DEBIT MODELING USING ARIMA METHOD TO DETERMINE OPERATIONAL PATTERN OF SELOREJO DAM

Pitojo Tri Juwono 1 , Widandi Soetopo 1 and Bambang Pramujo 2 . 1. Lecturer of Water Engineering Department, Malang Brawijaya University. 2. Student of Water Engineering Magister, Faculty of Engineering, Malang Brawijaya University. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History


Partial Autocorrelation Function (PACF):-
Partial autocorrelation is used to measure the closeness rate of linear relationship between Z t and Z t+k if the effect of time lag 1, 2, .... k-1 is considered as separated (Makridakis et.al., 1988).

Time-Series Stationarity:-
As noted by Makridakis et.al. (1988), stationarity signifies a condition that there is no addition or reduction in the data. Data must be horizontal along with time axis. In other words, data fluctuation remains around constant mean, not depending on time and variance of fluctuation. Basically, it remains constant over times. Stationary time-series are those whose basic statistic is characterized by the constant mean and variant over times (Hanke et.al., 2003).
Seasonal ARIMA: -Hanke et.al. (2003) defined seasonal time-series as those with repeated change pattern. It changes every year. Seasonal prediction is then developed, usually involving selection between decomposition, multiplication, and addition, and then followed by estimating seasonal index from historical time-series. These indexes are used to put seasonal substance into the prediction or to separate certain effects of observation.
Seasonal ARIMA model is a part of flexible time-series model that is used to model seasonal data, including nonseasonal time-series. Seasonal ARIMA model is explained as following (Cryer, 1986;Box et.al., 1994): S is seasonal period length; B is back shift operator; and a t is white noise series with zero mean and constant variance.

Reservoir Operational Pattern:-
Pursuant to the Government Regulation No.37/2010 about Dam, it is said that dam management plan is aimed for managing water resource in the dam. This plan accomodates reservoir operational pattern. This reservoir operational pattern comprises of several items: 1. Dry-year operational pattern. 2. Normal-year operational pattern; and 3. Rain-year operational pattern.
Reservoir operational pattern explains about the method of releasing water from reservoir based on current volume, water elevation at reservoir, water demand, and river capacity at downstream part of the dam.

Operational Pattern of Selorejo Dam:-
Selorejo Dam is annual reservoir functioned as the controller of flood during rain season. The presence of reservoir is quite useful to provide irrigation water debit needed by Pare and Jombang Regions in dry season in 4 m 3 /second. It covers irrigation width of 5,700 ha in all irrigation systems of 22,000 ha, and it helps improving rice production of 7,500 tons/year. It is also useful to meet the demand of standard water for the industry. Water output from the dam is also useful for electric generation with power capacity of 1x4.500 kW, which then produces electrical power for ±49 millions kWh per year.  The graphic above showes time-series plot data of 10-days mean debit inflow in Selorejo Dam. It indicates seasonal pattern. Data are not yet stationary, and data variance is too big. Data are also not yet stationary against variance. Proving the stationarity against variance, and also the median, Box-Cox Plot is then made to identify data stationarity against variance adn ACF Plot to see data stationarity against median. Figure 2 shows that data of 10-days mean debit inflow of Selorejo Dam are tested with Box-Cox Test, and it obtains 1, meaning that data are not stationary against variance. Transformation of Box-Cox is then conducted resulting in =1. In first transformation, it is obtained that 10-days mean debit inflow of Selorejo Dam has been stationary against variance by =1 in Figure 3. Stationarity against Median:-When stationarity against variance has been proved, then stationarity test is performed against mean, and it is detected using ACF Plot.

Stationarity Test Against Variance:-
ACF Plot in Figure 4 indicates that data are not yet stationary, and therefore, it is made into stationarity with differencing at once (one-time). Stationary data are marked with unpatterned (random) lag which does not contain season.
ACF Plot is then made after one-time differencing. It is displayed in Figure 5 showing that data are stationary already. There is no seasonal substance and there is irregular pattern or also said as unpatterned.

ARIMA Model (P,D,Q)(P,D,Q) S For Water Debit Data:-
Based on ACF Plot after one-time differencing, there is cutoff after lag 2 and also seasonal cutoff at lag 36. Tentative model for non-seasonal AR is 1.2. Tentative model for seasonal AR is P = 1.2. One-time differencing process has produced orde d=1 and reasonal order D=1. Order q of MA Model can be determined from PACF Plot in Figure 6.
From PACF Plot, observation points of 10-days mean debit inflow are obtained and the characteristic of these points is dies-down. There are alternative models for MA Model at order q = 1.2 and also at seasonal MA order Q = 1.2 (assumed that it follows non-seasonal order). Based on ACF and PACF Plots, seasonal ARIMA tentative model is then prepared with seasonal order S = 36.  Model is feasible.
Model is feasible.
Model is feasible.
Model is not feasible.
Model is not feasible.
Model is feasible.
Model is not feasible.
Model is not feasible.
Model is not feasible.
Model is not feasible.
Model is not feasible.
Model is feasible.
Model is not feasible.
Model is feasible.
Model is not feasible.    Table 2.

The Comparison Between Data of Debit and Volume in Selorejo Dam:-
The following is a graphic that compares debit inflow across measurement results, precisely between those from ARIMA prediction and those from the planning of Perum Jasa Tirta I. The comparison is displayed in Figure 8.  Operational pattern with debit inflow as predicted from ARIMA (1,1,1)(2,1,1) 36 .    3. The following is the comparison between the reservoir operational pattern in Selorejo Dam using debit inflow from ARIMA Prediction, the plan made by Perum Jasa Tirta (PJT) I and all realizations of both. The comparison is shown in Table 7.  Table 7 indicates that reservoir operational pattern that is determined using debit inflow prediction from ARIMA model is closer to relization if understood from the difference in terms of volume and electric production.