FORCASTING OF RENEWABLE ENERGY LOAD WITH

RADIAL BASIS FUNCTION (RBF) NEURAL NETWORKS

Otilia Elena Dragomir

Automation, Computer Science and Electrical Engineering Department, Electrical Engineering Faculty

Valahia University of Targoviste, 18 Unirii Av., Targoviste, Romania

Florin Dragomir

Automation, Computer Science and Electrical Engineering Department, Electrical Engineering Faculty

Valahia University of Targoviste, 18 Unirii Av., Targoviste, Romania

Eugenia Minca

Automation, Computer Science and Electrical Engineering Department, Electrical Engineering Faculty

Valahia University of Targoviste, 18 Unirii Av., Targoviste, Romania

Keywords: RBF, Neural networks, Load renewable energy, Forecasting.

Abstract: This paper focus on radial- basis function (RBF) neural networks, the most popular and widely-used

paradigms in many applications, including renewable energy forecasting. It provides an analysis of short

term load forecasting STLF performances of RBF neural networks. Precisely, the goal is to forecast the

DPcg (difference between the electricity produced from renewable energy sources and consumed), for

short- term horizon. The forecasting accuracy and precision, in capturing nonlinear interdependencies

between the load and solar radiation of these neural networks are illustrated and discussed using a data

based obtain from an experimental photovoltaic amphitheatre of minimum dimension 0.4kV/10kW.

1 INTRODUCTION

Research efforts on artificial neural networks

(ANNs) for forecasting are considerable. The

literature is vast and growing. In the forecasting

works, the term “forecasting” is called also

prediction or prognosis. This reveals that there is no

consensual acceptation of term. Due to these facts,

in this article the forecasting will be associated with

the notion of prediction and will determine the

future state of the analyzed system the closest

possible to the future real state of the system (O.

Dragomir. 2010).

Different forecasting time horizons are employed

in prediction approaches (day-ahead, hour-ahead) in

relation with the application. Short term load

forecasting (STLF) samples the information on an

hourly (or half hourly) basis, or even a daily basis

(for load peak prediction) so is defined as varying

from a few minutes up to a few weeks ahead. This

type of forecasting is important because the national

grid requires DPcg (difference between the

electricity produced and consumed) values at any

moment in the day. Traditionally, hourly forecasts

with a lead time between one hour and seven days

are required for the scheduling and control of power

systems. From the perspective of the system

operators and regulatory agencies, STLF is a source

of primary information for safe and reliable

operation of the system. For producers also, this type

of forecasting is a basic tool for determining the

optimal utilization of generators and power stations,

as some facilities are more efficient than others.

In this context, this paper provides architecture

of RBF, capable to forecast the DPcg for short- term

horizon. The proposed structures are applied on a

data based obtain from an experimental photovoltaic

amphitheatre of minimum dimension (0.4kV/10kW),

located in the east-centre region of Romania, more

precisely in the city of Targoviste (ICOP- DEMO.

409

Elena Dragomir O., Dragomir F. and Minca E..

FORCASTING OF RENEWABLE ENERGY LOAD WITH RADIAL BASIS FUNCTION (RBF) NEURAL NETWORKS.

DOI: 10.5220/0003534204090412

In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 409-412

ISBN: 978-989-8425-75-1

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

1998). The paper is organized as follows: first, it

provides an overview of RBF neural networks,

which are the most popular and widely-used

paradigms in many applications, including energy

forecasting. Second, a particular RBF architecture is

proposed to forecast the DPcg. The forecasting

accuracy and precision in capturing nonlinear

interdependencies between the load and solar

radiation of these one are illustrated and discussed.

2 RBF NEURAL NETWORKS

Actually, the systems are very complexes and the

conditioning parameters that influence system

functioning are significant. In these cases it is very

difficult to determine any sort of model for

forecasting purposes. The advantages and the

drawbacks of ANNs, leaded us to RBF neural

networks as reference tools for our approach of short

term energy balance forecasting.

The RBF network is commonly used for the

purpose of modeling uncertain and nonlinear

functions. Utilizing RBF networks or modeling

purposes could be seen as an approximation problem

in a high-dimensional space (Zemouri. 2002). A key

feature of RBF is that the output layer is nerely a

linear combination of the hidden layer signals, there

being only one hidden layer.Therefore, RBF

networks allow for a much simpler weight updating

procedure and subsequently open up greater

possibilities for stability proofs and network

robustness in that the network can be described

readily by a set of nonlinear equations

In RBF networks, determination of the number of

neurons in the hidden layer is very important

because it affects the network complexity and the

generalizing capability of the network. If the number

of the neurons in the hidden layer is insufficient, the

RBF network cannot learn the data adequately; on

the other hand, if the neuron number is too high,

poor generalization or an over learning situation may

occur (Liu, 2004). The position of the centers in the

hidden layer also affects the network performance

considerably (Simon. 2002), so determination of the

optimal locations of centers is an important task. In

the hidden layer, each neuron has an activation

function. The gaussian function, which has a spread

parameter that controls the behavior of the function,

is the most preferred activation function. The

training procedure of RBF networks also includes

the optimization of spread parameters of each

neuron. (Martinez. 2008) studied the best

approximation of Gaussian RBF neural networks

with nodes uniformly spaced. Afterwards, the

weights between the hidden layer and the output

layer must be selected appropriately. Finally, the

bias values which are added with each output are

determined in the RBF network training procedure.

In the literature, various algorithms are proposed for

training RBF networks, such as the gradient descent

(GD) algorithm (Karayiannis, 1999) and Kalman

filtering (KF) (Simon. 2002). (Ferrari. 2009) studied

the multiscale approximation problem with

hierarchical RBF neural networks. But these above

RBF methods have the same defects of the

backpropagation algorithm. They are either

instability or complicate and slow. They have

proved that the connection weight of RBF neural

networks can be obtained through various learning

algorithms; therefore the weight has certain

instability.

3 PERFORMING STLF WITH

RBF

The forecasting performances of RBF neural

networks in load forecasting, are illustrated using a

dataset with 240 data points {y(t), u(t)}, representing

the radiation [W/m2] (mean value=0.9255 and

standard deviation= 97.6705) and the DPcg [kW]

(mean value=0.8156 and standard deviation=

130.9313) , obtained from a Solar Amphitheatre

(ICOP-DEMO. 1998) and (F. Dragomir et al. 2010).

The data used are normalized before starting the

training session and de-normalized at the end of the

training.

RBF neural network, used for performing STLF,

has an input layer, one hidden layer and an output

layer. The neurons in the hidden layer contain

Gaussian transfer functions, whose outputs are

inversely proportional to the distance from the center

of the neuron (see Table 1).

Table 1: RBF parameters.

Architecture RBF

Number inputs 1

Number layers 1 hidden layer with 5 radbas neurons

1 output layer with with purelin neurons

Transfer functions gaussian - hidden layer

purelin- output layer

Performance

functions

MSE (Mean Squared Error)

MAE (Mean Absolute Error)

Initial MSE goal 0.0098

Initial spread 0.02719

For the dataset, simulations are repeated 8 times.

ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics

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Dataset is divided into train and test subsets. 60% of

the data set is selected as the training data and

remained data set is selected as the testing data. For

each run, the number of neurons, deviations of the

radial units, MAE (Mean Absolute Error) and MSE

(Mean Square Error) are computed in order to reach

the MSE goal 0.0115. The measurements based on

MSE are suggestive, because it penalizes the huge

forecasting errors. The MAE is considered that

would be an adequate error measure if the loss

function were linear (and linear in percentage, not in

absolute error); however, recent studies and the

experience of system operators indicates that the loss

function in the load forecasting problem is clearly

nonlinear, and that large errors may have disastrous

consequences for a utility.

The goal of the tests is, given training and test

data, to choose the input parameters MSE goal,

spread and Hmax (hidden layer neurons number) to

minimize MSE value.

The input parameters have been initialized with:

MSE goal= 0.0098, the minimum distance between

clusters of different classes MNDST =0.8156,

spread0 = 0.2719 and Hmax0 = 60.

In training phase the following steps are repeated

until the network's mean squared error falls below

goal or the maximum number of neurons are

reached: 1) the network is simulated, 2) the input

vector with the greatest error is found 3) a radbas

neuron is added with weights equal to that vector

and 4) the purelin layer weights are redesigned to

minimize error.

Table 2: Train and test results of RBF simulations.

Train

Test

Spread

H MSE

(*10-3)

MAE

(*10-1)

MSE

(*10-1)

0.027 25 7.4966 4.9542 6.4668

0.127 15 7.8996 1.9653 1.4197

0.227 11 9.6653 0.6805 0.1354

0.327 8 6.4121 0.5751 0.0766

0.427 7 9.3638 0.5456 0.0786

0.527 7 8.6819 0.8884 0.1448

0.627 6 8.8989 0.5706 0.0730

0.727 7 6.1955 0.9046 0.1332

The general characteristics of the RBF training

are illustrated in Table 2.

Firstly, it was investigated how the spread of the

hidden layer base function affects the network’s

performance (see Figure 1). The initial downward

trend of MSE due to spread growth isn’t the same all

over training set. This indicates the need for

consideration of a second parameter in the

evaluation of RBF training performance. This is the

number of neurons in the hidden layer.

Figure 1: MSE in relation with spread for training phase.

The number of neurons in the hidden layer is

very important in design issue of an RBF network.

Therefore, the experiments have been conducted on

different RBF networks which has 6 neurons to 25

neurons located in the hidden layer. Using more

neurons than that is needed causes an over learned

network and moreover, increases the complexity of

the RBF network (see Table 2).

The predictions made by RBF neural network

over the test dataset in relation with the measured

outputs (targets) are illustrated in Figure 3. The

small number of test data has a bad influence over

the forecasting accuracy (see Figure 2). The output

of the RBF network is a measure of distance from a

decision hyper plane, rather than a probabilistic

confidence level The quality of the possible

solutions are calculated using MSE and MAE.

Figure 2: RBF outputs vs. targets in testing phase.

Figure 3 indicates the RBF testing errors with the

help of MAE and MSE. The Figure 3 shows that, the

growth of spread values until 0.3 has a big influence

over MAE and MSE values. These ones decreas a

lot, from 4.9542 to 0.5751 MAE and from 6.4668 to

0.0766 MSE. The trend change when the spread

reach 0.327 value. The error values increase and

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.002

0.004

0.006

0.008

0.01

spread

MSE

RBF training

0 2 4 6 8 10 12

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0.2

0.4

RBF prediction (testing phase)

input

output

real output

RBF output

FORCASTING OF RENEWABLE ENERGY LOAD WITH RADIAL BASIS FUNCTION (RBF) NEURAL NETWORKS

411

indicate that the optimal values for spread and

number of neurons in hidden layer has to be locate.

Figure 3: MAE and MSE values of RBF in testing phase.

At the beginning, Hmax was equal with the

number of training points. The training tests with

variable number of neurons in hidden layer have

showed that 8 is the optimum number for the

neurons in hidden layer, much less than the number

of training points. At the end of RBF training, the

optimum spread value found is 0.327.

4 CONCLUSIONS AND WORK IN

PROGRESS

This paper focus on a particular neural network, the

radial basis function neural network. Considering a

data based obtain from an experimental photovoltaic

amphitheatre and MSE and MAE metrics for

forecasting performance evaluation , the simulations

and tests made in this article, put in evidence the

accuracy and precision of the particular proposed

RBF structure in capturing nonlinear

interdependencies between inputs and outputs. Due

to its good capabilities to forecast the DPcg in

relation with solar radiation, this architecture in well

suited in STLF energy applications.

The work is still in progress and the developments

are at present extended to: training the radial layer

(the hidden layer) of RBF using the Kohonen and

LVQ training algorithms, which are alternative

methods of assigning centres to reflect the spread of

data, training the output layer (whether linear or

otherwise) using any of the iterative dot product

algorithms and improving the interpretability of the

obtained predictive system.

REFERENCES

European Research Programm ICOP-DEMO 4080/98,

“Building Integration of Solar Technology”

(http://dcem.valahia.ro)

J. I. M. Martínez (2008), Best approximation of Gaussian

neural networks with nodes uniformly spaced, IEEE

Trans. Neural Netw., Vol. 19(2), pp.284–298.

Karayiannis N. B. (1999), Reformulated radial basis

neural networks trained by gradient descent,. IEEE

Trans. Neural Netw. Vol.3, pp.2230–2235.

R. Zemouri, D. Racoceanu and N. Zerhouni (2002),

Réseaux de neurones récurrents à fonctions de base

radiales: RRFR. Application au pronostic, RSTI- RIA.

Vol.16(3), pp. 307-338.

S. Ferrari, M. Maggioni & N. A. Borghese (2004),

Multiscale approximation with hierarchical radial

basis functions networks, IEEE Trans.Neural Netw.,

Vol. 15(1), pp. 178–188.

Simon D.(2002), Training radial basis neural networks

with the extended Kalman filter, Neurocomputing,

Vol. 48, pp. 455–475.

The 18th IEEE Mediterranean Conference on Control and

Automation, Marrakech, Morocco (2010), Adaptive

Neuro Fuzzy Forecasting of Renewable Energy

Balance on Medium Term, Dragomir O., Dragomir F.

& Minca E.

The 25th European Photovoltaic Solar Energy Conference

and Exhibition (25th EU PVSEC) and 5th World

Conference on Photovoltaic Energy Conversion

(WCPEC-5, Valencia, Spain (2010) Control Solution

Based on Fuzzy Logic for Low Voltage Electrical

Networks with Distributed Power from Renewable

Resources, Dragomir F., Dragomir O.. Olariu N and

Minca E.

Y. Liu; Q. Zheng; Z. Shi & J. Chen (2004), Training radial

basis function networks with particle swarms,

Comput. Sci., Vol. 3173, pp. 317–322.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

spread

error

MAE and MSE for RBF testing phase

MSE

MAE

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