. The paper is organized as follows. The RBF Network is presented in Section II. In Section III details on training and validation of the network are discussed. Experimental setup and results on experimental tests are reported in Section IV. The paper ends with comments on the performance of the proposed approach. II. RBF N ETWORK Different neural network structures have been proposed for identification and control of nonlinear systems (see, e.g. [1] and [16]). A good tradeoff between complexity and accuracy is generally obtained using Radial Basis Function Networks (RBFNs). These nets avoid the nonlinear optimization techniques used in the learning algorithm of Multilayer Neural Networks (MNNs) and the related problems of local minima. Moreover in [11] has been proved that RBFNs do have the best approximation property with respect to MNNs and this significant result provides theoretical support for favoring RBF networks. A. Radial basis function networks A RBFN with input pattern 𝒘 ∈ ℝ𝑚 and a scalar output 𝜓ˆ ∈ ℝ implements a mapping 𝑓𝑟 : ℝ𝑚 → ℝ according to 𝜓ˆ = 𝑓𝑟 (𝒘) = 𝜅0 +

𝑛𝑟 ∑ 𝑖=1

𝜅𝑖 𝜙 (∥𝒘 − 𝒄𝑖 ∥) ,

(1)

where 𝜙(⋅) is a given function from ℝ+ to ℝ, ∥ ⋅ ∥ denotes the Euclidean norm, 𝜅𝑖 , 𝑖 = 0, 1, ⋅ ⋅ ⋅, 𝑛𝑟 are the weights or parameters, 𝒄𝑖 ∈ ℝ𝑛 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑟 , are the radial basis functions centers and 𝑛𝑟 is the number of centers [8] , [7], [16]. The block diagram representing the implementation of the considered network (1) is shown in Figure 1.

w1 w2

wn

φ1 ( w - c1

)

φ2 ( w - c2

)

φn

Fig. 1.

r

( w-c )

1

κ1

κ0

κ2

Σ

ψl

κn

r

nr

Radial basis function neural network.

Note that, if a RBFN is used for the estimation of the output of a dynamical system, the system dynamics can be taken into account through the input pattern 𝒘, that must be composed of a proper set of system input and output samples acquired in a finite set of past time instants. In the RBFN the functional form 𝜙(⋅) and the centers 𝒄𝑖 are suitably fixed. By providing a set of network input pattern 𝒘(𝑘) and the corresponding desired output 𝜓(𝑘) to be approximated by the net, for 𝑘 = 1, 2, ⋅ ⋅ ⋅, 𝐿, the values of the weights 𝜅𝑖 , 𝑖 = 0, 1, ⋅ ⋅ ⋅, 𝑛𝑟 can be determined using the linear Least Squared (LS) method. The choices of centers 𝒄𝑖 must be carefully considered in order the RBFN closely matches the performance of MNNs, while theoretical investigation and practical results show that the choice of the non-linearity 𝜙(⋅), a function of the distance 𝑑𝑖 between the current input 𝒘 and the centre 𝒄𝑖 , does not significantly influence the performance of the RBFN [8]. Therefore, the following gaussian function is considered: ) ( 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝑛𝑟 𝜙(𝑑𝑖 ) = exp −𝑑2𝑖 /𝛽 2 , (2) where 𝑑𝑖 = ∥𝒘 − 𝒄𝑖 ∥ and the real constant 𝛽 is a scaling or “width” parameter [8]. The centers 𝒄𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝑛𝑟 are generally chosen from the data set {𝒘(𝑘)}𝐿 𝑘=1 . The Orthogonal Least Squares (OLS) algorithm [8] is an efficient method for selecting centers from the data set obtaining adequate and parsimonious RBFN thus reducing computational complexity and numerical ill-conditioning. The application of this method requires to rewrite the RBFN expressed by (1) as a linear regression model: 𝑀 ∑ 𝜋𝑖 (𝑘)𝛿𝑖 + 𝑒(𝑘) (3) 𝜓(𝑘) = 𝑖=1

where 𝑀 := 𝑛𝑟 + 1, 𝜓(𝑘) is the desired output to be ˆ approximated by the net, 𝜖(𝑘) := 𝜓(𝑘) − 𝜓(𝑘) is the error signal, 𝛿𝑖 := 𝜅𝑖−1 are the parameters to be estimated, 𝜋𝑖 (𝑘) := 𝜙(∥𝒘(𝑘) − 𝒄𝑖 ∥) are given fixed function of 𝒘(𝑘) where the centers 𝒄𝑖 have to be fixed and 𝜋1 (𝑘) = 1. The

error signal 𝜖(𝑘) is assumed to be uncorrelated with 𝜋𝑖 (𝑘). The OLS method, is considered for the selection of centers 𝒄𝑖 from the data set {𝒘(𝑘)}𝐿 𝑘=1 , with a reduced number 𝑀 of 𝜋𝑖 (⋅). These significant regressors 𝜋𝑖 (𝑘), 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝑀 can be selected using the OLS algorithm operating in a forward regression way [8]. Arranging (3) for 𝑘 = 1, 2, ⋅ ⋅ ⋅, 𝐿 in the following matrix form: Ψ = ΓΔ + 𝑬 (4) where Ψ := [𝜓(1) ⋅ ⋅ ⋅ 𝜓(𝐿)]𝑇 , Δ := [𝛿1 ⋅ ⋅ ⋅ 𝛿𝑀 ]𝑇 , 𝑬 := [𝜖(1) ⋅ ⋅ ⋅ 𝜖(𝐿)]𝑇 and Γ := [𝝅1 ⋅ ⋅ ⋅ 𝝅𝑀 ] with 𝝅𝑖 := 𝑇 [𝜋𝑖 (1) 𝜋𝑖 (2) ⋅ ⋅ ⋅ 𝜋𝑖 (𝐿)] , 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝑀 the OLS method involves the transformation of the set of 𝝅𝑖 into a set of orthogonal basis vectors, 𝒛𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑀 , where the space spanned by the set of orthogonal basis vectors 𝒛𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑀 , is the same space spanned by the set of 𝝅𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑀 , and (4) can be rewritten as Ψ = 𝒁𝒉 + 𝑬,

(5)

with 𝒁 = [𝒛1 , 𝒛2 , ⋅ ⋅ ⋅ , 𝒛𝑀 ]. This algorithm makes it possible to calculate the individual contribution to the desired output energy from each basis vector. Because 𝒛𝑖 and 𝒛𝑗 are orthogonal for 𝑖 ∕= 𝑗, the sum of squares or energy of Ψ(𝑘) is 𝑀 ∑ Ψ𝑇 Ψ = 𝒉2𝑖 𝒛𝑖𝑇 𝒛𝑖 + 𝑬 𝑇 𝑬, (6) 𝑖=1

and this relation suggests to consider as the error reduction ratio due to 𝒛𝑖 the following quantity: /( ) (7) [𝑒𝑟𝑟]𝑖 := 𝒉2𝑖 𝒛𝑖𝑇 𝒛𝑖 Ψ𝑇 Ψ , 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝑀. The regressors selection procedure terminates at the 𝑀𝑠 th step when 𝑀𝑠 ∑ [𝑒𝑟𝑟]𝑗 < Ξ (8) 1− 𝑗=1

where 0 < Ξ < 1 is a chosen tolerance. This gives rise to a model containing only 𝑀𝑠 significant regressors. The orthogonal property makes the whole selection procedure simple and efficient and the tolerance Ξ is an important parameter in balancing the accuracy and the complexity of the final network. B. 𝐾-Means clustering algorithm The OLS method can be employed as a forward regression procedure [8] to select a suitable set of centers 𝑛𝑠 ≤ 𝑛𝑟 , from a large initial set of candidates, for the RBFN 𝑓𝑟 : ℝ𝑚 → ℝ. In the developed solution the initial set of centers is obtained using the 𝑘-means unsupervised clustering algorithm that starting from a reasonable high number of centers randomly chosen in the input space, moves them in the most significative regions of the input space [4] , [5]. The 𝑘-means clustering algorithm is given by the sequential execution of the following steps: (20)

Step 1 Initialize the cluster centers 𝒄𝑗 , 𝑗 = 1, 2, ⋅ ⋅ ⋅, 𝑛𝑐 . This centers are randomly chosen from the input data set {𝒘(𝑘)}𝐿 𝑘=1 . Step 2 Group all the inputs of the data set {𝒘(𝑘)}𝐿 𝑘=1 in the sets Ω𝑗 , 𝑗 = 1, 2, ⋅ ⋅ ⋅, 𝑛𝑐 , where Ω𝑗 is the set of input vectors 𝒘(⋅) closest to the cluster center 𝒄𝑗 , i.e. Ω𝑗 := 𝐿 {𝒘(⋅) ∈ {𝒘(𝑘)}𝐿 𝑘=1 ∣ ∥𝒘(𝑘) − 𝒄𝑗 ∥ = min𝑖=1 ∥𝒘(𝑘) − 𝒄𝑖 ∥}. ∑ 𝒘(𝑘), where Step 3 For all 𝑗 = 1, 2, ⋅⋅⋅, 𝑛𝑐 do 𝒄𝑗 = 𝑀1𝑗 𝒘(𝑘)∈Ω𝑗

𝑀𝑗 is the number of input vectors 𝒘(𝑘) ∈ Ω𝑗 . Step 4 If there is no change in the cluster assignments of step 3 from one iteration to the next the algorithm is stopped otherwise go back to step 2. C. “Width” of the radial basis functions Once the centers have been selected, the normalization parameter 𝛽 2 of Equation (2), that represents a measure of the spread of the data associated with each centre, has to be determined. No rigorous method exists to calculate this parameter. In the developed solution an Akaike-type criteria [8] is proposed for choosing the normalization parameter obtaining a good compromise between estimate accuracy and network complexity [4]. Akaike-type criteria which compromises between the performance and the number of parameters has the following form: 𝐴𝐼𝐶(𝜒) = 𝑁 log(𝜎𝜀2 ) + 𝑀𝑠 𝜒

(9)

where 𝜒 is the critical value of the chi-squared distribution with one degree of freedom and for a given level of significance, 𝑁 is the number of data set, 𝜎𝜀2 is the variance of the net estimate residuals and 𝑀𝑠 is the number of significant regressors. The procedure terminates when 𝐴𝐼𝐶(𝜒) reaches its minimum.

performed on data to distribute them evenly and to scale them into an acceptable range for the input neurons of the NN. This contribute to increase the NN ability to learn the association between inputs and outputs as well as to fasten significantly the calculations. In particular the set of experimental data is given by the pairs (𝒙(𝑛), 𝑦(𝑛)), 𝑛 = 1, 2, . . . where 𝒙(𝑛) is composed by the whether forecast, the number of day of the year, the hour of the day, and 𝑦(⋅) is the solar irradiation. The whiteness test on the prediction errors 𝑒(⋅) (residuals) has been also used for network validation [26]. The whiteness of residuals is usually evaluated by computing the sample covariances 𝑁 ∑ ˆ 𝑒𝑁 (𝜏 ) = 1 𝑒(𝑛)𝑒(𝑛 + 𝜏 ) 𝑅 𝑁 𝑛=1

(10)

with 𝜏 = 1, . . . , 𝑃 . If 𝑒(⋅) is a white-noise sequence, then the quantity 𝜁𝑁,𝑃 =

𝑁

ˆ 𝑁 (0))2 (𝑅 𝑒

𝑃 ∑

ˆ 𝑁 (𝜏 ))2 (𝑅 𝑒

(11)

𝜏 =1

will have, asymptotically, a chi-square distribution 𝜒2 (𝑃 ) [26]. The independence between residuals can be verified by testing whether 𝜁𝑁,𝑃 < 𝜒2𝛼 (𝑃 ), the 𝛼 level of the 𝜒2 (𝑃 )-distribution, for a significant choice of 𝛼. Typical choices range from 0.05 to 0.005. In Figure 2 the sample covariance is reported for the first data set considered; the Anderson whiteness test passes with 𝛼 = 0.05.

III. N ETWORK TRAINING AND VALIDATION Tests are based on data acquired from August 2010 to July 2011 during PV plant standard working. In particular two different data sets have been considered; one is relative to the first 12 days of each month and the second is relative to remaining days. These data sets are acquired with a significative time lag between them in order to assure independent data sets. This allows to test the considered network on significative data corresponding to two different time periods and gives a good test bed to evaluate the performance of the proposed algorithm on different conditions. As measure of the performance of the proposed algorithm the Root Mean Square of the Error 𝑒(⋅) (RMSE) and its Standard Deviation (SD) have been calculated. Only hours with daylight (irradiance greater than zero) are considered for the calculation of RMSE, night values with no irradiance are excluded from the evaluation. The first and the second set of experimental data are composed of 3564 and 5184 pairs of input and output samples, respectively. Sampling time is one hour and the data have been also normalized, between 0 and 1 in order to have the same range. Normalization is a widely used preprocessing method

Fig. 2.

Result of whiteness test.

The structure of the network has been confirmed by tests on the second data set (relative to the last 18 days of every month considered). These tests have been used to monitor the performance generalization of the network in order to avoid the overfit of the first data set. The network trained on the first data set (first 12 days of every month) has shown good capability to generalize the prediction of the irradiation on the second data set. Samples of the results on training the network on the first data set are shown in Figures 3 and 4, while the behaviour of

the network on the second set (validation) is shown in figures 5 and 6. The red-continuous line is the predicted irradiation, while the blue-dashed line is the measured one.

Fig. 6. Predicted (continuous red line) and Measured (blue dashed line) Irradiation for 20-23 May 2011

Fig. 3. Predicted (continuous red line) and Measured (blue dashed line) Irradiation for 1-6 February 2011

IV. E XPERIMENTAL I MPLEMENTATION The proposed RBF Neural Network forecast has been implemented to predict energy production of one of the PV plants of the italian company Energy Resouces, in Jesi (AN), Italy. The experimental setup is shown in fig. 7. It is a combination of hardware and software and includes nine solar inverters (8 of them connected to Renergies solar panel and the last to the battery pack), a lithium battery pack, a Battery Management System (BMS) and a software platform to develop forecast and management of the produced power. All communication between software and hardware is done through TCP/IP protocol, using serial to TCP/IP conveters. Using a simple model

Fig. 4. Predicted (continuous red line) and Measured (blue dashed line) Irradiation for 13-17 February 2011

Fig. 7.

Fig. 5. Predicted (continuous red line) and Measured (blue dashed line) Irradiation for 5-9 May 2011

TABLE I P REDICTION RESULTS OF THE RBF ALGORITHM . T HE FIRST AND SECOND DATA SETS ARE RELATIVE TO THE FIRST TWELVE AND THE LAST DAYS OF EACH MONTH BETWEEN AUGUST 2010 AND J ULY 2011, RESPECTIVELY.

DATA

RMSE

SD

MSE %

First data set

0.0667

0.0701

9.70

Second data set

0.0951

0.0997

11.22

Scheme of the whole system.

to estimate the power production, as described in [24], the hourly irradiation prediction can be used to generate power reference for every panel-inverter combination, in order to give generality to the forecast. The input parameters of the PV model are the irradiation (forecasted with the RBFN) and the module temperature; an approximate expression for calculating the cell temperature is given by: 𝑇𝑐𝑒𝑙𝑙 = 𝑇𝑎𝑖𝑟 +

(𝑁 𝑂𝐶𝑇 − 20)𝐼𝑟𝑟 800

(12)

where 𝑇𝑐𝑒𝑙𝑙 is the cell temperature, 𝑇𝑎𝑖𝑟 the air temperature provided by weather forecasts, Irr is the forecasted irradiation in [𝑊/𝑚2 ] and NOCT is the Nominal Operating Cell Temperature, a specif parameter for each kind of PV module (see [27]).

Defining 𝑅(𝑡) the power reference, 𝑃 (𝑡) the measured power and 𝜖(𝑡) the difference between them, after inizializing the system variables 𝑅(𝑡), calculated offline for every t value, 𝑢(𝑡0 ) and the sampling time 𝑇 𝑐 for the cycle (default Tc=5 seconds), the algorithm used to control the power flux for the 𝑘 − 𝑡ℎ cycle is the following: Step 1 Inizialize counter 𝑖 = 1; Step 2 Send the control signal to battery charger 𝑖𝑓 𝑢(𝑡) < 0 in order to charge the exceding energy; Step 3 Send the control signal to siac inverter 𝑖𝑓 𝑢(𝑡) > 0 in order to provide the lacking energy; Step 4 Measure the istantaneous power production 𝑃 (𝑡) of the PV plant; Step 5 Evaluate and store the difference 𝜖(𝑡) = 𝑅(𝑡) − 𝑃 (𝑡); Step 6 𝑖𝑓 i = 10 Calculate the control signal for the next 12 steps: 𝑢(𝑘 ∗ 10) = .. = 𝑢((𝑘 + 1) ∗ 10) =

10 ∑

𝜖(𝑛)/10 (13)

𝑛=1

and go back to step 1, otherwise 𝑖 = 𝑖 + 1 go back to step 2. The adopted solution is useful to limitate the cycles of charge/discharge (DOD) of the battery but at the same time introduces an error on the hourly energy provided to the power line of about 1/60 of the hourly energy reference. It’s possible to use this solution because the law proposal, discussed in section I admitted an error of maximum 10% on the effective hourly energy provided. In this section, the plant structure, the experimental setup and the results are discussed. A. Experimental Setup Experimental tests have been performed on a PV plant composed by 8 strings of Renergies 220P/220 polysilicon panels [28]. Each string is connected to a SMA Sunny Boy 1700IT solar inverter [29]. A lithium battery pack is composed by the series of two sub-module with 80 ThunderSky modules 40 Ah, a Battery Management System (BMS) and a battery charger for each module [31]. A solar inverter (model SIAC soleil 10Kw) is connected to this pack [30]. The parameters of the system are listed in table II , III, IV and V. TABLE II T ECHNICAL DATA I NVERTER S UNNY B OY 1700IT Input Data

Output Data

Max DC Pow

1850 W

Nominal AC Pow

1550 W

Max Efficiency

0.935

Max AC Pow

1.7 KW

Max Current

12.6 A

Max Current

8.6 A

B. Experimental Results In the following table are listed the results with the difference between the real energy produced on the day and the relative predicted one. Those results shows that, with our experimental setup (14 KWp of PV plant and 20 KWh of

TABLE III T ECHNICAL DATA I NVERTER S IAC S OLEIL 10 Input Data

Output Data

Recommended DC Pow

8 − 12 KW

Max AC Volt

400 V

Min-Max DC Volt MPPT Max Current

139-320 V

Max AC Pow

9 KW

29 A

Max Efficiency

0.93

TABLE IV BATTERY PACK S PECIFICATIONS Max Module Volt

300 V

Max Charge Curr

25 A

Min Module Volt

200 V

Max Discharge Curr

40 A

Module Capacity

10 Ah

Life Cycles (80% DOD)

3000 cycles

TABLE V T ECHNICAL DATA R ENERGIES 220P/220 Max Module Pow

220 W

Temp Coeff Pmax

−0.43 %/∘ 𝐶

Isc Current

8.5 A

Temp Coeff Isc

4.00 mA/∘ 𝐶

Voc Voltage

36.6 V

Temp Coeff Voc

−126 mV/∘ 𝐶

Max Pow Volt

28.11 V

Max Pow Current

7.87 A

TABLE VI RMSE AND PERCENTAGE ERROR BETWEEN TOTAL DAILY ENERGY PREDICTED AND MEASURED . DATA

RMSE

SD

ERR%

First data set

3.8667 KWh

550.4 W

7.70

battery pack) and with a simple PV model, we are able to provide the desired power reference to the power line. Only on 4 of the 120 days considered for the tests, the battery pack was not able to meet the prediction error. In the figure 8 are shown the measured and predicted power output of the PV plant and the corresponding trend of the battery State Of Charge (SOC), between 24 and 27 august 2011. Notice that the initial SOC of the battery is 50 %. V. CONCLUDING REMARKS We have presented a Neural Network based approach to derive hourly site-specific irradiance forecast on the horizontal plane (the same of the panels) from daily wheather forecasts as a basis to predict PV power output. The results of hourly forecast for the city of Jesi (AN), Italy, show a percentage RMSE of 12% on the validation set. The main factor that reduces accuracy of the forecast is, of course, the correctness of the wheather forecast provided by the web, moreover on what regards suddenly cloudness or cloudless of a sunny or a cloudy day; foregone weather changings don’t affect in a significative way the performances of the RBF network. The paper proves the possibility to provide the power line with a scheduled profile from PV plant’s production using a simple system of lithium batteries. An optimization of the capacity of the batteries and a new control strategy is under

Fig. 8. Power production (blue dashed line), predicted power reference (red continuous line) and Battery State Of Charge (blue continuous line) during considered period (24 to 27 August 2011)

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