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CHAPTER 4

METHODOLOGY

The methodology for the development of forecasting models to

predict the energy requirement in India for the years 2010, 2020 and 2030 is

presented in this chapter. Also the development of an Optimal Electricity

Allocation Model (OEAM) for allocating electricity through various energy

sources is described.

4.1 DEVELOPMENT OF FORECASTING MODELS TO

PREDICT ENERGY REQUIREMENT IN INDIA

Since energy becomes a vital factor for future developments of the

country, a system of models has to be developed to provide forecasts of the

energy demand in various sectors. Over the past two decades, forecasting has

gained a widespread acceptance as an integral part of business planning and

decision-making. Forecasting is the predicting of future values of a variable

based on historical values of the same or other variables. Recent literature on

energy forecasting offers a broad range of forecasting tools. The energy

consumption is characterized by its large variations over time period. It varies

widely from year to year and exhibits seasonal variations. Formulating a

forecasting model that can accurately forecast the energy consumption is of

prime importance in energy system planning.

In the present work, an attempt was made to develop forecasting

models for consumption of commercial energy sources in India. This analysis

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utilizes regression techniques, double moving average method, double

exponential smoothing method, triple exponential smoothing method, Auto

Regressive Integrated Moving Average (ARIMA) model and Artificial Neural

Network (ANN) model (Univariate and Multivariate) for the energy

forecasting purpose of commercial energy sources such as coal, oil, electricity

and natural gas consumption in India. Figure 4.1 illustrates the schematic

representation of the development of forecasting models. The methodologies

of above-mentioned models are discussed below.

4.1.1 Time Series-Regression Methods

Regression is a statistical method of fitting a line through data to

minimize squared error. The actual number of data required depends mainly

upon the nature of the data. The objective of the regression is to estimate the

equation of a line through a set of data that minimizes the sum of the squared

differences between the actual data and the line. Regression analysis is used

to estimate the energy demand in different sectors. In time series technique,

past consumption data was used to predict future energy demand. These are

statistical techniques used when several years of data are available and when

relationships and trends are both clear and relatively stable. Time series

analysis helps to identify and explain any regularity or systematic variation in

the series of data which is due to seasonality, cyclical patterns that repeat any

two or three years or more, trends in the data and growth rates of these trends.

The following time series – regression techniques such as linear model,

exponential model, power model and quadratic model were used in the

analysis. The methodologies of these models are given below.

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89

4.1.1.1 Linear Model

In the linear model, the relationship that is used to fit the ‘n’

number of data is a linear equation in one variable. The linear model is of the

form,

baxy (4.1)

where y is the consumption of particular energy source in specific sector in a

particular time period, x is the time period in year and a and b are constants.

4.1.1.2 Exponential Model

Exponential growth models are appropriate for long-term

forecasting when growth rates are constant over time. An exponential growth

curve has a constant rate of growth at any point in time, resulting in

staggering increases over longer periods as the increase is compounded

exponentially. This model can be approximated by the exponential curve of

the following form,

)( btat eY (4.2)

where Yt is the time series at time t, e = 2.71828, a is the intercept and b is the

equivalent of the slope and denotes the growth rate. The above equation can

be transformed to make it linear, by taking natural logarithms of both sides of

the equation. The forecasting is obtained by the following equation,

)()(

btamt eF (4.3)

where m is the number of periods ahead to be forecasted.

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4.1.1.3 Power Model

The power model is also one of the time series regression

techniques considered for the forecasting. The power model is of the form,

baxy (4.4)

where y is the consumption of particular energy source in specific sector in a

particular time period, x is the time period in year and a and b are constants.

4.1.1.4 Quadratic Model

If the data appear to follow a quadratic pattern, then the non-linear

regression has to be applied. The quadratic model is of the form,

cbxaxy 2 (4.5)

where y is the consumption of particular energy source in specific sector in a

particular time period, x is the time period in year and a, b and c are constants.

4.1.2 Double Moving Average Method

The moving averages are applicable to time series data and in many

situations they are more appropriate and easier to use than regression

methods. The term “moving average” is used because, as each new

observation in the series becomes available, the oldest observation is dropped

and a new average is computed. One of the chief difficulties in applying

regression is the need to update coefficients of the least square equation

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whenever a new data point is obtained. Because a relatively large number of

calculations are required to update the equations, more suitable forecasting

techniques such as moving average methods have been developed. With the

moving average methods, forecasting equations can be quickly revised with a

relatively small number of calculations as each new data point is collected.

Furthermore, moving average methods are able to deal directly with serial

correlation in time series data while regression techniques are not. A moving

average is simply a numerical average of the last N data points that are used

for purposes of making a forecast. As each new data point is acquired, it is

included in calculating the average and the data point for the Nth period

preceding the new data point is discarded. If the data have a linear or

quadratic trend, the simple moving average will be misleading. In the present

work double moving average method was used as one of the methods for

forecasting.

To calculate the double moving average Mt[2], the simple moving

average Mt[1] was treated as an individual data points and a moving average of

these averages was obtained. The values of Mt[2] were used to determine the

equation for forecasting future energy demand. The forecasting equation is of

the form,

Tbay ttTt (4.6)

where T is the number of time periods from the present time, t, to the period

for which the forecasting was made. The values of at and bt were determined

by the equations,

]2[]1[2 ttt MMa (4.7)

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]MM[)]1N/(2[b ]2[t

]1[tt (4.8)

4.1.3 Exponential Smoothing Method

Exponential smoothing methods are one of the popular methods of

forecasting because they are easy to use, require very little computer time, and

need only a few data points to obtain future predictions. The exponential

smoothing method also eliminates the difficulties faced in the regression

methods. Another advantage of exponential smoothing methods over

regression methods is that it permits the forecaster to place more weight on

current data rather than treating all data points with equal importance.

Exponential smoothing methods also minimize data storage requirements

when calculations are performed manually or with a computer. The moving

average technique requires a relatively large amount of data storage.

Exponential smoothing methods have many of the same advantage as moving

average technique but require a minimum amount of data storage. The basic

exponential smoothing model is,

]1[

1]1[ )1( ttt SXS (4.9)

or

New estimate = [new data] + (1 - ) [Previous estimate]

The term (alpha) is called smoothing constant normally ranges between

0.01 to 0.3. The value of is obtained by trial and error, starting with a

certain value of and then increasing or decreasing it to find the value that

minimizes the percentage error. The double exponential smoothing method

and triple exponential smoothing method were used in the analysis.

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4.1.3.1 Double Exponential Smoothing Method

The single exponential smoothing cannot deal with non-stationary

data. The linear exponential smoothing is an attempt to deal with linear non-

stationary data. Its only difference from single exponential smoothing is that it

introduces extra formulas that can estimate the trend and subsequently use it

for forecasting. To develop an equation that takes account of a linear trend in

data, double exponentially smoothed statistics St[2] was calculated. The value

of St[2] was determined by the following relation,

]2[1

]1[]2[ )1(ttt SSS (4.10)

The initial values of S0[1] and S0

[2] were assumed in these

calculations. The initial estimates were not important since relatively large

amount of data was available for forecasting. The forecasting equation is of

the form,

Tbay ttTt (4.11)

where T is the number of time periods from the present time, t, to the period

for which the forecasting was made. The values of constants at and bt were

determined by the following relation,

]2[]1[2ttt SSa (4.12)

]SS[)]1/([b ]2[t

]1[tt (4.13)

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4.1.3.2 Triple Exponential Smoothing Method

The triple exponential smoothing is also called as quadratic

exponential smoothing. It is an extension of linear exponential smoothing. It

aims at dealing with trend of a higher order than linear trend. This is achieved

by introducing triple exponential smoothing which, in addition to the double

exponential smoothing, is used to remove quadratic trends. If the data used for

the analysis exhibits curvature in nature, then double exponential smoothing

is inadequate. In such cases, triple exponential smoothing is used. Triple

exponential smoothed data are sufficient for almost all practical applications.

The triple exponentially smoothed statistics were calculated as follows,

]3[1

]2[]3[ )1( ttt SSS (4.14)

The initial values of S0[2] and S0

[3] were also assumed in these calculations.

The forecasting equation is of the form,

2TcTbay tttTt (4.15)

where T is the number of time periods from the present time, t, to the period

for which the forecasting was made. The values of constants at, bt and ct are

determined by the following relation,

]3[]2[]1[ 33 ttt SSSa (4.16)

])34()45(2)56][()1(2/[ ]3[]2[]1[2tttt SSSb (4.17)

]2][)1(2/[ ]3[]2[]1[22tttt SSSc (4.18)

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4.1.4 Auto Regressive Integrated Moving Average (ARIMA) Model

A common approach to the forecasting is the Box-Jenkins time

series approach, which is used here for the forecasting of commercial energy

sources in India. It has the attracting features because it is an economical

approach, which can represent stationary and non- stationary stochastic

processes. The objective here was to build an Auto Regressive Integrated

Moving Average (ARIMA) model, which adequately represents the data

generating processes. The Box-Jenkins method involves the following four-

step iterative cycle. They are:

1. Model identification,

2. Model estimation

3. Diagnostic checking, and

4. Forecasting with the final model.

Forecasting with the estimated model is based on the assumption

that the estimated model will hold in the horizon for which the forecasts are

made. The AR (Auto Regressive) part of the model indicates the future values

of yt are weighted averages of current and past realizations. Similarly, the MA

(Moving Average) part of the model shows how current and random shocks

will affect the future values of yt. Since AR models are simple to estimate,

have well developed model selection criteria, and require limited pre testing,

they are the form of ARIMA used here.

In an autoregressive integrated moving average model, the future

value of a variable is assumed to be a linear function of several past

observations and random errors. That is, the underlying process that generate

the time series has the form

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yt 0 1yt-1 2yt-2 + · · · pyt-p t - 1 t-1 - 2 t-2 - · · · - q t-q

(4.19)

where yt t are the actual value and random error at time period t,

i j (j=0, 1, 2...q) are model parameters. p and q

are t,

are assumed to be independently and identically distributed with a mean of 2.

Equation 4.19 causes several important special cases of the ARIMA

family of models. If q = 0, then the model becomes an AR model of order p.

When p = 0, the model reduces to an MA model of order q. One central task

of the ARIMA model building is to determine the appropriate model order (p

and q). Based on the earlier work Box and Jenkins developed a practical

approach to building ARIMA models, which has the fundamental impact on

the time series analysis and forecasting applications. The Box–Jenkins

methodology includes three iterative steps of model identification, parameter

estimation and diagnostic checking. The basic idea of model identification is

that if a time series is generated from an ARIMA process, it should have some

theoretical autocorrelation properties. By matching the empirical

autocorrelation patterns with the theoretical ones, it is often possible to

identify one or several potential models for the given time series. Box and

Jenkins proposed to use the autocorrelation function and the partial

autocorrelation function of the sample data as the basic tools to identify the

order of the ARIMA model.

In the identification step, data transformation is often needed to

make the time series stationary. Stationary is a necessary condition in building

an ARIMA model that is useful for forecasting. A stationary time series has

the property that its statistical characteristics such as the mean and the

autocorrelation structure are constant over time. When the observed time

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series presents trend, differencing and power transformation are often applied

to the data to remove the trend and stabilize the variance before an ARIMA

model can be fitted.

Once a tentative model is specified, estimation of the model

parameters is straightforward. The parameters are estimated such that an

overall measure of errors is minimized. This can be done with a nonlinear

optimization procedure. The last step of model building is the diagnostic

checking of model adequacy. If the model is not adequate, a new tentative

model should be identified, which is again followed by the steps of parameter

estimation and model verification. Diagnostic information may help to

suggest alternative model(s).

This three-step model building process is typically repeated several

times until a satisfactory model is finally selected. The final selected model

can then be used for prediction purposes. For applying the above model a

package called Statistical Package for Social Sciences (SPSS) was used in the

present work and the step-by-step procedure is explained below.

4.1.4.1 Step by step procedure

The systematic procedure for applying ARIMA (Box-Jenkins)

model through SPSS (Statistical Package for Social Sciences) software

package for forecasting is given below.

Select one dependent variable and move it into the ‘Dependent

Box’.

Optionally, select one or more independent variables and

move them into the ‘Independent(s) Box’.

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Transform the series before estimation by choosing one of the options from the ‘Transform List’. The available transform options are:

o None. No transformation is performed.

o Natural log. Log transforms the series before estimation using the natural logarithm (base e).

o Log base 10. Log transforms the series before estimation using the base 10 logarithm.

The model group allows specifying the three parameters of the ARIMA model, autoregressive, difference and moving

average. These parameters are commonly referred to as p, d, and q, respectively. The corresponding seasonal parameters can also be defined if required.

Deselect the option named ‘Include Constant’ in equation if there is no need to estimate a constant term.

By clicking over ‘Save’, new variables were created, containing predicted values and residuals.

By clicking on ‘Options’ one can select convergence criteria, set initial values for the model and to choose how to display parameters in the output.

By following the above-mentioned steps, the ARIMA model was applied to the data and the forecasting of the commercial energy sources in India were arrived.

4.1.5 Artificial Neural Network (ANN) Model (Univariate and

Multivariate)

In general ANN’s are computational paradigms that implements

simplified models of their biological counter part, biological neural structures.

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Biological neural network are the local assemblage of neurons and their

dendrite connections that form the human brain. Accordingly, ANN’s are

characterized by local processing in artificial neuron i.e parallel processing,

which is implemented by the rich connection pattern between processing

elements. The basic building block of the ANN is artificial neuron. The

neurons are grouped together in parallel to form layers. The layers are

interconnected through the weighting factors. Signals can flow from the input

layer through to the output layer in two ways that is unidirectional or

bi-directional. In unidirectional connections the neurons are connected from

one layer to next but not with in the same layer. The first and last layers of

Feed Forward Neural Network (FFNN) are called the input and output layers

and those in between are termed as hidden layers.

4.1.5.1 Application of Neural Network in Forecasting

The data set involves inputs and outputs of the network. Inputs

were past energy consumption data, GNP (Gross National Product) and

population in the case of multivariate Artificial Neural Network (ANN) model

and only past energy consumption data in the case of univariate Artificial

Neural Network (ANN) model and energy demand of different energy sources

like coal, petroleum, electricity and natural gas as output in both the cases. In

the total data available, 80 % was used for training and remaining data for

validation purpose. Once the network is trained, it was used for forecasting

the future energy demand.

4.2.5.2 Simple Neuron Physiology

The simple neuron physiology is illustrated in Figure 4.2 that

depicts the major components of the typical nerve cell in the central nervous

system. The membrane is a permeable to certain ionic species and acts to

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maintain a potential difference between the intracellular and extra cellular

fluid. It accomplishes this task primarily by the action of sodium – potassium

membrane.

Figure 4.2 Simple neuron physiology

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4.1.5.3 General processing element

The general processing element of the Artificial Neural Network

(ANN) is shown in Figure 4.3. The individual computational elements that

make up the most artificial neural system models are rarely called artificial

neurons. They are more often referred as nodes, units or processing elements

(PE’s). It is not always appropriate to think of the processing elements in a

neural network as being one to one relationship with actual biological

neurons.

Figure 4.3 General processing element of ANN

Each processing element is numbered, the one in the Figure 4.3

being ‘i’. For example, like a real neuron the processing element has many

inputs but it has only a single output, which can fan out to many other

processing elements in the network. The input, ith receives from the jth

processing element as indicated as Xj. Each connection to the ith-processing

element has associated with it a quantity called a weight or connection

strength (from here both words are used interchangeably). All these quantities

have analogous in the standard neuron model. The output of the processing

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element corresponds to the strength of the synaptic connection between

neurons. In the present models, these quantities were represented as real

numbers.

Each processing element determines a net input value based on all

its input connections. In the absence of special connections, the net input have

been calculated by summing the input values gated (multiplied) by their

corresponding weights. In other words the net input to the ith unit can be

written as

Netij = Xj x Wij (4.20)

Where the index j runs all over connections to the processing element.

4.1.5.4 Network properties

Some of the important neural network properties are listed here.

The topology of a neural network refers to its framework as well as its

interconnection scheme. The framework is specified by a number of layers or

slabs and the number of nodes per layer. The types of layers include the input

layer, the hidden layer and the output layer.

4.1.5.4.1 The input layer

The nodes in the input layer are called input units, which encode

the instance presented to the network for processing. For Example, each input

may be designated by an attribute value possessed by an instance of the value.

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4.1.5.4.2 The hidden layer

The nodes in the hidden layer are called hidden units, which are not

directionally observable and hence hidden. They provide non-linearity for the

network.

4.1.5.4.3 The output layer

The nodes in the output layer are called output units, which encode

possible concepts or values to be assigned to the instance under

considerations. For example, each output units represent a class of objects.

4.1.5.4.4 Feed forward network

All connections point in one direction (from the input toward the

output layer).

4.1.5.4.5 Symmetrical connections

If there is a connection pointing from node ‘i’ to node ‘j’, then there

is also a connection from node ‘j’ to node ‘i’ and the weights associated with

the two connections are equal to notational Wij = Wji.

4.1.5.4.6 Asymmetrical connections

If connections are not symmetrical as defined above, then they are

asymmetrical. Connection weights can be real numbers or integers. They are

adjustable during network training, but some can be fixed deliberately. When

training is completed all of them should be fixed.

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4.1.5.5 Node Properties

The activation levels of nodes can be discrete (e.g. 0 & 1) or

continuous across in the range or unrestricted. This depends on the activation

(transfer) function chosen. If it is a hard limiting function then the activation

levels are 0 (or -1) and 1. For a sigmoid function the activation levels are

limited to a continuous range of real [0, 1]. The Sigmoid function at can be

mathematically given as

)e1/(1a xt (4.21)

where x is the input variable. In the case of a linear activation function, the

activation levels are open. The activation function is mentioned again in the

case of the system dynamics.

4.1.5.6 Back Propagation Neural (BPN) Network Algorithm

The schematic of the feed forward back propagation neural network

is shown in Figure 4.4. The network learns a predefined set of input-output

variable pairs by using two phases propagate- adept cycle. After an input has

been applied as stimulus to the first layer of neural network units, it is

propagated through each upper layer until an output is generated. This output

pattern is then compared to the desired output and an error signal is computed

for each output unit.

The error signals are then transmitted backward from the output

layer to each node in the intermediate layers organize themselves such that

different nodes learn to recognize different features of the total input space.

After training, when presented with an arbitrary input pattern that resembles

the feature the individual units learned to inhibit their outputs if the input

pattern does not contain feature that they trained to recognize.

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Figure 4.4 Schematic of back propagation neural network

As the signals propagate through the different layers in the network,

the activity present at each upper layer can be thought of as a pattern with

features that can be recognized by units in the subsequent layer. The output

pattern generated can be thought as a feature map that provides an indication

of the presence or absence of many different feature combinations at the

input. The total effect of this behaviour is that the BPN provides an effective

means of allowing a computer system to examine data may be incomplete or

noisy, and to recognize delicate patterns from the partial input.

The above theory can be summarized and given as step-by-step

algorithmic procedure, which is to be used for developing a C++ code, and

developed as general-purpose user interactive program for this present work

analysis. The various steps involved in the Feed Forward Back Propagation

Network (FFBPN) algorithm, which is used for the forecasting of commercial

energy sources, has been presented below.

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o Apply the input vector to the input units

Xp = (Xp1, Xp2, Xp3…Xpn) (4.22)

o Calculate the net-input values to the hidden layer units

netpjh = Wji

h Xpi (4.23)

o Calculate the outputs from the hidden layers.

ipj = fjh (netpj

h ) (4.24)

o Move to the output layer. Calculate the net-input

values to each unit.

netpko = Wki

o ipj (4.25)

o Calculate the outputs from the output layer.

Opk = fko ( neto

pk) (4.26)

o Calculate the error terms for the output units.opk = (Ypk - Opk) f k

o (netopk) (4.27)

o Calculate the error terms for the hidden units.opj = f jh (netpj

h) opk Wo

kj (4.28)

o Update the weights on the output layer.

Wkjo (t+1) = Wkj

o (t) + opk ipj (4.29)

o Update the weights on the hidden layer.

Whji (t+1) = Wh

jiopj Xi (4.30)

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where

h - Number of the hidden layer in the network.

j - Number of nodes in the hidden layer

k - Number of nodes in the output layer.

i - Number of nodes in the input layer.

w - Connection strength or Weight.

- Error between actual and predicted value.

- Learn Rate of the network.

O - Output demand calculated by the network

Xp - Input variable to the neural network

Y - Actual demand

The following sample calculations explain the step by step working

of C++ Artificial Neural Network (ANN) code used for energy demand

forecasting in the present study. The procedure was based on the Feed

Forward Back Propagation Network (FFBPN) algorithm.

Step 1: Generation of initial weights and applying the input vector

to the input units

Random numbers generate initial connection strengths or weights

by using the user defined functions. In general these weights may have any

value. At the end of the each iteration these weights will be modified

according to error calculated in the output layer and later they may assume

either positive or negative values, until the least Mean Square Error (MSE) is

arrived. Figures 4.5 and 4.6 show the modification of error in the starting and

final stages of the network training, respectively.

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0

0.05

0.1

0.15

0.2

0.25

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 Iterations

Figure 4.5 Error propagation in initial stages of weights modification

0.00E+00

1.00E-05

2.00E-05

3.00E-05

4.00E-05

5.00E-05

6.00E-05

7.00E-05

8.00E-05

9.00E-05

1.00E-04

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183

Iteration Number from last

Figure 4.6 Error propagation in final stages of weights modification

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In the present total electricity demand case, the network consists of

three input neurons, six hidden neurons and one output neuron. So, eighteen

weights are present between input to hidden layer and six between the hidden

to output layer.

Input to the neural network should always be in the range 0.0 and 1.0.

So, all the actual input values should undergo a process of transformation

called Normalization. Normalization is a process similar to interpolation

defined as the process of conversion of all inputs into the defined zone. In this

present work, intentionally the range was set between 0.1 and 0.9. This is

done to accommodate any further forecast beyond the year 2030. Here 0.1

was made equivalent to year 1950 data and 0.9 to year 2030 data. Due to the

unavailability of 2030 demand some arbitrary value was chosen from

regression analysis, whose validation error for the network was calculated.

Depending on the validation error and training set results, 2030 demand was

changed (similar to a trail and error procedure). The same process was

continued till the validation error was minimized. This way of the input data

optimization is required for the cases where the maximum value in input data

is not available.

As an illustration one data input is chosen and its propagation

through the network for the Artificial Neural Network (ANN) (Multivariate)

model is presented below. Table 4.1 shows the actual and normalized data for

the year 1950 consisting year, GNP, population and total electricity

consumption.

The weights for the input to hidden layer nodes are given in

Table 4.2 and the weights of the hidden to output layer nodes are listed in

Table 4.3.

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Table 4.1 Actual and normalized input data for year 1950 for the total

electricity consumption

Data Type YearGNP, Billion Rupees

Population Total Electricity Demand, GWh

Actual Data 1950 100 36321100 4156.61

Normalized Data 0.1 0.1 0.1 0.1

Table 4.2 Connection strengths of the input layer to hidden layer

W1 W2 W3 W4 W5 W6

W1 0.492965 0.235099 0.394299 0.001205 0.309076 0.063707

W2 0.029893 0.066073 0.491241 0.057970 0.483245 0.389294

W3 0.192404 0.194601 0.029237 0.178014 0.426878 0.260628

Table 4.3 Connection strengths of hidden layer to output Layer

W11 W21 W31 W41 W51 W61

0.379208 0.135166 0.472167 0.030244 0.201987 0.408155

The following Figure 4.7 gives the pictorial representation of the

multivariate ANN model.

Step 2: Calculate the net-input values to the hidden layer units.

The connection strengths or weights generated in the step 1 were

multiplied with the normalized data and were given as the input to the

corresponding node in the hidden layer.

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netpjh = Wji

h Xpi

netpjh (First node of the hidden layer) = 0.071526

netpjh (Second node of the hidden layer) = 0.049577

netpjh (Third node of the hidden layer) = 0.091478

netpjh (Fourth node of the hidden layer) = 0.023719

netpjh (Fifth node of the hidden layer) = 0.12192

netpjh (Sixth node of the hidden layer) = 0.071363

Figure 4.7 Pictorial representation of the multivariate ANN model

Step 3: Calculate the outputs from the hidden layers.

The net value calculated in the step 2 was sent into the hidden

neuron, where an activation function (sigmoid function) is applied. Figure 4.8

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f( netpj) Xp1 x W11

Xp2 x W21

Xp3 x W31

ipj

illustrates the transformation of data in the hidden layer node. After

activation, net value is the output of the hidden neuron to the nodes in the

output layer and is given as

ipj = fjh (netpj

h )

ipj = 1 / ( 1+ e - netpjh

)

Figure 4.8 Transformation of data in the hidden layer node

ipj (First node of the hidden layer) = 0.517874

ipj (Second node of the hidden layer) = 0.512392

ipj (Third node of the hidden layer) = 0.522853

ipj (Fourth node of the hidden layer) = 0.505929

ipj (Fifth node of the hidden layer) = 0.530442

ipj (Sixth node of the hidden layer) = 0.517833

Step 4: Move to the output layer. Calculate the net-input values to

each unit.

Each hidden node of the hidden layer gives single ipj value and the

same is multiplied with the weights of hidden to output layer.

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netpko = Wki

o ipj

netpko = 0.846314.

Step 5: Calculate the outputs from the output layer.

The value obtained from the previous step is subjected to the

activation or transformation by activation function, which gives the output

from the neural network.

Opk = fko ( neto

pk)

The output from the network was renormalized to get the predicted

value.

Step 6: Calculate the error terms for the output units.

The error terms for the output units were determined by the

following relation.

opk = (Ypk - Opk) f k

o (netopk)

Step 7: Calculate the error terms for the hidden units.

The error terms for the hidden units were determined by the

following relation. Since there are six nodes in the hidden layer, the number

of error terms for the hidden unit is also six.

opj = f jh (netpj

h) opk Wo

kj

op1 = -0.051185

op2 = -0.051219

op3 = -0.051144

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op4 = -0.051243

op5 = -0.051061

op6 = -0.051186

Step 8: Update the weights on the output layer.

During the training phase, weights were changed continuously until

the optimization of the weights was reached. Modification of the weights

depends on the error of the network. This part will be taken care of by the

back propagation algorithm. Modification of the weights depends on the learn

rate, which is defined as a parameter taking care of the magnitude of change

in the values of connection strengths. The weights for the hidden to output

layer were modified by the following equation

Wkj o (t+1) = Wkjo (t) + o

pk ipj

Table 4.4 Updated weights for the hidden to output layer

W11 W21 W31 W41 W51 W61

0.336742 0.093149 0.429292 -0.011243 0.15849 0.365692

Step 9: Update the weights on the hidden layer.

The weights for the input to hidden layers are modified by the

following equation and the values are given below.

Whji (t+1) = Wh

ji opj Xi

.

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Table 4.5 Updated weights for the input to hidden layers

W1 W2 W3 W4 W5 W6

W1 0.490918 0.23305 0.392253 -0.00084 0.307034 0.06166

W2 0.027846 0.064024 0.489195 0.05592 0.481203 0.387247

W3 0.190357 0.192552 0.027191 0.175964 0.424836 0.258581

In similar fashion, weights were modified until the least Mean

Square Error (MSE) is reached.

As per the methodologies of the various forecasting techniques

such as regression techniques, double moving average method, double

exponential smoothing method, triple exponential smoothing method,

ARIMA model and Artificial Neural Network (ANN) (Univariate and

Multivariate) models, energy forecasting was carried out for the commercial

energy sources such as coal, oil, electricity and natural gas consumption in

India.

4.2 DEVELOPMENT OF AN OPTIMAL ELECTRICITY

ALLOCATION MODEL (OEAM)

The utilization of commercial energy sources is increasing

enormously which will inevitably lead to a tremendous amount of

environmental pollution and global warming due to the green house effect.

Hence, it is essential to seek for non-polluting renewable energy sources for

the power generation in addition to the commercial energy sources. An

electricity model would facilitate the effective utilization of energy sources

for the power generation in India. In this chapter, an Optimal Electricity

Allocation Model (OEAM) was developed and presented in detail.

116

4.2.1 Fuzzy Linear Programming

In many practical situations, it is not reasonable to require that the

constraints or the objective function in linear programming problems be

specified in precise, crisp terms. In such situations, it is desirable to use some

type of fuzzy linear programming. In general fuzzy linear programming

problems are first converted into equivalent crisp linear or non-linear

problems, which are then solved by standard methods. The results of a fuzzy

linear programming problem are thus real numbers, which represent a

compromise in terms of the fuzzy numbers involved.

In the present research work, the constraints efficiency, emission

and carbon tax were considered as fuzzy linear constraints. These constraints

have the linguistic variables rather than exact quantitative variables to

represent imprecise concepts.

The membership function used in the fuzzy linear programming

model characterizes the fuzziness in a fuzzy set. In the present research study,

the data for the constraints such as efficiency, emission and carbon tax were

not available in single value. But it is available in the form of specific ranges.

Hence, it was identified to use the fuzzy logic concept for these three

constraints. In addition, it was identified to use the trapezoidal shaped

membership function. The intuition method was used in the present work to

assign membership functions to fuzzy variables. Intuition involves contextual

and semantic knowledge about an issue.

The Matlab code was generated to solve the fuzzy linear

programming. The input values such as unit cost of power generation from

different energy options, the predicted energy demand for the year 2020, the

potential of the various energy sources, the efficiency of the various energy

117

systems, the emission released from the different power plants and the carbon

tax imposed for the carbon emission were fed into the fuzzy linear program.

The program first converts the data into fuzzified values and the solution is

arrived, which was then defuzzified to get the actual output of the program.

The model was run for the electricity distribution to meet the electric energy

gap in India for the year 2020.

The Matlab code generated to solve the fuzzy based linear programming problem was programmed in such away that it can solve the problem with any number of variables in the objective function. In addition, it is possible to solve problems with any number of fuzzy linear constraint equations and variables.

4.2.2 Variables in Optimal Electricity Allocation Model (OEAM)

The possible energy options were considered in the model to meet the electricity demand in India. There are 20 energy options considered in the Optimal Electricity Allocation Model (OEAM) as shown in Table 4.6.

The model optimizes and selects the appropriate energy options for the power generation based on the factors such as cost, potential, demand, efficiency, emission and carbon tax. The objective function of the model is minimizing the unit cost of power generation. The other factors were used as constraints in the model.

4.2.2.1 Cost Factor

Since India is a developing country, the cost of power generation is very important in the economic point of view. The renewable energy systems typically have higher capital costs than fossil-fuelled systems, since all the fuel, equivalent over the useful lifetime was purchased at the beginning of the

118

Table 4.6 Different energy options (i) for the power generation (j) in

India

S.No Xi,j Different Energy Systems (i)

1.

2.

3

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

X1, 1

X2, 1

X3, 1

X4, 1

X5, 1

X6, 1

X7, 1

X8, 1

X9, 1

X10, 1

X11, 1

X12, 1

X13, 1

X14, 1

X15, 1

X16, 1

X17, 1

X18, 1

X19, 1

X20, 1

Coal based power generation

Diesel based power generation

Gas turbine power generation

Nuclear based power generation

Hydro based power generation

Wind based power generation

Biodiesel based power generation

Biomass gasifier based power generation

Biogas based power generation

Solid Waste based power generation

Cogeneration based power generation

Ethanol based power generation

Solar PV based power generation

Solar thermal based power generation

OTEC based power generation

Tidal based power generation

Geothermal based power generation

Mini Hydel based power generation

Fuel cell based power generation

MHD based power generation

system life. Emphasis on life-cycle costs and reduction of the risks of high

capital investments would be necessary for the success of renewables. Costs

for energy from renewable energy systems are expected to be reduced over

119

the next few decades, due primarily to higher production volume. But the cost

of power generation by means of commercial energy systems are found to be

increasing every year considerably. Incremental improvements are also

expected in efficiency, materials, reliability and application of the energy

systems.

4.2.2.2 Potential Factor

The magnitude of each energy resource was dependent on local

conditions. To optimize the use of these resources, better data was needed,

especially on their variation. The comprehensive understanding of local

conditions throughout the world, will take extensive effort, but each country

has indigenous energy resources that should be understood as part of the

national or regional energy planning.

4.2.2.3 Demand Factor

The World Energy Council (WEC) report made a conservative

projection for renewable energy systems, and according to the minimum

possible scenario for 2020, renewable would meet 3-4 percent of the total

energy demand. Under the maximum possible scenario with major policy

initiatives, renewable might provide 8-12 percent of the total energy demand

by the year 2020. The recent Energy Policy review concluded that renewables

offer considerable potential for displacing conventional energy sources, and in

some cases was already competitive with them, in addition to offering

environmental advantages. This emphasized the need to make detailed studies

at the country level, both to estimate the potential of each energy source, to

meet the future demands, and to highlight environmental and socioeconomic

benefits.

120

4.2.2.4 Efficiency Factor

The efficiency factor plays a major role in the selection of the

energy systems for power generation. Many of the technologies necessary to

make efficient use of various energy systems were quite immature and

relatively costly. Renewable energy systems have a relatively low energy

density in their raw form. To convert this to high energy density requires

tremendous costs. Research and development of the emerging, renewable

energy technologies need to be continued and expanded, if these options were

to be rapidly moved to maturity. Also, necessary R&D is required to increase

the efficiency of the commercial energy systems.

4.2.2.5 Emission Factor

The emission factor is very important in the environmental point of

view. The emission released by the power plants leads to the increase in

global warming and Green House Gases (GHG) emissions. Also, it affects the

health of the people living in and around the location of the power plants.

Also, the increase in emission level leads to the ozone layer depletion. It is the

need of the hour to curtail the emission released by the power plants. Since

the emissions are mainly from the fossil- fuelled power plants, these plants

should be replaced with non-polluting plants to some extent.

4.2.2.6 Carbon Tax Factor

Emissions of Green House Gases (GHG) are thought to be a serious

threat to the well being of humankind and other species. The principal culprit

is CO2, generated largely by the extensive use of fossil fuels for the power

generation. The CO2 concentrations in the atmosphere have risen from

285 ppm at the beginning of the industrial revolution to around 370 ppm

121

today, and are still rising. Carbon tax is the tax on CO2 emissions from major

energy sources, and in particular on the burning of coal, oil and natural gas,

unless the CO2 released from such plants are prevented from entering the

atmosphere through sequestration. The eventual rate of tax should be

calibrated to the desired reduction in CO2 emissions. The tax would thus

discharge the use of all fossil fuels relative to alternative renewables. The tax

rate should be set high enough to reduce CO2 emissions significantly.

The mathematical representation of the Optimal Electricity

Allocation Model (OEAM) is given in the following equations:

Minimizel

iijij XCZ

1 (4.31)

Subject to constraints

Potential ][k

P)X(ij

19

1k

m

1i

(4.32)

Demand l

1ijij DX (4.33)

Efficiencyl

ijijij DX

1 (4.34)

Emission ]][[ nnE T)X(ik

19

1k

m

1i

(4.35)

Carbon Tax ]][[ RnnE T )X( rik

19

1k

m

1i

(4.36)

122

where

C = Unit cost of the energy system

= Efficiency of the energy system

l = Number of energy systems for power generation = 20

m = Number of system in respective resources = 20

D = Energy demand (GWh)

k = Resources

P = Potential of sources (GWh)

En = Emission constant (g/GWh)

Tn = Target emission level (g/year)

r = Carbon tax (Rs/ton)

R = Projected carbon tax in 2020 (Rs/ton)

X = Quantum of energy (GWh)

i = Various energy systems

j = Power generation

Among the five constraints considered in the model, efficiency,

emission and carbon tax were considered as fuzzy linear constraints while

demand and potential were considered as ordinary linear constraint.

Here, the constraints such as efficiency, emission and carbon tax do

not have crisp values and hence they were treated as fuzzy linear constraints.

On the other hand, potential and demand were considered as ordinary linear

constraints since they have exact values.

The demand for electricity consumption during 2020 in India would

be 993385 GWh, which was predicted by the Artificial Neural Network

(ANN) forecasting model. The present electricity demand is 565102 GWh.

Hence; the electric energy gap for the year 2020 was calculated as

123

428283 GWh. This electric energy gap should be met by the various energy

options with special consideration to the emission level and carbon tax and

minimizing the unit cost as the objective function. Even though abundant

potential of solar, wind and biomass energy are available in India, factors like

quality of the resources, intermittent nature and technical feasibility would

decide the quantum of electricity utilization from different energy sources. In

this OEAM model, this was considered as potential constraint.

4.2.3 Various Energy Options

The various energy options for the power generation are shown in

Figure 4.9. There are twenty different energy systems namely coal, diesel,

gas, nuclear, hydro, wind, biodiesel, biomass gasifier, biogas, solid waste,

cogeneration, ethanol, solar PV, solar thermal, OTEC, tidal, geothermal, mini

hydel, fuel cell and MHD were considered in the model for the allocation of

electricity to meet the electric energy gap (demand) for the year 2020. The

various inputs to the OEAM model are listed in Table 4.7. The input values

have been fed in the fuzzy linear programming and the model was run for the

electricity distribution to meet the electric energy gap in India for the year

2020. Figure 4.10 shows the schematic representation of the Optimal

Electricity Allocation Model (OEAM). The inputs for different variables were

fed into the model and the model was run to obtain the optimal electricity

distribution pattern for the year 2020.

124

Centralized andDecentralized

PowerGeneration

Diesel

Ethanol

Biodiesel

Nuclear

Wind

Hydro

Biomass Gasifier

Solid Waste

Bio gas

MHD

Fuel Cell

OTEC

Solar

CoalGas

Solar PV Cogeneration

Tidal

Geothermal

Mini Hydel

Figure 4.9 Various energy options for power generation

125

Table 4.7 Inputs to the OEAM model

S.No. Different Energy Systems (i)

Unit cost,Rs. / kWh

Potential,*1010

kWh/year Efficiency,

%

CarbonEmission,

g/kWh1. Coal based power

generation 1.80 34194 29-38 960-1300

2. Diesel based power generation 4.50 767.36 34-42 690-870

3.

Gas turbine power generation 3.30 750.36 28-38 460-1230

4. Nuclear based power generation

3 131.88 33-43 9-100

5. Hydro based power generation

1 19.11 40-50 2-41

6. Wind based power generation

2.75 2.24 30-40 11-75

7. Biodiesel based power generation

4.60 5.067 30-35 550-690

8. Biomass gasifier based power generation

2 4.5518 20-28 37-166

9. Biogas based power generation

1.25 1.4112 22-26 6-10

10. Solid Waste based power generation

2.75 0.84 15-20 30-150

11. Cogeneration based power generation 1.50 3.36 70-80 23-41

12.

Ethanol based power generation 4 6 30-35 500-620

13. Solar PV based power generation 12

500000 10-17.5 Negligible

14.

Solar thermal based power generation 11.70 14-20 Negligible

15. OTEC based power generation 28 24.893 2-6 Negligible

16. Tidal based power generation 9 1.204 15- 25 Negligible

17. Geothermal based power generation 5 5.107 10-20 Negligible

18. Mini Hydel based power generation 2 1.197 40-45 Negligible

19.

Fuel cell based power generation 5 400000 50-60 Negligible

20. MHD based power generation 6 0.005913 50-60 Negligible