# Forecasting of the monthly electric power supply in Birnin Kebbi, Kebbi State from (2010-2017)

Forecasting of the monthly electric power supply in Birnin Kebbi, Kebbi State from (2010-2017)

Introduction

1.1 Background of the study

One of the fundamental challenges of an electric utility company is to distribute the load required to various places. The foremost reason behind this task is that power demand in a given place does vary with growth in population and economic activities. Outcomes obtained from load distribution progressions are used in different endeavors of power sector such as planning, system expansion, maintenance and operational schedule. For example, there was a newly distribution station constructed along Sani Abacha bye pass road in Birnin Kebbi, which was applied for resolution of some basic power sector challenges. Some of the problems can be expansion planning, inter-tie-tariff setting and long-term modeling of capital investment. Short-term load distribution domino effects of few days to months ahead are considered necessary in unit commitment analysis, maintenance schedule and diagnosis of economic dispatch, Abraham A. and Nath, B. (2001).

Distribution is an essential and integrated process in planning and operation of electric power utilities. It involves the accurate forecast of both the magnitudes and geographical location of electric load over the different periods of planning horizon. The basic quantity of interest of distribution is typically the time period in relation to load demand studied. However, load distribution is also concerned with the prediction of hourly, daily, weekly, monthly and yearly increase in demand of electric distributions, Akaike, (1974).

A system network or grid is that part of power system which consists of the sub-stations and transmission lines of various voltage levels. The Nigerian electricity sub-sector is called Power Holding Company of Nigeria (PHCN). PHCN has a system comprising twenty six buses, forty transmission lines and fourteen generating stations which make up the Nigerian National Grid System; the system is developed upon 330kV and 132kV transmission infrastructural network. At various point of time, power demand in the country has been constantly out-weighing the available capacity. Power demand developed from 3233 MW in 2002 to 3479 MW in 2003 and 3403 MW in 2004. The maximum power demand in 2003 exceeded the available capacity of 3477 MW. A national target for electricity production has been advocated at different time from one regime of government to another. The more recent is the 20,000MW defined by the National Economic Empowerment and Development Strategy (NEEDS) in 2014. Incontrovertibly, some technical constraints needed to be resolved to strike this target, this includes upgrading of generation, transmission and distribution capacities, Darbellay, G.A. and Slama, M.( 2000).

A well identified situation of insufficient maintenance, regression of new investments as well as high rate of auto-generation has result in several catastrophic system failures, and eventually frequent power outages. Considering a point of fact, the demand of electricity in the country shall continue to increase due to continuous growth in population and increase quest for development. Consequently, this paper addresses an issue of electric load distribution in Birnin Kebbi Local Government, Kebbi State using ARIMA model. NEED. (1997).

1.2 Statement of the problem

In most locations in Nigeria, the distribution network is poor, the voltage profile is poor and the billing is inaccurate. As the department, which inter-faces with the public, the need to ensure adequate network coverage and provision of quality power supply in addition to efficient marketing and customer service delivery cannot be over emphasize. In summary some of the major problems identified are:

Weak and Inadequate Network Coverage;

Overloaded Transformers and bad Feeder Pillars;

Substandard distribution lines;

Poor Billing System;

Unwholesome practices by staff and very poor Customer relations;

Inadequate logistic facilities such as tools and working vehicles;

Poor and obsolete communication equipment;

Low staff morale and lack of regular training; and

Insufficient funds for maintenance activities.

1.3 Aim and Objectives

The aim of this research work is to forecast the electric load, which might be consumed between 2010 to 2017 in Birnin Kebbi, kebbi Statae.

The major objectives are as follows:

To determine the level of stationarity.

To determine the best model out of the models.

1.4 Significance of the study

This project will serve as reference for subsequent researchers who may intend to carry out studies related to this topic. It will also serve as reference point to aid government and private

1.5 Scope and limitation of the study

The validity of the result is to the secondary data obtained and covers the period of eight years (2010-2017) of electric load distributed, consumed and recorded at the power transmission station in Birnin Kebbi, Kebbi State. shares holders in their efforts to control the unnecessary power failure and provide adequate remedies in the forth coming years.

2.0 literature review

The energy reaching the Earth's surface from the sun is called solar energy. Most of this energy is in the form of radiation from the "visible" wavelengths. Visible radiation and radiation with shorter wavelengths, such as ultraviolet radiation are called "shortwave" (Graham, 1999). The Earth's climate system constantly adjusts in order to maintain a balance between the incoming and out-going radiation. This is referred to as Earth's "radiation budget". The components of the Earth system that are essential to the radiation budget are the Earth's surface, atmosphere, and clouds (Graham, 1999). Based on the principle of conservation of energy, the Earth's radiation budget comprises of the incident, reflected, absorbed, and emitted energies by the Earth system. This radiation budget represents the accounting of the balance between incoming radiation, which is almost completely solar radiation, and out-going radiation, which is partly reflected solar radiation and partly terrestrial radiation, including the atmosphere (National Aeronautics and Space Administration, 2010).

Solar energy is the most sustainable renewable energy source (Pettazzi & Salsón, 2012). It is more predictable than wind energy and less vulnerable to changes in seasonal weather patterns (Muneer et al., 2005). However, comprehensive information on the spatio-temporal pattern of monthly mean values of global solar radiation reaching the Earth's surface is required in the design and development of solar energy systems (Rehman & Ghori, 2000). This information on renewable energies such as solar is generally scarce in Africa (Monforti, 2011). Consequently, the level of solar energy development and utilization as an alternative source of power is very low resulting to millions of Africans lacking access to electricity.

Nigeria is endowed with abundant solar energy potentials which vary spatially and temporally. These potentials can be exploited to increase the energy supply mix in the country.Several studies have been carried out on solar energy potentials in different parts of Nigeria using empirical models such as Akpabio, Udo and Etuk (2004), Chiemeka (2008), Augustine and Nnabuchi (2009), Sanusi and Abisoye (2011), Falayi, Rabiu and Teliat (2011), Ibeh et al. (2012), Musa et al. (2012), Ogbaka and Silikwa (2016). These studies have focused mostly on southern Nigeria and some parts of northern Nigeria.

Few studies have assessed the solar energy potentials for various locations in northwestern Nigeria. Such studies include Sambo (1988) in Kano; AbdulAzeez (2011) in Gusau; Gana and Akpootu (2013) and Gana et al. (2014) in Kebbi; Mohammed et al. (2015) in Sokoto.These studies employed various statistical techniques to model solar radiation directly from insolation data, or, indirectly from other climatic variables.

An alternative approach to analyze solar energy potentials is the application of geospatial techniques (Rehman & Ghori, 2000; Ramachandra, 2007). Geospatial technique has the ability to analyze the spatial variation in solar energy potentials, and to determine the most suitable locations for harnessing solar energy in an area or region (Ramachandra, 2007).This study employs geospatial technique to map the potential sites for exploiting solar energy in Kebbi state; and to determine the spatial extent and the amount of solar energy that can be exploited in the study area.

3.0 Methodology

One mathematical approach for forecasting time series is known as the Box-Jenkins method and was suggested by Box and Jenkins (1970). Technically, the Box-Jenkins technique is an integration of the autoregressive and the moving average methods, so it is also named ARIMA (Autoregressive, Integrated, Moving Average) model. Since its first introduction, this ARIMA approach has become widely used in many fields such as specification, estimation, and diagnostic (Thomas 1983).

3.2 Statistical tools

3.2.1 Autoregressive Integrated Moving Average Process {ARIMA (p,d,q)}

This process was developed by BOX and JENKINS (1976) to help remove trends and uncover hidden patterns in non-stationary data. Once more, in the ARIMA model, the diagnostic involves checking the residuals between the forecast and actual series and determine if they are small, randomly distributed, and uncorrelated, if so the chosen ARIMA model is said to be a good fit.

ARIMA process first transforms non-stationary time series by differencing it a number of times, which is then modeled with an ARMA (p,q) process. As described above, non-stationary time series can often be converted to a stationary by simple differencing. Differencing a non-stationary time series a number of times into a stationary, this is a form of detrending the series i.e. to remove the trend component. Knowing that every stationary time series is integrated of order d (1(d)) and if by differencing the terms an ARMA(p,q) process, then the un-differenced process can have an ARIMA(p,d,q) representation in this case, the time series {} can be expressed as

(L) d X t = (L)t

A time series {} is said to be integrated of order d(1(d))

d=1,2,3,------, if is not stationary butis stationary and has the representation of the form

=

Where = is a stationary series and also represent the number of regular difference.

Therefore, ∆ = (1-L) which is the difference

= First () difference

=-

=-

=

Also the second () difference is given by

=

=-

= --(-)

=-2+

Where the difference operator ∆ is defined by

=

= -

= ∆(- = -2+

So in general,

= (-)

Therefore the process of applying the difference operator is called differencing and

ф(L) = 1-фL--____-

θ(L) = 1+θL++------+

and further assumed that ф(L) and θ(L) are co-prime ( i .e they do not have any factor in common order than 1) and none of them has (1-L) as factor, therefore letting

=

Therefore, the ARIMA representation now becomes

Ф(L) = θ(L)

Where is the white noise

P is the order of ф(L) of the AR characteristics polynomial and it donated the number of autoregressive terms

d donates the number of differencing required to remove trend

q is the order of θ(L) of the MA characteristics polynomial and

it donates the number of moving average terms.

3.2.2 Stationarity

A test of stationary (or non-stationary) that has become widely popular over the past several years is the unit root test. This test has been used to carry out or to know the order of integration. To use a data for analysis, this time series should be subjected to stationary conditions. A time series {} is said to be weakly stationary or wide sense stationary or covariance stationary or second order stationary if it satisfies the following three conditions.

E() = (i.e constant mean)

Var () = (i.e constant variance)

Cov() = (i.e covariance is independent of t) and

=

The most common unit root tests are ADF (Augmented Dickey-Fuller) test, KPSS (Kwiatkowski Phillip Schmidt Shin) test, PP (Phillip – Perron) test, DF-GLS test and NP test. It is important to know the above named tests.

Augmented Dickey-Fuller (ADF) test

This test was first introduced by Dickey and Fuller in 1979 to test for the presence of unit root and the ADF equation is given as

= +

The hypothesis testing is

(The series contains unit root)

(The series is stationary)

The test statistic:

Decision rule: Reject is less than the asymptotic critical value (tabulated)

Where

is the differenced series

is the immediate previous observations

is the optional exogenous regressor which may be constant or constant and trend.

and are the parameters to be estimated

is the coefficient of the lagged difference term up to lag p

Kwiatkowski Phillip Schmidt Shin (KPSS) test

The integration properties of a series may also be investigated by testing the null hypothesis that the series is stationary against the unit root. This test was introduced by Kwiatkowski, Phillip, Schmidt and Shin in 1992 to test for the present of unit root and then hypothesis testing is Vs

That is, the null hypothesis that the data generating process is stationary is tested against the alternative hypothesis (unit root). Kwiatkowski et al. (1992) have derived the test for this pair of hypothesis. If there is no linear trend term, they start from the data generating process

, where is a random walk i.e and and is a stationary process

Test statistics:

Where

is an estimator of

That is, is an estimator of the long-run variance ()

is a Bartlett window with truncated lag .

Decision rule: Reject the null hypothesis if the test statistic is greater than the asymptotic critical values (tabulated).

3.2.3 Model Checking

After estimation of model, the Box – Jenkins model building strategy entails a diagnosis of the adequacy of the model. More specifically, it is necessary to ascertain in what way the model is adequate and in what way it is inadequate. This stages of the modeling strategy involves several steps (Kendal and Ord, 1990).

A good way to check the model adequacy of an overall Box – Jenkins model is to analyze the residual from the model. The statistic have suggested determining whether the first k sample autocorrelation indicate the adequacy of the model and they are the Box – pierce statistics and the Ljung – Box statistic (portmanteau test). In spite of this, we can also check the model adequacy by examining the sample autocorrelation function of the residual (ACF) and the sample partial autocorrelation function of the residual (PACF). We can conclude that the model is adequate if there are no spikes in the ACF and PACF. We can also employ the Jargue – Bera test for non – normality of residual.

Portmanteau test

A portmanteau test used for investigating the presence of autocorrelation in time series. The Portmanteau test checks the pairs of hypothesis

i.e all lags correlations are zero

for at least one is tested and at least one lag with non-aero correlations

The test statistics has an approximate distribution if the null hypothesis holds.

An adjusted version with potentially better small sample properties was proposed by Ljung and BOX (1978) and is given as

We reject if the P-value is less than the significant level

Lomnicki-Jarque-Bera test for non-normality

Jarque and Bera (1987) proposed the test for non-normality based on the third and fourth moments or in the order words, on the Skewness and kurtosis of a distribution denoting by the standardized mode residuals i.e. , the test checks the pair of hypothesis:

And E i.e the distribution is symmetry and hence normal.

or i.e the distribution is asymmetry and hence non normal. This checks whether the third and fourth moments of the standardised residuals are constant with standard normal distribution. Denoting the standardised estimation residuals by , the test statistics is

Where

is the measure for the skewness of the distribution.

is the measure of kurtosis of the distribution.

The test statistics has an asymptotic - distribution if the null hypothesis is corrects and the null hypothesis rejected if LJB is large.

We reject the null hypothesis if the P-value is less than the critical value .

ARCH (Autoregressive Conditional Hetroscedasticity) test

The arch test of the residuals is performed to check if the residuals are consistent with a standard normal distribution. The ARCH test checks the pair of hypothesis

(The distribution is symmetric)

(The distribution is asymmetric)

The test statistics has an asymptotic - distribution. The null hypothesis is rejected if the test statistics is greater than the significant level

4.0 Discussion of the result

4.1.1 A graphical representation

Time plot of the series before difference

Figure 4.1: Time series plot before difference

There are possible shifts in both the mean and the dispersion over time for this series. The mean may be edging upwards, and the variability is increasing. This plot indicates that the data is not stationary.

ACF and PACF plots before difference

Fig 4.1 is the plot of ACF and PACF of Electric Load data from 2010 to 2017. The visual inspection of the plot shows that there is presence of autoregressive parameter in the model since in ACF there positive spike and in the PACF there is cut off after lag one and the moving average part have been with greater distance.

Figure 4.2: ACF and PACF plots before difference

The above ACF declines (spikes down) very slowly and all PACFs after lags one are statistically significantly different from zero while the other lags are statistically insignificant. This signifies that the above plots are non-stationary.

4.1.2 Unit root and stationary test before differencing

We used two methods to determine the order of integration of the series, ADF (Augumented Dicckey-Fuller) and KPSS tests. The test ADF test checks the null hypothesis of unit root against the alternative of Stationary for the data generating process. The KPSS test checks the null hypothesis of Stationarity against the alternative of a unit root for the data generating process. Table 4.1: Unit Root/Stationary Test

Test

Test Statistics

p-value/critical value

ADF Test

With constant

3.44336

0.9999

KPSS Test

At lag 2

3.90703

0.349 *

0.465 **

0.735 ***

KPSS Test

At lag 4

2.40179

0.349 *

0.465 **

0.735 ***

Key* at 10 **at 5 ***at 1

The above Result is for ADF and KPSS test before differencing. The ADF test of the result shows that the test statistic is greater than the tabulated value; this indicates that the data is non stationery.

KPSS test also reveal that at 10, 5 and 1 critical values they are all less than the test statistics at both lag 2 and 4. This shows that the data is non stationary and suggesting possible transformation for the data in order to make it stationary.

First Difference of the Electric Load Data

Time plot of the series after first difference

The series now appears stationary with respect to central tendency, so second differencing does not appear to be necessary. However, the variability seems to be increasing over time. Transformation is considered for series in which variance changes over time and differencing does not stabilize the variance. The transformed difference is plotted to see if both mean and variance are now stabilized, as seen in Figure 4.4.

Although the scale has changed, the transformed difference does not appear to have less variability than the untransformed difference. There is also the usual problem of increased difficulty of interpretation of transformed variables. Therefore, the untransformed difference is used in future analyses.

ACF and PACF plot after first different

Figure 4.5: ACF and PACF plot for electric load data after first difference

The plot of ACF and PACF of electric distribution data from 2010 to 2017 after first difference. The visual inspection of the plot shows that there is few if any autoregressive parameter in the model, since in ACF there is no such much spike and the PACF does not cut off after lag one and the moving average part have been with less distance.

3.1.3 Unit root and stationary test after first difference

Table 4.2: Unit Root/Stationary Test

TEST

LAG

TEST STATISTIC

CRITICAL VALUE/P-VALUE

ADF test with constant

2

-7.84855

1.181e-013

KPSS, without trend

2

0.213186

0.349*

0.465**

0.735***

KPSS, without trend

4

0.268431

0.349 *

0.465**

0.735***

Key* at 10 **at 5 ***at 1

Result for the ADF and KPSS test after differencing Test after the first difference, it shows that the data is stationary. ADF test statistics is less than critical value or p-value. Also KPSS test statistic is greater than the respective critical values at.

4.1.4 Model Identification

By the use of three information criterion, AIC, SCHW and HQ the following models were found and we are to determine the minimum value and consider it as our best model.

Table 4.3: Model identification using information criteria.

Model

Criteria

Akaike information criterion(AIC)

Hannan-Quinn (HQ)

Schwarz criterion(Sc)

ARIMA(3,1.0)

968.9919

974.6168

982.8454

ARIMA(3,1,1)

936.4775

943.2274

953.1016

ARIMA(3,1,3)

934.5738

943.5737

956.7393

ARIMA(2,1,0)

982.3901

986.8900

993.4728

ARIMA(2,1,1)

936.5242

942.1491

950.3776

ARIMA(2,1,2)

936.8948

943.6446

953.5189

ARIMA(2,1,3)

932.8744

940.7493

952.2692

ARIMA(1,1,0)

997.5544

1000.929

1005.866

ARIMA(1,1,1)

938.7516

943.2515

949.8343

ARIMA(1,1,2)

935.5415

941.1664

949.3949

ARIMA(1,1,3)

937.2789

944.0288

953.9030

ARIMA(0,1,1)

937.5886

940.9636

945.9007

ARIMA(0,1,2)

938.2008

942.7008

949.2836

ARIMA(0,1,3)

935.5898

941.2147

949.4433

From the above table we found that ARIMA (2,1,3) MODEL has the minimum value of all the criterion, there for ARIMA (2,1,3) MODEL is the model that best fit our data.

4.1.5 Model Estimation

ARIMA (2,1,3) model

Table 4.4: Model estimation

coefficient

Std. error

Z

P-VALUE

Const

0.0396855

0.0270288

1.468

0.1420

phi_1

1.06378

0.0643391

16.53

2.09e-061 ***

phi_2

-0.911028

0.0661152

-13.78

3.39e-043 ***

theta_1

-2.18479

0.0580705

-37.62

9.01e-310 ***

theta_2

2.11027

0.126213

16.72

9.38e-063 ***

theta_3

-0.925477

0.0843562

-10.97

5.26e-028 ***

The model parameters were found to be significant by comparing the choosing alpha () at 5 with respective p-value. It can be seen that the p-value is less than at 5 in almost all the estimates. This confirmed that the model parameters are significant.

4.1.6 Models Checking

Before the interpretation and use of the fitted model, we are to look at some tests to check whether the model is specified correctly. The following residuals test are applied for diagnostics model checking.

Test for autocorrelation, partial autocorrelation and correlograms.

Portmanteau test for residual, ARCH test for residual and Jarque-Bera test for non normality

ACF and PACF of the residuals

In testing the autocorrelation and the partial autocorrelation, if the residual is uncorrelated then the residual is adequate.

Figure: 4:6 Residual ACF and PACF plot for ARIMA (2,1,3)

These is the second part for the diagnostic checking, which contains the residual analysis of portmanteau test, jarque bera test and ARCH-LM test

Portmanteau test at lag 12

Portmanteau: 13.1983

p-Value (Chi^2): 0.0048

Ljung & Box: 14.3603

p-Value (Chi^2): 0.0083

ARCH-LM test at lag 12

test statistic: 2.1704

p-Value(Chi^2): 0.0291

F statistic: 0.1847

p-Value(F): 0.0488

JARQUE-BERA TEST:

test statistic: 2025.6509

p-Value(Chi^2): 0.0000

skewness: -2.0928

kurtosis: 23.0395

Interpretation

From the three tests that have been carried out to check for the adequacy of our model, we found that both the models pass the test. By looking at their test statistics comparing with their respective p-value, we found that p-value in all the cases are less than the relevant test statistics.

4.1.7 Model Forecast

ARIMA(2,1,3) MODEL

For 95% confidence intervals, z(0.025) = 1.96

Obs

v1

prediction

std. error

95% interval

2020:01

Undefined

2005.79

11.5991

(1983.06, 2028.53)

2020:02

Undefined

2011.71

15.1412

(1982.03, 2041.38)

2020:03

Undefined

2019.03

25.5192

(1969.01, 2069.04)

2020:04

Undefined

2024.92

40.9816

(1944.60, 2105.25)

2020:05

Undefined

2027.96

53.5662

(1922.97, 2132.95)

2020:06

Undefined

2029.22

59.7035

(1912.20, 2146.23)

2020:07

Undefined

2031.21

61.6237

(1910.43, 2151.99)

2020:08

Undefined

2035.67

62.5609

(1913.06, 2158.29)

2020:09

Undefined

2042.12

64.6488

(1915.41, 2168.83)

2020:10

Undefined

2048.43

69.8563

(1911.52, 2185.35)

2020:11

Undefined

2052.73

76.9926

(1901.83, 2203.63)

2020:12

Undefined

2054.97

82.3765

(1893.52, 2216.43)

2021:01

Undefined

2056.86

84.8711

(1890.52, 2223.21)

2021:02

Undefined

2060.28

85.9645

(1891.80, 2228.77)

2021:03

Undefined

2065.70

87.1959

(1894.80, 2236.61)

2021:04

Undefined

2071.87

89.9124

(1895.64, 2248.09)

2021:05

Undefined

2076.96

94.4828

(1891.78, 2262.15)

2021:06

Undefined

2080.21

99.0988

(1885.98, 2274.44)

2021:07

Undefined

2082.45

102.011

(1882.51, 2282.39)

2021:08

Undefined

2085.31

103.453

(1882.55, 2288.07)

2021:09

Undefined

2089.79

104.524

(1884.93, 2294.66)

2021:10

Undefined

2095.46

106.232

(1887.25, 2303.67)

2021:11

Undefined

2100.90

109.253

(1886.77, 2315.03)

2021:12

Undefined

2104.98

112.994

(1883.52, 2326.44)

2022:01

Undefined

2107.80

116.022

(1880.40, 2335.20)

2022:02

Undefined

2110.52

117.823

(1879.59, 2341.45)

2022:03

Undefined

2114.31

118.976

(1881.12, 2347.50)

2022:04

Undefined

2119.35

120.275

(1883.61, 2355.08)

2022:05

Undefined

2124.75

122.371

(1884.91, 2364.60)

2022:06

Undefined

2129.39

125.286

(1883.83, 2374.94)

2022:07

Undefined

2132.84

128.157

(1881.66, 2384.02)

2022:08

Undefined

2135.73

130.212

(1880.52, 2390.94)

2022:09

Undefined

2139.11

131.545

(1881.28, 2396.93)

2022:10

Undefined

2143.55

132.718

(1883.43, 2403.67)

2022:11

Undefined

2148.68

134.294

(1885.47, 2411.90)

2022:12

Undefined

2153.58

136.543

(1885.96, 2421.20)

Interpretations

ARIMA(2,1,3) was been used to make a forecast or predict the future expected value of monthly load for the people in Birnin Kebbi. We have forecasted only three years with the totals month of 24 using previous observation of 120 month which range from 2018 to 2020.

4.2 DISCUSSION

This study attempts to outline the practical steps which need to be undertaken to use the Autoregressive integrated moving average (ARIMA) in time series to models and forecast the electric load distribution in Birnin Kebbi. After the first difference our unit root test show that the data is stationary, because ADF test Statistic is less than critical value or P-value also KPSS test statistic is lees that its respective critical value at 10%, 5% and 1%. The model parameters were also found to be significance by comparing the choosing alpha at 5% with their respective P-value. To check the model adequacy, different tests were been use such as Jarque-Bera test, ARCH-LM test and Portmanteau test. Then ARIMA (3,1,2) was found to be the most parsimonious and make best used of it to forecast the electric load distribution in Birnin Kebbi.

CONCLUSION

The use of autoregressive integrated moving average (ARIMA) modeling strategy is now among the most popular way of analyzing time series data. We employed this method to model the Electric distributions in Birnin kebbi recorded from 2010 to 2018. The order of difference d was found to be one (i.e I(1)). ARIMA (2,1,3) model was found to be the most interesting consistent model to described and forecast the future values of Birnin kebbi electric distribution. The model was found to be adequate and best to describe the monthly electric distributions rate in Birnin kebbi Kebbi State . Jarque-Bera test for non normality was used and the forecast indicate that the model is adequate.

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