Five forecasting algorithms for energy consumption in Vietnam

TRAN Van Giang1,2

1Grenoble University, G2Elab 2Electric Power University, Vietnam

vangiang.tran@gmail.comvan-giang.tran@g2elab.grenoble-inp.fr

DEBUSSCHERE Vincent1

1Grenoble University, G2Elab vincent.debusschere@g2elab.grenoble-inp.fr

BACHA Seddik1

1 Grenoble University, G2Elab seddik.bacha@g2elab.inpg.fr

Abstract— This paper presents a comparative study of five forecasting models for electric consumption data from Hanoi city, Vietnam. These five forecast models are Adaptive Network Based Fuzzy Inference System (Anfis), Seasonal Auto Regressive Moving Average (Sarima), Multi layer Perceptron (MLP), Elman Recurrent Neural Network (Elman) and Cascade-Correlation Neural Network (CCNN). This study presents an efficient approach in selecting the input variables and the training data structure. The criteria of performance evaluation are computed to estimate and compare these five models. The results indicate in our case a preference for the Multilayer Perceptron, Elman and Cascade-Correlation model. In that the best accuracy prediction is given by the Cascade-Correlation model.

Index Terms-- Anfis, MLP, Elman, Cascade-Correlation, Seasonal Arima, Load Forecasting, Power System, Energy consumption, Vietnam.

I. INTRODUCTION

Electric load forecasting has been in recent years one of the major topics of research in power systems and Smart Grids. In the energy industry, load forecasting allows to make important decisions like purchasing and generating electric power, load switching, infrastructures development, etc. [1]. The methodologies for electric consumption forecasting are almost all based on two different techniques: statistical analysis and computational intelligence techniques [2].

The former include traditional statistical approaches such as nearest neighbor technique [3-4], linear and nonlinear regression analysis [5-6], autoregressive integrated moving average models (Arima) [7-8], Seasonal ARIMA (Sarima) [9] and Kalman Filtering models [10-11]. Most of the statistical based methods for load forecasting are based on linear models that make some assumptions about the characteristics of the load series. However, the relationship between the variables that affect load demand and the actual load demand is complex and nonlinear making the accuracy of the different statistical models system variable [12].

Box and Jenkins then have presented the seasonal time series Arima (Sarima) model [13]. It was successfully used in forecasting economic, marketing and social problems. While this model has the advantage of accurate forecasting over short periods, it also has the limitation in that at least 50 and preferably 100 observations or more should be used [13].

Another perspective for solving the load forecasting problem emerged with the growing of computational intelligence techniques including Artificial Neural Networks (ANN) [4,14-22], Support Vector Machine (SVM) [23], Fuzzy Systems (FS) [24], Wavelet-Arima models [25], Wavelet–ANN [26] and Neuro-Fuzzy System (ANFIS) [27-28]. From these models, the SVM model is adequate for predicting small samples of data, for which it is able to get the global optimum, but its forecast is very sensitive to the kernel function [29].

Wavelet - ANN models or Wavelet-ARIMA models can be used for very short-term prediction (several steps ahead) but not for long horizon prediction because they need a long period of time in training the model and easily fall into divergence [30].

FS models offer a very powerful framework for approximate reasoning as it attempts to model the human reasoning process at a cognitive level [31], with better accuracy than the forecasting models based on classical prediction methods and on the models applying unconventional techniques [32-33]. Takagi and Sugeno (1985) [34] have tried to apply human knowledge to reasoning processes without employing precise quantitative analyses. Unfortunately, there is no standard method for transforming the human knowledge or experience into the rule base of a fuzzy inference system. Besides, an effective method should be defined for tuning the membership functions in order to minimize the measure of the output error or maximize the performance index [35].

The ANN model is most widely used for load forecasting. It can learn the past behavior of variables and recognize the complexity and nonlinearity in the pattern of dataset [36].

ANFIS was proposed by Jang [35] with the combination of the advantages of both fuzzy logic methodology and neural network architecture, with the ANN good learning capability and adaptation, while conserving the FS strategy for dealing with reasoning at a high level.

The main objective of this study is to apply the performance of intelligent techniques like “Adaptive Network Based Fuzzy Inference System (ANFIS)”, “Multi layer Perceptron (MLP)”, “Elman Recurrent Neural Network (Elman)” and “Cascade-Correlation Neural Network (CCNN)” in comparison with the statical technique “Seasonal Auto Regressive Moving Average (Sarima)” to estimate hourly electric consumption values based on different electric load dataset. The next section shows the database that is used in our computation process. Section III summarizes the basics mathematics of ANFIS, MLP, Elman, CCNN and Sarima models. Section IV presents a procedure to build the models. Section V proposes some results of model parameters and discussion. Section VI concludes the paper and open perspectives.

II. REAL DATABASE

Four years hourly electricity load database from year 2004 to 2007 of the Hanoi city (Vietnam) were provided by EVNHanoi (Hanoi Electric Power Corporation, http://evnhanoi.vn). EVNHanoi’s main objectives are to work on electric power transmission, electricity distribution and electricity retailing.

Fig. 1 presents five weeks of Hanoi’s historical hourly load data from Monday, 1st March 2004 to Sunday, 4th April 2004. It shows that the load patterns during weekdays (Monday to Friday) differ from those on weekends. Other factors such as seasonal daily, weekly, monthly and yearly or holidays are also considered. The weather data is not used.

III. MODEL SYSTEM THEORIES

A. Multi-layer Perceptron Neural networks can be classified into dynamic (e.g.

Elman) and static (e.g. MLP) categories. MLP is the most widely used static networks, in which the input is presented to the network along with the desired output, and the weights

(bias) are adjusted iteratively for adjusting the desired network output. The MLP model is made up of a layer R of input neurons, a layer S of output neurons and one or more hidden layers; although it has been shown that for most problems one layer of hidden neurons is enough (Fig. 2) [37][50].

This figure shows that each layer has its own weight matrix W, its own bias vector b, a net input vector n and output vector a. The weight matrix for the first layer and the second layer are written as W1 and W2, respectively. There are R inputs, s1 neurons in the hidden layer and s2 neurons in the output layer. The output of the hidden layer is the input for the output layer.

The output layer can be considered as a network layer with s1 inputs, s2 neurons and s1*s2 weight matrix W2. The input to the output layer is a1 and the output is a2. This network has “logsig” neurons in its hidden layer, and “purelin” neurons in its output layer. The optimal topology can be detected with an only requirement that the hidden layer must have enough neurons. Too many neurons in the hidden layers may result in over-fitting.

B. Elman Recurrent Neural Network The Elman network (Fig. 3) [38][50] is commonly a two-

layer network with feedback from the first-layer output to the first layer input. This recurrent connection allows the Elman network to both detect and generate time-varying patterns. In this figure, the Elman network has “tansig” neurons in its hidden layer and “purelin” neurons in its output layer. This combination is special in that two-layer networks because its transfer functions can approximate any function (with a finite number of discontinuities) with arbitrary accuracy. The only requirement is that the hidden layer must have enough neurons. More hidden neurons are needed as the function being fit increases in complexity [39-40].

Note that the Elman network differs from conventional two-layer networks in that the first layer has a recurrent connection. The delay in this connection stores values from

Figure 2. Two layers of the Multi-layer Perceptron

W1

a1 = logsig(W1p-b1)

b1 -1

p R x 1

s1 x R

s1 x 1

n1

s1 x 1

s1 Rb2

a1

s1 x 1s2 x s1

W2

+ +

s2 x 1 s2

a2

s2 x 1 n2

s2 x 1

a2 = logsig(W2a1-b2)

-1

Input Hidden layer Output layer

Figure 1. Hanoi’s hourly load time series for five weeks from Monday, 1st

March 2004 to Sunday, 4th April 2004.

0 100 200 300 400 500 600 700 800 900150

200

250

300

350

400

450

500

550

600

time (hour)

load

(MW

h)

Figure 3. Elman recurrent neural network

a1(k) = tansig(IW1,1p+LW1,1a1(k-1)+b1)

-1

p R x 1

s1 x R

s1 x 1

n1

s1 x 1

s1 RD

+ + s2 x s1

s2

a2(k) s2 x 1 n2

s2 x 1-1

Input Recurrent tansig layer Output layer

IW1,1

b1

LW2,1

b2

LW1,1

a1(k) s1 x 1

a1(k-1) s2 x 1

a2(k) =purelin(LW2,1a1(k)+b2)

the previous time step, which can be used in the current time step. Thus, even if two Elman networks, with the same weights and biases, are given identical inputs at a given time step, their outputs can be different due to different feedback states. Because the network can store information for future reference, it is able to learn temporal patterns as well as spatial patterns. The Elman network can be trained to respond to, and to generate both kinds of patterns.

C. Cascade-Correlation Neural Network (CCNN) Cascade-Correlation (Fig. 4) [41] is an algorithm

developed by Scott Fahlman. To build feed-forward neural networks, a cascade-correlation learning algorithm which generates hidden neurons as they are needed can be used.

A cascade network differs from fully connected feed forward neural networks exploiting a fixed architecture in the way that the procedure begins with a minimal network that has some inputs and one or more output nodes as indicated by input/output considerations, but no hidden nodes. During the learning phase, the hidden neurons are added to the network and trained, one by one, while the training error decreases, thereby obtaining a multi-layer structure.

As a result, the neural network grows through the training algorithm up to a near optimal complexity which can be well generalized. The advantages of this kind of ANN are well known. First, no structure is predefined. The network is automatically built up from the training data. Secondly, it learns very fast because each of its neurons is trained independently to each other [40]. However, cascade-correlation networks can be over-fitted in the presence of noisy features [41-42].

D. Adaptive Neuro-Fuzzy Inference System An adaptive Neuro-Fuzzy inference system is a cross

between an artificial neural network and a fuzzy inference system. It is a fuzzy Takagi-Sugeno model put in the framework of adaptive systems to facilitate learning and adaptation [35]. The ANFIS architecture for two input variables x and y with two fuzzy sets A1, A2, and B1, B2 is shown in Fig. 5.

The ANFIS network is composed of five layers. Each layer contains several nodes described by the node function. Let j

iO denote the output of the i th node in layer j .

In layer 1, every node i is an adaptive node with an appropriated membership function corresponding to the input of node i :

)(1 xAO ii μ= , i = 1, 2 (1) or )(2

1 yBO ii −= μ , i = 1=3, 4 (2) Where x (or y ) is the input to node i and iA (or 2−iB ) is

a linguistic label associated with this node. 1iO is the

membership grade which specifies the degree to which the given input satisfies the quantifier iA . All the parameters in this layer are referred to as antecedent parameters.

In layer 2, every node i is a fixed node whose output is the fire strengths of the rules. For instance:

)()(2 yBxAO iiii μμϖ == , i =1, 2 (3)

In layer 3, every node i is a fixed node whose output is called normalized firing strength which represents the ratio of the i th firing strength rule to the sum of all firing strength rules:

21

3

ϖϖϖϖ+

== iiiO (4)

In layer 4, each node computes the contribution of the i th

rule to the overall output:

)(4iiiiiii cybxafO ++== ϖϖ , i =1, 2 (5)

Where iϖ is the output of layer 3 and { ia , ib , ic } is the parameter set. Parameters of this layer are referred to as consequent parameters.

In layer 5, the single node is a fixed node which computes the overall output as the summation of all incoming signals:

==i i

i ii

iiii

ffO

ϖϖ

ϖ5 (6)

E. Sarima modeling A time series can be defined as a set of values of a variable

during consecutive time increments. Time increments vary from series to series. The most comprehensive of all popular

Figure 4. Cascade Correlation Neural Network

Freezed

Trained

Activation

Output units

Inputs

Hidden

Figure 5. General ANFIS architecture.

A1

Layer 1

A2

B1

B2

∏

∏ N

N 1ϖ1ϖ

2ϖ 2ϖ

11 fϖ

22 fϖ

f

x

y

x

x

y

y

Layer 2 Layer 3 Layer 4 Layer 5

and widely known statistical methods used for time series forecasting are Box-Jenkins models [13].

The ARIMA model, which was introduced by Box and Jenkins, originates from the autoregressive (AR) model, moving average (MA) model, and the combination of AR and MA (ARMA) models, while AR, MA and ARMA are applied to stationary series. ARIMA models are used for non-stationary series. For seasonal time series, SARIMA models are used.

When a time series exhibits potential seasonality indexed by s, using a multiplied seasonal ARIMA(p, d, q)(P, D, Q)smodel is advantageous. The seasonal time series is transformed into a stationary time series with non-periodic trend components. A multiplied seasonal ARIMA model can be expressed as

ts

QqtDs

dPp eBBxBB )()()()( Θ=∇∇Φ θφ (7)

Where tDs

tDs xBx )1( −=∇ , P, D, Q are the order of

seasonal AR, differencing, and MA respectively. d is the differencing of non-seasonal ARIMA, )( S

P BΦ and

)( sQ BΘ are the seasonal AR(p) and MA(q) operators,

respectively, which are defined as: Ps

Psss

P BBBB Φ−−Φ−Φ−=Φ 2211)( (8)

QsQ

sssQ BBBB Θ−−Θ−Θ−=Θ 2

211)( (9)

Where 1Φ , 2Φ , …, PΦ are the parameters of the

seasonal AR(p) model, 1Θ , 2Θ , …, QΘ are the parameters of the seasonal MA(q).

IV. FORECAST PROCESS

A. MLP, Elman, CCNN and ANFIS model building To remove the annual seasonality (period = 12), we

reconstructed the original load data time series into 12 new time series sets corresponding to 12 months of the four years from 2004 to 2007.

From each new hourly load time series data, i.e. z(t), we use the z(t-120), z(t-96), z(t-72), z(t-48), z(t-24) and z(t) to predict z(t+24), where t 121. Therefore the format of the training data is: [z(t-120) z(t-96) z(t-72) z(t-48) z(t-24) z(t) z(t+24)].

Fig. 6 shows the selection of inputs and outputs for MLP, Elman, CCNN and ANFIS forecast models. Fig. 7 presents a forecast process for those modes.

For training those four neural networks models (MLP, Elman and CCNN) and hybrid (ANFIS), the most common learning algorithm is the back-propagation algorithm [43], in which the input is passed layer through layer until the final output is calculated, and is compared to the real output to determine the error. The error is then propagated back to the input adjusting the weights and biases in each layer. The

standard back-propagation learning algorithm is a steepest descent algorithm that minimizes the sum of square errors.

However, the standard back-propagation learning algorithm is not efficient numerically and tends to converge slowly.

In order to accelerate the learning process, two parameters of the back-propagation algorithm can be adjusted: the learning rate and the momentum.

The learning rate is the proportion of error gradient by which the weights should be adjusted. Larger values can give a faster convergence to the minimum but also may produce oscillation around it.

The momentum determines the proportion of change of the past weights that should be used in the calculation of the new weights [43].

An algorithm that trains an artificial neural network 10 to 100 times faster than the usual back-propagation algorithm is the Levenberg-Marquardt algorithm. While back-propagation is a steepest descent algorithm, the Levenberg-Marquardt algorithm is a variation of Newton's method [44].

For MLP, Elman and CCNN, a three-layered artificial neural network (only one hidden layer) trained by the Levenberg-Marquardt algorithm is proposed. All those models (MLP, Elman, CCNN and ANFIS) are developed for forecasting the next 24 hours or 168 hours of electric consumption data.

Figure 7. MLP, Elman, CCNN and ANFIS forecast process

Figure 6. Inputs and Outputs for MLP, Elman, CCNN and ANFIS models

Forecasting performance is expressed with the Mean Absolute Percentage Error (MAPE). It is usually expressed as a percentage:

100)(

)(ˆ)(11=

−=

n

i

xiz

izizn

MAPE (10)

Where )(iz is the actual data of the hourly electric consumption time series, )(ˆ iz is the forecasted data at the time step i , and n is the number of samples.

B. Sarima model building Spectrum Analysis [45], Unit Root Test (URT) [46] and

Augmented Dickey-Fuller Test (ADF) [47] allowed determining the cyclical patterns of time series data (the s) and detecting the orders of differencing (d and D). Autocorrelation function (ACF) and Partial autocorrelation function (PACF) allowed to select the orders p and q: ACF observation allowed to detect the order q of MA(q), and PACF is used to select the order p of the AR(q). The Akaike Information Criterion (AIC) [48] and the Bayesian Information Criterion (BIC) [49] are used with this selection defining the AIC as a priority.

AIC is a measure of the fits accuracy for statistical models. The AIC is formulated as

++=n

RSSnkAIC ln2 (11)

Where k is the number of parameters or regressors (k=p+q+P+Q), n is the number of time series database and RSS is the residual sum of square.

The BIC (Schwartz’s BIC) is expressed as

−=n

RSSnkBIC ln2)ln( (12)

The procedure needed to build a Seasonal ARIMA model is proposed in 5 steps [30]:

Step 1: To obtain the period s through spectrum analysis.

Step 2: To determine the coefficients d and D using the URT and ADF test.

Step 3: By differencing the coefficients d and D of the series, to obtain a stationary data using the ACF and PACF in order to detect the order of the series. The set of orders (p, q, P, Q) should be started with small values between 0 and 6 (the maximum of order is 6) but where p, P and q ,Q should not be null simultaneously in one set. The set of (p, d, P, Q) is optimized when the AIC (or BIC) value is minimum.

Step 4: To have the set of p, d, q, P, D, Q in order to estimate the coefficients of the model using an approximate

LSE to estimate: 1θ , 2θ , …, qθ of the non-seasonal MA(q),

1φ , 2φ , …, pφ of the non-seasonal AR(p), 1Φ , 2Φ , …,

PΦ of the seasonal AR(p), 1Θ , 2Θ , …, QΘ of the seasonal MA(q).

Step 5: To use the obtained model to forecast some future values of the time series. In our case it is used to forecast one day or one week of hourly electric consumption time series.

Forecasting performance of the Sarima model is compared with MLP, Elman, CCNN and ANFIS model using the Mean Absolute Percentage Error (MAPE) such as the equation (10).

V. RESULTS AND DISCUSSIONS

In order to reduce the computation time for training and coefficients identification of the models, we use 2688 hourly load data observations for training and validating the models of each test week.

Six first datasets from 12 new time series datasets (IV.A) will be considered in this paper. In each dataset, 2520 observations were used for training and identifying the models, and 168 observations time series data were used for validation.

In this study, six test weeks are considered corresponding to six months from January 2007 to June 2007. The chosen test week is the last week of those months. For example, the first test week for January 2007 has 168 hourly electric consumption observations from Monday (January 22, 2007) to Sunday (Jan 28, 2007).

Weekly Mean Error (WME) is considered such as equation (10) in which the number of forecasting points varies from n = 1 to n = 168:

100)(

)(ˆ)(168

1 168

1=

−=

i

xiz

izizWME (13)

Where )(iz and )(ˆ iz are actual and forecasted load of hour i , respectively.

Table I shows the best topologies of neural network models (MLP, Elman and CCNN) and the parameters of and Sarima models. The number of neurons in the hidden layer varies from 6 to 20. Different properties of the test week datasets have leaded to different numbers of hidden neurons of the neural network models or different coefficients of Sarima model. Note that the model parameters for the ANFIS model is not included in this table because it was fixed for all ANFIS models for six test weeks: number of nodes is 161, number of linear parameters: 448, number of nonlinear parameters: 36, number of training data pairs: 1188, number of checking data pairs: 1044 and Number of fuzzy rules: 64 are obtained.

Table II shows a comparison of weekly mean error for six test weeks using MLP, Elman, CCNN, ANFIS and Sarima models. Table III presents a comparison of time period for the training phase (in second). It shows that the fastest training time is given by CCNN, then MLP and Elman.The longest training time are given by ANFIS and then the Sarima models.

Note that the dataset of February, 2007 consists of an electric load time series for several days on the Vietnam Lunar New Year, which can cause difficulties in forecasting and lower the forecast accuracy. For the test week of this dataset, the Anfis and Sarima models generate poor results while the three neural network models can generate more accurate results. The CCNN model gives always better prediction accuracy.

These results allow to highlight the generalization ability of the various ANN used such as MLP, Elman and CCNN where they can give better prediction accuracy with a very short time of training phase. In that, the best accuracy is almost given by the CCNN methodology. Fig. 8 shows one week forecasting for the first test week with CCNN model. It demonstrates that this CCNN model can generate good forecast results for the working days and weekends from the second forecast day to the sixth forecast day, but less accurate results for the first forecast day and the seventh forecast day.

All the computations are run on a PC Dell-Optiplex 745, 2 core duo, 2.13Ghz, 3Gigabyte RAM, Windows XP SP3.

VI. CONCLUSION

In this paper we have presented a comparative study of five forecast models on a Hanoi’s electric load dataset. The proposed approach is based on Anfis, Sarima, MLP, Elman and CCNN methodologies. The application of neural network techniques such as MLP, Elman, CCNN presented better accuracy than the hybrid network technique (Anfis) or the statical method (Sarima). The average WME results of the neural network models (MLP, Elman, CCNN) are acceptables while those training and validating times are very shorts. The best results were given by the Cascade-Correlation Neural Network model with the best WME, fastest training and validating time. Future work will be focused on applying those models on developing the Neural Network Artificial Model Predictive (ANNPredictor) for electric load time series prediction in electrical network management in Vietnam. Furthermore, those techniques will be developed in order to build the data center workload prediction model (using neural network) for the Green Data Center project (EnergeTIC-FUI, 2010-2013) in France.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for their valuable comments, which helped them to clarify and improve the contents of the paper.

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TABLE III. TRAINING TIME FOR THE SEVEN FORECASTS (SECOND)

Test Week MLP Elman CCNN ANFIS Sarima

Jan.2007 4 2 2 264 302Feb.2007 4 2 4 323 349 Mar.2007 2 15 2 368 530 Apr.2007 7 19 2 289 611 May.2007 3 10 2 243 367 Jun.2007 3 4 3 327 369 Average 3.83 8.67 2.50 302.33 421.33

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300

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800

Time (hours)

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Actual CCNN

TABLE II. WME (%) FOR ONE WEEK ELECTRIC LOAD FORECASTING

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Jan.2007 4.77 4.78 5.08 6.14 5.69 Feb.2007 9.44 8.87 8.49 32.43 34.78 Mar.2007 5.01 4.71 4.27 9.11 6.00 Apr.2007 13.11 13.78 13.16 17.59 12.90 May.2007 10.44 9.10 7.99 23.81 13.94 Jun.2007 5.57 6.14 5.35 13.46 12.60 Average 8.06 7.90 7.39 17.09 14.32

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Jun.2007 6-9-1 6-6-1 6-14-1 (4,1,3)x(0,1,0)168

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