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Application of Data Mining Techniques as a Complement to Natural Inflow Uni-variable Stochastic Forecasting – A Case Study : The Iguaçu River Basin

Marcio Cataldi1, Carla da C. Lopes Achão2 and Luiz Guilherme Ferreira Guilhon1

1Operador Nacional do Sistema Elétrico (ONS) 2Planning Engenharia e Consultoria

E-mail: [email protected]

Abstract

This paper presents the results obtained from the utilization of a public dominion software that, through Data Mining and Neural Networks with Bayesian training is capable of laying the foundation for the selection of the most appropriate natural inflow forecast used in the PREVIVAZ stochastic modeling system. This technique utilizes precipitation information, forecasted and observed, a well as verified natural inflow data recorded over the weeks that precede the actual forecast target made at the water courses at the Foz do Areia and Jordão hydroelectric plants located in the Iguaçu River Basin. The results obtained indicate that the usage of these tools can provide a simple and efficient solution to reduce natural inflow forecast errors on a weekly forecast basis for the Iguaçu River Basin

Keywords: Data Mining; Bayesian Networks; Stochastic Models; Inflow Forecasts .1. Background

In May of 1993, at the Technical Meeting on Time-based Forecasts – SCEN, some companies in the electrical sector initiated the first steps for the contracting of the development of the PREVIVAZ Model by CEPEL. At that time, the participating companies in the GCOI (Coordinator Group for the Interconnected Operation), financed the project. The development of the project was started in the second semester of 1994 resulting in a first version (Version - 1.1) that was delivered in December of 1996. Since that time, the PREVIVAZ model has been used while undergoing improvements which terminated in March of 1998 when validation tests were conducted leading to a posterior writing of the Report entitled “The PREVIVAZ Model – Validation Tests – August/1998”. The document was approved by GCOI Resolution RS-G-2946/98 dated October 6th, 1998. The system was implanted by ONS, the national

independent electric system operator, in the watershed basins of the Brazilian National Interconnected Electric system, the so-called SIN. In February of 1999, ONS, the National Operator, began using the model in the periodic reviews made of the PMOs, the Monthly Operation Planning Reports. On April 4th, 2000, the Operator presented a comparative study made between the PREVIVAZ model used for elaborating the 1999 monthly stochastic forecasts by ELETROBRÁS and the monthly horizon PREVAZ model. General census conclusions of the study led to the substitution of the PREVAZ model by weekly inflow forecast reports based on the PREVIVAZ model used for elaborating the PMO reports as of May, 2000. The PREVIVAZ program is a model designed for making weekly average basin inflows consisting of alternative forecast methodologies for horizons up to six weeks on a stochastic basis utilizing combinations of stationary and periodic stochastic models with different estimation parameters and different kinds of transformations. The program calculates the forecasts for designated base hydro plants with at least 20 years of historical weekly forecasts.

2. Presentation of the Problem

The stochastic methodologies contained in the PREVIVAZ [CEPEL, 2004] model contemplate the autoregressive models and floating average models, with periodic and stationary structures, or in other words, AR(p) models and PAR(p) models where the value of “p” is up to the order of 4, and PARMA(p,q) models and ARMA(p,q) models where the value of “p” is up to the order of 3 and “q” equals 1. The transformations can be either logarithmic, Box & Cox or without transformation [Guilhon, 2003]. The parameter estimation methods are based on the Maximum Likelihood (ML) method which is used to measure general moments, single regressions and regressions in relation to forecast origins. The methods used for ascertaining estimation parameters are based on the ML method and the

Proceedings of the Fifth International Conference on Hybrid Intelligent Systems (HIS’05) 0-7695-2457-5/05 $20.00 © 2005 IEEE

GMM, the General Moments Method, the Single Regression Method and regressions related to the origin of forecasts.

The PREVIVAZ model divides historic data into two halves, estimating, for each week, the parameters for all the models for the first half and verifying the quadratic average error of the second half.

In the next step, the PREVIVAZ estimates the parameters for all the models for each week of the second half of the series and verifies the quadratic average error for the first half of the series. The averages, of the average quadratic errors, are then calculated for the two halves of all the models and are listed so that the model demonstrating the smallest average value is evidenced. After selecting the best model, the PREVIVAZ program once again, estimates the parameters, now considering the complete weekly historical data base and starts using the selected model with the new estimated parameters for each specific week. This modeling system however does not incorporate either observed or forecasted precipitation data, information which is fundamental for the composition of the natural affluent water inflow.

This paper will show however, that it is possible to improve the performance of the PREVIVAZ system, while maintaining its characteristic use as input, only natural water inflows, based on a selection criterion capable of facilitating the detection of the best model within a range of forecasted inflows by an auxiliary system that incorporates other data such as the information related to observed and forecasted precipitation.

With this aim in mind, Data Mining resources were applied to classify natural inflow forecasts from two sources located in the Iguaçu River basin, namely, the Foz do Areia and Jordão hydro plants.

A software application was used for this purpose, the WEKA1 Data Mining – Waikato Environment for Knowledge Analysis, a public dominion software, known for being easily portable and implemented in Java language which makes possible the application of a varied number of distinct technologies used for study and classification purposes. Among the techniques employed by the WEKA software used for the forecasting of the natural inflow ranges, test were run using Neural Networks and Artificial Intelligence such as ID3 and J48 decision trees of the Multi Layer Perception and Lazy types, classifying agents based on the rules of association and neural networks with Bayesian training and algorithm search automation based on the Hill-Climbingtechnique. In all the tests conducted, this last technique 1 http://www.cs.waikato.ac.nz/ml/weka/

presented the best results, when the main results of these applications, considering the correct natural inflows band, in Table 1. A brief description of the Bayesian classifier available in the WEKA software used in this study is presented below.

Table 1. Summary of mean errors for neural networks techniques application (8 years of training and 4 years for test, considering perfect precipitation).

TechniqueTraining

(% true prediction) Test

(% true prediction) Bayes network 74 73

Lazy Bayes 70 71 PART Rules 77 65 NNG Rules 84 68

J48 Tree 75 65 J48 Tree 75 65

Randomic Tree 94 67

3. Bayesian Theory

A basic premise of this theorema proposes that items and relations of interest are a manifestation of the laws of background probability distribution laws. It is therefore an approach that is essentially quantative to learning, looking at the problem as a choice of the best hypothesis within a space of hypotheses, or in other words, the choice that is most coherent with the data presented by the problem [Friedman et al. 1997]

The Bayes theorema, seen as one of the most important contributions to the theory of probabilities is also the main principal behind Bayesian learning.

The theorema can be summed up as follows: Theorema: If {A1, A2, ..., Am} is a partitioning of

the space of the results and B is a random happening, where P(B)>0, and for each i P(Ai)>0, therefore:

=

= m

iii

iii

ABPAP

ABPAPBAP

1)|()(

)|()()|( (1)

),...,1{ mi ∈where: P(A) Is a probability of occurrence of the event A P(A | B) is a probability of A conditioned by B,

defined as P (A B)/P(B)In a general way, we understand that P(A | B) as a

probability of event A happening, in view of event B, already happened.

An immediate consequence of this theorema can be applied to two different events, A and B, so that P(A)>0 and P(B)>0. In this case we can assume that:

)()()|()|(

BPAPABPBAP = (2)


In a Bayesian network training exercise, supposing that a set of data is designated by “D”, then, based on the previous theorema, we can calculate the probability of an occurrence of a hypothesis “h”, using as a base, the training data through the following relation:

)()()|()|(

DPhPhDPDhP = (3)

In this case, probability P(h | D) is the denominated “probability à posteriori” , an occurrence of an event in “h”, given that a determined event did in fact occur within the group of data “D”. The term P(h) is a “probability à priori” of the “h” hypothesis. The “à priori” probability represents a probability of occurrence not conditioned to the training itself but linked only to the group of data making up the training program. This probability is calculated by evaluating the initial probability of occurrence of each class within the training group.

Considering that we have a space of possible hypotheses “H”, then in a Bayesian network we can determine which is the best hypothesis “à posteriori”, taking into consideration the group of data observed in “D”. If we interpret as being the best hypothesis, (the most probable hypothesis), satisfying the data, that is – the hypothesis with the best “probability à posteriori”value, the value in question should be:

)|(maxarg DhPhMAP = (4) where h ∈ H.

Therefore, in equation number 4 (above), we can obtain the highest probability value for the occurrence of an event “à posteriori” - (hMAP) – taking into consideration the Bayesian probabilities training. The Bayesian classifier will create a group of probability tables organized in the form of a tree of probability tables that when unified through various knots, will form an uncyclical search group as shown in Figure 1. In this figure, the knots represent the dominion variables and the arcs represent the direct probabilistic dependency relationships between the variables that they connect. The probability of each table will be aligned with the Bayesian network configuration so that the group of variables “D”, is made up of “k” variables, where D = {xi, ..., xk}, with k>1. The Bayesian probability (PBS) of occurrence for each table will therefore be [Bouckaert 2004]:

}|))(|({ DddpadPPBS ∈= (6)

where - pa(d) - is the probability of each subgroup of “d” that composes the table.

Figure 1 – Example of a “Bayesian probability tree” for a section of the Iguaçu River upstream from the Foz de Areia

hydro power plant.

The highest Bayesian probability or the class most probable to occur will be identified through an uncyclical search through all the “k” variable tables. The search is made by using the Hill Climbingalgorithm. More details can be seen in [Buntine 1996] literature.

Attention is called to the fact that for resolution of the sixth equation, the Bayesian classifier requires three hyper parameters, insofar as the information contained in one of them – the α parameter – is sufficient in the Bayesian classifier version available in the Weka program. According to the value attributed to this parameter, the weight that each table of probabilities will have in the selection of the class showing the greatest probability of occurrence. Further information about the Bayesian classifier can be seen in Witten and Frank (2000) literature.

4. Description and Data Justification

With the objective of obtaining a satisfactory performance in the WEKA software tests, various configurations were used based on the combination of observed and forecasted inflow variables and observed and forecasted precipitation values. Its worth noting that the forecasted precipitation variables included in this study correspond to the values generated by the ETA model [Black, 1994] from the Centro de Previsão do Tempo e Estudos Climáticos – CPTEC, (Weather Forecasting and Climatic Studies Center – CPTEC) related to the water inflows in the Iguaçu River Basin (Foz do Areia and Jordão hydro plants).


The data used covered the period from 1994 to 2003. In order to validate the proposed methodology, data for two years was selected (2002 and 2003). The training period corresponding to the 2002 test was 1994 - 2001. The 2003 test utilized material from 1994 to 2002.

To compose the observed precipitation data, information was used coming from the upstream section of the Iguaçu River at the Foz de Areia plant, registered at twelve pluviometric control points. Additionally, data was gathered from two control points from the equivalent river section upstream from the Jordão plant.

Figure 2 below, shows the topographical region selected for the study as well as the control points, the ETA grade points and finally, the main hydro plants located along the Iguaçu River.

0 200 400 600 800 1000 1200 1400 1600Centro da grade do modelo ETAPostos pluviométricos

Aproveitamentos hidroelétricos

Figure 2 – Topography of the Iguaçu River Basin from the headwaters to the plant at Salto Santiago.

It is important to note that for each water course the best performance was achieved by applying Data Mining software, using a determined configuration, that is to say, based on the use of different groups of variables, observing the dependence of each group in relation to the other variables studied. The definition of the optimum inflow configuration - that allowed for the best performance in the tests held with the Data Mining program - resulted from the application of a group of tests, when various combinations among the available variables were evaluated. The selection of variables was made by using standard statistical analyses in the Data Mining studies, namely: Co-relation Matrixes, Dendograms and Main Component Analyses. The ranges of classes of these variables were selected starting with the analysis of permanence curves for each variable for the purpose of obtaining the classes that best characterize the flood and low-water periods, where the errors in the PREVIVAZ model are greatest due to the fact that this

model does not take precipitation data into consideration.

Some of the figures utilized in the analyses are presented below that represent a section of the Iguaçu River upstream from the Foz de Areia hydroelectric plant. It is important to mention that similar analyses were conducted so that a final composition of the variables presented could be obtained, variables presented in Figures 4 through to 7.

Based on these analyses and the running of various tests with the training group, a configuration was obtained that facilitated an improved forecast performance at the Foz de Areia hydro plant. This configuration contained the following variables:

- Observed average natural inflow for the week, previous to the forecasted week (Q-1).

- Observed average natural inflow for the forecasted week (Q)

- Forecasted average natural inflow for the week following the forecasted week (QPREV).

- Accumulated daily precipitation observed for a seven-day period (P).

- Accumulated daily precipitation forecasted for a 7-day period was divided in two blocks: PdM4 and PdM3, where PdM4 is the total accumulated forecast for the first 4 days from the forecast date and PdM3 is the last 3 days of the forecast period.

The average precipitation was pondered by the Kriging method. The calculation methodology and the analysis and discussion of their use were discussed by Cataldi and Machado (2004).

Figure 3 – Co-relation Matrix.

In Figure 3, the co-relation between variables used in the study can be observed. The dendogram representing the obtained groupings is shown in Figure 4. The analyses of the main components by variable and by historic period are presented in Figures 5 and 6.

Foz do Areia

Jordão


Q_1 Q P PDM4 PDM3 QM1

QM1 Q_1 Q PDM3 PDM4 P

Weeks

Figure 4 – Dendogram.

The analysis of Figures 5 and 6 demonstrates that the group of main components formed by the inflows, in general, is only capable of representing the group of data for the cases where only small variations are observed in the weekly natural inflows.

Figure 5 – Analysis of the Principal Components by Variable.

Figure 6 – Distribution of Principal Components in the Historic Series.

The ranges that make possible, the best performance of the Bayesian classifier were obtained from the probability distribution curves (Figure 7). The “optimal” configuration at the Jordão hydro plant not only contained the variables described above, it also included the observed average inflow recorded at the Foz do Areia plant. Indeed, the inclusion of this variable was justified in virtue of the high co-relation

between the inflows at these two plants. The number of inflow ranges and the precipitation recorded at theJordão plant were reduced, based on the analysis of the probability distribution curves representative of this water course.

0200400600800

100012001400160018002000220024002600280030003200340036003800

0 10 20 30 40 50 60 70 80 90 100

Figure 7 – Permanence Curve in percentiles of the total natural inflows at the Foz do Areia Plant (m3/s)

5. Results

The results presented below were obtained from the implementation of a new selection criterion that acted as a platform for the selection procedure used to identify the best model in the PREVIVAZ system, within a range of forecasted inflows, formed by the WEKA software Bayesian classifier.

Beginning with the observed and forecasted precipitation data and the observed and forecasted inflows related to the weeks preceding the forecasts, ranges were established for the variables used in the Bayesian classifier so that a classification could be made available for each forecast.

Starting with the forecasted weekly inflow ranges presented by the Bayesian classifier, interference was made in the selection of the best model in the PREVIVAZ system for all the weeks in which the forecast was observed to exceed the limits of the suggested range. In these cases a search was made to identify the forecast of the best model positioned within the ranking of the models used in the PREVIVAZ system, listed in the results report that was within the inflow range determined by the anchor model developed by the WEKA software. The alternative found to overcome this phenomenon was to search the models in the ranking whose forecast most closely approximated the forecasted range determined by the Bayesian classifier. This methodology was tested using “perfect and real” precipitation data. As explained previously the real forecast precipitation data was generated by the model ETA/CPTEC.

Comp 1 Comp 2 Comp 3

P PDM4 PDM3 Q_1 Q QM1


In this manner, for each water course and for each year studied (2002 and 2003), two sets of results were developed: one set considered as being the perfect precipitation forecast (obtained by the interpolation of observed data at the pluvial control points) and the other considered as being the real precipitation forecast as will be presented below. In Tables 2 and 3, the main results of this study are demonstrated. Table 3 presents the comparison of the quadratic average errors related to the natural inflow forecasts in the PREVIVAZ system for the years studied - 2002 and 2003, without the usage of the methodology developed in this work. Table 3 presents the result only for the weeks where it was possible to apply the selection criterion as proposed.

Table 3. Summary of quadratic average errors for 2002 and 2003 .

Plant Year Precipitation Forecast

Previvaz (%)

Previvaz with DM

(%) Perfect 22.8 2002 Real 28.5 26.2 Perfect 35.4

Foz do Areia 2003 Real 50.0 36.9

Perfect 37.2 2002 Real 39.6 34.9 Perfect 39.3 Jordão

2003 Real 47.0 37.6 Table 3. Summary of quadratic average errors for the weeks

in which the application of this methodology did not interfere in the results.

Plant Year Precipitation Forecast

Previvaz (%)

Previvaz with DM

(%) Perfect 34.7 26.6 2002 Real 35.7 31.1 Perfect 57.3 29.2

Foz do Areia 2003 Real 63.3 38.1

Perfect 40.6 36.1 2002 Real 40.2 31.2 Perfect 51.8 40.7 Jordão

2003 Real 51.8 38.2

6. Conclusions

This study demonstrated that the application of Data Mining techniques can be seen as an important tool for analyzing non-linear interactive variables such as those that make up the structure of hydrologic phenomenon. Among the techniques that were examined, the Bayesian classifiers were the ones that presented the best performance in the prediction of natural water inflow classes in the majority of the cases studied. In the selected years chosen for validation of the methodology, the results obtained with the interference of the Bayesian classifier showed an improved index of correct choices made by the PREVIVAZ model, in all situations, including those where precipitation

forecasts were used from the ETA model. One of the highlights of this study demonstrated that the forecast made for 2003 at the Foz do Areia water course, showed a reduction of 40% of errors in the weeks where the classifier interred directly in the results. It must also be mentioned that the alternative presented in this study requires only a very simple application that may be utilized at an extremely low computational cost.

Continued investigation of this methodology will now be replicated making use of the Iguaçu River section incremental to the Salto Santiago hydroelectric plant.

7. Bibliography

[1] CEPEL, “Modelo de Previsão de Vazões Semanais Aplicado ao Sistema Hidroelétrico Brasileiro – Modelo Previvaz”, Manual de Referência, 2004. [2] Guilhon, L.G.F. “Modelo Heurístico de Previsão de Vazões Naturais Médias Semanais Aplicado à Usina de Foz do Areia”, - Tese de Mestrado, UFRJ, 2003 [3] Friedman, N., Geiger, D., Goldszmidt, M., “Bayesian network classifiers”. Machine learning, 29:131-163, 1997. [4]Bouckaert, R. B., “Bayesian Network Classifiers in WEKA”, Internal Notes, 2004 [5]Buntine, W.L. “A guide to the literature on learning probabilistic networks from data”, IEEE Transactions on Knowledge and Data Engineering, 8:195-210, 1996. [6]Witten, I.H., Frank, E., “Data Mining: Practical machine learning tools and techniques with Java implementations”, Morgan Kaufmann Publishers, 2000. [7] Black T.L., 1994: NMC Notes: The New NMC mesoscale Eta model: Description and forecast examples. Weather and Forecasting, 9, 256-278[8] Cataldi, M., Machado, C.O., “Avaliação da previsão de precipitação utilizando a técnica de Downscale do modelo ETA e suas aplicações no setor elétrico”, XIII Congresso de Meteorologia, 2004


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