peak+load+forecasting

8/6/2019 Peak+Load+Forecasting

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Peak Load Forecasting

CHAPTER 1

INTRODUCTIONForecasting is the process of making statements about the events whose actual

outcomes has not been yet observed. It is a systematic effort to anticipate future eventsand conditions. As power system planning and construction requires a gestation period

of four to eight years or even longer for the present day super power stations energyand load demand forecasting plays an important role. Related to Power system

forecasting means estimating active load at various load buses ahead of actual load

occurrences. The importance of demand forecasting needs to be emphasized at alllevels as the consequences of under or over forecasting the demand are serious and will

affect all stakeholders in the electricity supply industry. If under estimated, the result is

serious since plant installation cannot easily be advanced, it will affect the economy,business, loss of time and image. If over estimated, the financial penalty for excess

capacity (i.e., over-estimated and wasting of resources). Nature of forecasts, lead times

and application are summarized in the table.TABLE 1. TYPES OF FORECAST AND ITS APPLICATION

Nature of forecast Lead time Application

Very short term A few seconds to severalminutes

Generation, distributionschedules. contingency

analysis for system security

Short term Half an hour to a few hours Allocation of spinning

reserve, operational planning

and unit commitment,maintanence and scheduling

Medium term A few days to few weeks Planning for seasonal peak winter, summer

Long term A few months to a few years Planning generation growth

Two approaches used for load forecasting is total load approach and component

approach. Advantage of total load approach is, it is much smoother and indicative ofoverall growth trends and it is easy to apply. On the other hand component approach

shows the abnormal conditions in growth trends of a certain component which prevents

the misleading forecast conclusions. All the forecasting techniques are based on theassumption that the load supplied will meet the system demand at all points of time. A

statistical analysis of previous load data is used to set up a demand pattern. Once this

has been done, this load model is used for predicting the estimated demand for selectedtime. The major task associated with forecasting is to select the best load model this canbe done by decomposing the given load demand at any given point of time into number

of distinctive components. The load is depending on industrial, commercial and

agricultural activities and the weather conditions like temperature, cloudiness, windvelocity, visibility and precipitation.

PEPS EEE Department AJCE 1


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Since there are few comparative studies available to help the forecaster to choose

wisely among the various forecasting schemes, some of which are already in use and

some of which are not yet developed to the point of practical implementation. Thispaper reviews six possible approaches to peak-demand forecasting but places greatest

emphasis on probabilistic monthly or weekly peak demand forecasting. Due to the vast

extent of relevant literature, only a few contributions to weather sensitive peakforecasting, will be pointed out here. A methodology for forecasting annual or weekly

and monthly peak loads, based on the decomposition of peak loads into nonweather

sensitive and weather sensitive components, was developed in [1], [2] and [3]. In [4], aweather-sensitive model for summer afternoon peak loads, applied to both long and

short term forecasting, using a normal distribution was developed. In [5], a different

regression model, used in forecasting is described. This methodology was modified in

order to apply for a practical system in [6].



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CHAPTER 2

METHODS FOR LOAD FORECASTING

Forecasting techniques may divide into three main classes. Techniques may be basedon extrapolation or on correlation or on a combination of both. Techniques may be

further classified as deterministic, probabilistic or stochastic. Methods are selected inorder to improve.

1. The accuracy of forecast

2. Comprehensiveness of forecast.

Six forecasting techniques used are

1. Energy and Load factor method

2. Extrapolation of annual peak demand

3. Extrapolation of demand data for a sampling of potential peak days

4. Separate extrapolation of trend and weather sensitive components of annual peakdemand.

5. Determination of annual peak demand, from weekly or monthly peak demand

forecast.6. Stochastic methods.

Method 1,2 and 3 are deterministic method and simple but do not provide estimates

of variance. Other 3 methods are probabilistic method and its advantage is theavailability concerning the uncertainty of the forecast. The methods are described

below.

2.1 ENERGY AND LOAD FACTOR METHOD

Energy forecasting can be done with more reliability than demand forecasts. Annual

energy can be converted into demand by multiplying it with load factor. Therefore it is

possible to obtain a demand forecast by combining the forecasting of energy and loadfactor which is superior in accuracy to a peak demand forecasted by direct methods. It

is easy to obtain energy forecasting because of the availability of data but forecasting of

load factor is just as difficult as the annual peak demand. The accuracy of load factor

forecast can be maintained but the work involved in forecasting is comparable withpeak demand forecasting.

Advantage of energy forecasting is the availability of data from which the energy

consumption of different classes of customers can be known.

Limitation of this method is that it does not provides estimates of the variance ofannual peak demand forecasts. This method is used for finding the seasonal peak

demands.



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2.2EXTRAPOLATION OF ANNUAL PEAK DEMANDS.

Extrapolation technique involves fitting a trend curves to basic historical dataset ofannual peak demand adjusted to reflect the growth trend itself. With the trend curve a

forecast is obtained by evaluating the trend curve function at the desired future point.

The effect of weather in peak demand is neglected by assuming that the same weathercondition prevails at the time of annual peak demand. The effect of economic

conditions on peak demand may be used in analysis by including an economic variable

when fitting the trend curve but this step is practically difficult.

The variance of annual peak demand can be computed in this manner but itsreliability is unsatisfactory because of limited amount of data. Twenty years of data will

provide twenty points for fitting the trend curve which is not enough for estimating the

variance. Extending the database is not practical even if adequate amount of historicaldata is available because the phenomena which produce the annual peak demand

changes with time. By this method seasonal peak can be obtained by fitting two trend

lines one to summer peak and other for winter peak.

2.3 MODIFIED EXTRAPOLATION OF ANNUAL PEAK DEMAND

The advantage of using this method is the use of extra data without resorting to a

longer data base. If we are taking a sample of six days on which peak demand occurs,

we can find out that the conditions on these days will be similar. Statistically these sixvalues may be treated in the same class as the annual peak demand, and a six fold

increase in the data results. The number of extra demand value used is arbitrary but

going beyond 10 or 12 risks including data points for which peak demand conditions do

not apply.

Typically modified procedure uses six data points over a period of 10 or 12 years.

The time of incidence associated with each demand value, a trend curve is fitted to thedata points using least square minimization. The effect of weather on the trend curve is

neglected by assuming that the same weather condition prevails at the time of peakdemand. It is not easy to treat effect of weather in this method because the dependence

of weather on peak demand changes from year to year. The effect of economic

conditions can be included by using an economic variable while fitting the trend curve.The forecast can be obtained by extrapolating the trend curve at desired time and

adjusting for expected economic conditions. By using this method it is expected to

provide variance of peak demand but the method fails to provide it when factorscontributing the peak demand changes with time. Another point is the reliability of

variance estimated will increase with number of data points used but its value will

increase with increase in data points.

The above three methods are simple and produces reasonable results in someinstances. Such techniques are called deterministic extrapolation since the random

errors in the data or in analytical model are not accounted.



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2.4 SEPARATE TREATMENT OF WEATHER- SENSITIVE COMPONENT OF

ANNUAL PEAK DEMAND

It is possible to observe the system demand at hourly, daily, weekly, monthly, byseasons, or annually the difference is in the rate of sampling of this continuously

varying demand. The sampling rate is important in order to differentiate the weathersensitive and nonweather sensitive components, which is obtained from the weatherload model which is determined from the data which is sampled more frequently in a

year. The weather load models must be updated annually due to the time variation of

weather sensitive demand. Hourly load data or weekly load data can be used for findingthe weather load model but weekly sampling of data gives an absolute minimum in the

number of data points also it gives adequate load model when correlated with

coincident dry bulb temperature. Once the weather load model have been determined itis possible to remove weather sensitive component from the annual peak demand data.

Modified extrapolation method can be used for the separate treatment of weather

sensitive and non weather sensitive data as follows.

1. Week day peak demand and weather data are used to determine weather

load model year by year or season by season2. The weather load models are used to separate the weather sensitive and non

weather sensitive components.

3. A trend curve is fitted for non weather sensitive components and isextrapolated to obtain the mean and variance of annual peak demand at

desired time.

4. Growth curves are fitted for the changing coefficients of weather load model

and are extrapolated to obtain the variance at desired time.5. The historical demand and weather data are used to determine the mean and

variance of the weather variable corresponding to annual peak demandconditions. It is assumed that the weather variable is normally distributed.

6. The forecasted weather load model obtained at the 4th step and the weather

statistics determined in the step 5 is combined to forecast the mean and

variance of weather sensitive component of future peak demands.7. The forecast of the non weather sensitive component obtained in step 3 is

combined with the forecast of weather forecast obtained in step 6 to make

the total peak demand forecast.

This forecasting scheme is acceptable on the basis of our knowledge in power systemgrowth.

2.5 FORECASTING ON A MONTHLY OR WEEKLY BASIS

Monthly or weekly peak demand forecast is necessary in situations where there is a

planning of Interchange energy requirement, Peaking capacity and Maintenance ofmajor plant as well as for economic studies.

Determining annual and seasonal peak demand forecast from weekly and monthly

forecast is considered as a superior approach, since it is possible to obtain annual peak

demand from monthly peak demand by applying monthly peak demand ratio. As



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weekly peak demand data is more suitable for obtaining weather load model it is used

over monthly peak demand data also a week is a precisely defined time period whereas

a month possesses an irregular number of potential peak demand days. The procedure issimilar with that of the method specified by the 4th method but in method describes

above it is assumed that the weather sensitive component is normally distributed but in

actual case it is not strictly normally distributed and it gives no indication of thedifficulties caused by the non Gaussian distribution of the weather sensitive component

of weekly peak demand. If it is assumed that the nonweather sensitive component of

weekly peak demand is normally distributed, and that its variance is several times thevariance of weather sensitive component, it is a reasonable approximation to assume

that the total load is normally distributed. The total peak demand forecast at some

future time has a mean value

D (t) = B (t) + S (t)

Where B (t) is the mean value of the non-weather sensitive component of weekly peakdemand and S (t) is the mean value of the weather sensitive component of the weekly

peak demand. Its variance is

2 D(t) = 2 B(t) + 2 S(t)

Although these results can be applied to any distribution, the usefulness of variance by

setting up a confidence interval for forecasts does depends on the total forecast beingnearly Gaussian.

Having obtained a weekly peak demand methods should be formulated to extract

information concerning the annual or seasonal peak demands. It should be noted that

the peak demand forecasted for a particular week is not a number rather a probabilitydistribution. Although we can represent the distribution by means of its mean and

variance but these two variables is not sufficient to describe the distribution. It is

temporarily assumed that the distribution of weekly peak demand is known and is notnecessarily Gaussian. Let Pk(x) be the cumulative probability distribution of peak

demand of kth week and pk(x) be the probability density function of kth week. The

problem is to determine the annual or seasonal peak demand from 52 differentprobability distribution. The probability density function for annual or seasonal peak

demand is given by

Where N=52 for annual peak and N=26 for seasonal peak. The result is quite generaland time consuming to evaluate using a digital computer. This can be simplified by

considering only those weeks where annual or seasonal peak occurs, in this way we can

reduce the value of N there by reducing the computational effort.

Procedure can be summarized as follows

1. Forecast both mean and peak demand for each week



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2. Assume the weekly peak demand is normally distributed and calculate the

probability density function of annual peak demand that includes only those

weeks in which annual peak demand occurs.3. Calculate mean and variance of the annual peak demand using

2.6. STOCHASTIC METHOD

Stochastic model involves developing a probabilistic model whose output is electrical

demand. Stochastic model take the form of difference equation since demand is

expressed in discrete time series perturbed by random inputs and the forecast can beobtained by solving these difference equations.

Stochastic process is concerned with the sequence of events governed by

probabilistic laws it may be defined as A stochastic process X = { X(t), t T } is acollection of random variables. That is, for each t in the index set T, X(t) is a random

variable. We often interpret t as time and call X(t) the state of the process at time t . If

T is a countable set then we have a discrete stochastic process and if T is a continuousset then we have a continuous time stochastic process.

In most case stochastic variable has both expected value term and a random

term the stochastic process forecasting for a random variable X, as a forecasted value

(E[X]) plus a forecasting error, where error follow some probability distribution. So:X(t) = E[X(t)] + error(t).

Gaussian process is a stochastic process whose realizations consist of random values

associated with every point in a range of times (or of space) such that each such random

variable has a normal distribution. Moreover, every finite collection of those randomvariables has a multivariate normal distribution.

Gaussian process may be defined as a stochastic process {Xt ; t TT} for which any

finite linear combination of samples will be normally distributed (or, more generally,

any linear functional applied to the sample function Xt will give a normally distributedresult). Although mathematical and computational complexity discourage the use of

stochastic methods, these techniques bring a new field of mathematics to bear on theproblem of demand forecasting.

The following three steps are used in stochastic forecasting procedure

1. specify the form of the stochastic model

2. use historical data to determine the unknown random input to thestochastic model

3. calculate the response of the stochastic model



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Stochastic model for monthly peak demand is given as

Where Xt is the estimate of monthly peak demand produced by the model, ut, vtand wt are variables with zero mean, Gaussian random process, is a parameter

varying between 0 and 1, and U is the backward shift operator. Four parameters

must be estimated before the model is complete, the variance of random inputs u2,

v2, w2 and a parameter of the model . However no meaningful statistics can bedeveloped from historical data due to non stationarity so a reversible transformation

which converts the non stationary process Xt to stationary process Yt.

The above equation represents a stochastic model producing the stationary

random process Yt.. The forecast is prepared by first manipulating Yt into

convenient form for forecasting Yt, performing necessary computation and thenusing the inverse transformation to convert Yt into forecasted values of monthly

peak demand Xt. Since the model is non stationary, it does not have to be updated

as frequently as the trend curves and weather load models used in other approaches.Regarding practical application it appears that a purely stochastic approach to

demand forecasting has a limited practical significance, but the combination of

simple stochastic models with more conventional techniques such as weather loadmodels may have great potential.

Out of these six methods the most applicable method to any type of system is

Separate treatment of weather sensitive component in annual peak demand. It is

more accurate and reliable too. For separate treatment of weather induced and nonweather induced demand it uses weather load model , by using weather load model

separate forecasting of weather induced and non weather induced demand is done

and both are combined to get the final total load forecast.



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CHAPTER 3

FORECASTING AND REGRESSION METHODS

3.1 REGRESSION

Regression is a statistical tool for finding the relationship among the variables, it is

used to investigate the causal relationship of dependent variable and one or more

independent variable .It helps us to understand how the value of dependent variableschange due to the change in one independent variable while the other independent

variables held as constant. The analysis starts by finding a function which defines the

dependent variables as a function of independent variable, by assembling data on the

underlying variable of interest, this function is termed as regression function. It iswidely used in prediction and forecasting.

Techniques available for performing regression are numerous. Most widely used

technique is Linear Regression or ordinary least square regression technique which isparametric which uses finite number of unknown parameters for defining the regressionfunction. Non parametric methods are also available which allows the regression

function to lie within a set of functions, which may be infinite dimensional.

Regression model

Regression model involves three variables

1. Unknown parameter denoted as which may be a scalar or a vector2. Dependent variable, Y

3. Independent variable, X

The dependent variable Y can be represented as a function of and X as

Y=f (, X)

For carrying out regression analysis the form of the function should be specified. The

form is obtained from the relationship between Y and X that does not rely on thedata. If no such knowledge is available then a flexible or a convenient form of

function is chosen.

Assume now that the vector of unknown parameters is of length k. In order to perform aregression analysis the user must provide information about the dependent variable Y:

IfNdata points of the form (Y, X) are observed, where N< k, most classical

approaches to regression analysis cannot be performed: since the system of

equations defining the regression model is underdetermined, there is not enough

data to recover.

If exactly N= k data points are observed, and the function f is linear, the

equations Y=f(X, ) can be solved exactly rather than approximately. Thisreduces to solving a set ofNequations with Nunknowns (the elements of),



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which has a unique solution as long as the X are linearly independent. Iff is

nonlinear, a solution may not exist, or many solutions may exist.

The most common situation is where N > k data points are observed. In this

case, there is enough information in the data to estimate a unique value for

that best fits the data in some sense, and the regression model when applied to

the data can be viewed as an over determined system in .

Statistical Assumptions

When the number of measurements,N, is larger than the number of unknown parameters,

k, and the measurement errors i are normally distributed then excess of information

contained in (N - k) measurements is used to make statistical predictions about the

unknown parameters. This excess of information is referred to as the degrees of freedomof the regression.

Classical Assumptions

Classical assumptions include

The sample is representative of the population for the inference prediction.

The error is a random variable with a mean of zero conditional on the

explanatory variables.

The independent variables are measured with no error. (Note: If this is not so,modeling may be done instead using errors-in-variables model techniques).

The predictors are linearly independent, i.e. it is not possible to express any

predictor as a linear combination of the other.

The errors are uncorrelated, that is, the variance-covariance matrix of the errors

is diagonal and each non-zero element is the variance of the error. The variance of the error is constant across observations (Note: If not, weighted

least squares or other methods might instead be used).

3.1.2 DIFERENT FORMS OF MULTIPLE REGRESSION

1. LINEAR REGRESSION

In Linear Regression the dependent variable is expressed as a linear combination of

the parameters. Linear Regression is of two types simple linear regression andmultiple linear regression. In simple linear regression there is only a single

independent variable and its analysis is simpler because the curve is approximatedas a straight line. Simple linear regression is of the form.

Where 0 and 1 are unknown parameters and xi is the independent variable.

Multiple regression model dependent variables are expressed as a function of more than

one independent variable or as a function of independent variable. For example



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there may be relationship between three variables X, Y, and Z that can be described

by the equation

Z= 0 + 1 X+ 2 Y, which is called a linear equation in the variables X, Y, and Z. In athree dimensional rectangular coordinate system this equation represents a plane.

2. QUADRATIC REGRESSION

Quadratic Regression is in the form of a parabola that means the function will

first increase then decrease or first decrease then increase. It can be described by the

equation

3. CUBIC REGRESSION

Cubic regression is described by the equation

Y= 3 X3 +2 X

2 + 1 X+ 0

Cubic regression can increase then decrease and then increase or it will decrease thenincrease and then decrease.

4. QUARTIC REGRESSION

It is described by the equation

Y= 4 X4+ 3X

3 + 2 X2+1 X+

Quartic regression can increase then decrease then increase then decrease or it will

decrease then increase then decrease then increase.

5. EXPONENTIAL REGRESSION.

It is described by the equationY= 0 + 1

x . It will strictly increase or decrease.

3.2. TERMINOLOGIES USED IN FORECASTING METHODOLOGY

3.2.1 WEATHER INDUCED DEMAND

Weather induced demand is affected by varying weather conditions and can be

calculated using weather load model given by

Dw = Ks(T-Ts) for T>TsDw=0 for TwTTs

Dw= Kw(T-Tw) for T


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Fig 3.1 Weather load model [2]

3.2.2 NON WEATHER DEMAND

It is otherwise called as the base load which is not affected by weather conditions. It

is assumed to be constant under the ideal conditions like no load growth due to

installation of new equipment and no change in the consumers users habit occur. It isobtained by subtracting the weather load model with weather induced demand.

3.2.3 SEASONAL COMPONENT OF DEMAND

It indicates the cyclic variation of demand throughout the year. Weather induceddemand contributes more for seasonal component of peak demand and is treated

separately in the forecasts. Non weather component also contributes for seasonal

demand for example demand due to grain drying, Festivals lighting loads etc

3.2.4 EXPONENTIALLY WEIGHTED REGRESSIONS

A least square curve fit where errors are discounted by multiplying by exponentially

decreasing weight factors as the data becomes older. The weight factor will be aparameter between zero and one which is selected by the user. A weighting factor of

one gives standard multiple regression where all errors are weighted equally.

3.3 DESCRIPTION OF METHODOLOGY

Application of the forecasting methodology Separate treatment of weather sensitivecomponent of Annual peak demand in Hellenic system is described here. Assuming for

a given month that economic activities are practically similar for every working day, it

becomes apparent that from day to day, the variation of the daily peak load is mainlydue to the impact of temperature (or other weather sensitive parameters like humidity).

The daily peak load, for every working day of the month, can be considered as the sum

of two components, a nonweather sensitive (base load) and a weather sensitive load.Decomposition of peak load into these two components requires weather load model.

3.3.1 DATA PREPARATION

Two sets of input data is required a daily summery of load and its corresponding

weather observations for developing weather load models and a weekly summary of

load and weather data which is the input to the weekly peak demand forecastingprogram.



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The demand data may be prepared on daily, weekly or hourly basis it may consists of

several useful variables, it is desirable to include only the most promising variables on

the data so that subsequent experimentation is possible without reprocessing the hourlydata. The results were produced with daily and weekly peak demand and coincident dry

bulb temperature.

3.3.2 WEATHER LOAD MODEL

The weather load model of every year is obtained by the scatter diagram of the daily

peaks vs one or more weather variables such as temperature, humidity etc. Weekends,major holidays and other abnormal days (such as major strikes, natural disasters,

blackouts etc) are excluded. A typical example of such a diagram for the Hellenic

system, considering only temperature as a weather variable, is shown in Fig.2

Fig 3.2 Typical scatter diagram for Hellenic system [3]

The scatter diagram is fitted to a non-linear model. After experimenting with several

forms of models it was seen that a quadratic model was the most suitable one, such asthe one depicted in Fig. 3

Fig 3.3 Quadratic weather load model [3]

Two methods are used to determine weather load model. First methods make use of

yearly load data to plot a scatter diagram with the values of Tw and Ts is given thevalue of Kw and Ks is estimated directly from the diagram. Second method makes use



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of linear regression to fit kw and Ks values of Tw and Ts have been specified. The

specific procedure is as follows.

D0 is the average value of all the demand values lying in the temperature range of

Tw


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that the temperature during peak hours and the weather sensitive load component are

normally distributed.

3.3.7 FORECAST OF THE MONTHLY DAILY PEAK LOAD

The total monthly peak is also assumed to be normally distributed. Its mean and

variance is given by the equationD (t) = L (t) + s (t) 2D(t) = 2L(t) + 2S(t) (4.2)



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CHAPTER 4

RESULTS

The above method is applied to Hellenic system. Historical data for the peak loadand the temperature variation is used. The mean monthly peak loads aredecomposed into base load and weather sensitive load components. The known

weather load models and base loads are extrapolated. The extrapolation result is as

shown in the figure4.1

Fig 4.1 Forecast of evening weather load model [3]

The forecast of the weather sensitive component requires the knowledge of thedistribution of the temperature during peak hours for every month. Forecast of non-

weather sensitive component is as shown in the figure 4.2



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Fig 4.2 Forecast of base load [3]

The dark lines in the above forecast indicate the forecast of the simulation period.

Since the weather conditions on future months cannot be predicted the choice ofmonthly temperature distribution can be on the basis of impact of different weather

conditions on peak load .The mean and variance of the weather sensitive components

are calculated using the techniques described earlier. The assumption that the totalmonthly peaks are normally distributed allows us to set up confidence intervals for the

forecast. The selection of the expected peaks that will be used for planning purposes

depends on the acceptable level of risk. As described in the methodology the total loadforecast is obtained by combining the forecast of weather sensitive and non weather

sensitive components which is indicated in figure 4.3



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Fig 4.3 Forecast of monthly peak load [3]



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CHAPTER 5CONCLUSION

The implementation of forecasting technique Separate treatment of weather

sensitive component is described in the paper which is quite straight forward and

also a widely used technique. The theoretical implementation of this loadforecasting approach for real system is not so straight forward so the method with

little modification is described here. The accuracy of this method is acceptable to

some extend. The method uses statistical extrapolation technique with lots ofassumptions which suffers from following drawbacks like annual peak load data of

particular element for example substation, can deviate much during the loading

history. These data shifts are mostly caused by switching as loads are routinelymoved from one substation to another during the course of utility operation. Hence

the extrapolation technique suffers badly due to these data shifts. This problem can

be significant, especially in electricity distribution systems with an incompleteRemote control Systems. In order to solve these problems improve techniques has

to be used. Forecasting using neural networks which models all these uncertaintiesis recommended.



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APPENDIX 1

DERIVATION OF PROBABILITY FUNCTION OF WEATHER SENSITIVE

COMPONENT

Fig 6.1. Derivation of probability function of weather sensitive component [1]

Fig 6.2. Probability density function for weather

sensitive component of demand [2]

From the theory of random variables it is known thatif a random variable Y is given by:

Y= g(X)

and fx(x) is the density distribution function of the

random variable X, then the density distribution

function fy(y)of the random variable Y is given by the

following

equation:

where xI, x2, . . . are the roots of the equation y = g (x).

Let C denote the normally distributed random variable of the temperature, with mean

value and variance ,Y the quadratic weather-load model and Cmin the

temperature that minimizes the value of the weather-load model. The random variable

X, given by:



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is a normally distributed random variable, with a mean value 0 and variance 1.

The mean value and the variance of the weather sensitive load component arecalculatedby applying the above theorem. It can be seen that the mean value s(t) is

where dY/dC is the value of the derivative of the weather-load model at temperature

and



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REFERENCES

[1] K.N Stanton and Pradeep C Guptha, Forecasting Annual or seasonal peak demand

in Electric utility systemsIEEE Transactions of Power Apparatus and systemspp.951

959 May/June 1970

[2] K.N Stanton, Medium Range, Weekly and Seasonal Peak Demand Forecasting byProbability methodsIEEE Transactions pp 1183-1189June 1970

[3] R.L Sullivan, Power System Planning, Tata Mcgraw Hill, Newyork. 1977, pp 19-

59



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[4] C.E Asburgy, Weather load model for electric demand and energy forecastingIEEE Transactions on Power Apparatus and Systems, vol.-94, no.4 July/August 1975

[5] Murray R. Spielgal and Larry J.Stephens, Shaums Outline of Statistics , Mcgraw

Hill ,Newyork , 1998 pp.282-286,311-318,434-439

[6] E.G Contaxi, Application of weather sensitive Peak load forecasting Model to theHellenic systemIEEE Melecon 2004 pp.819-822

[7] Slobodan M. Maksimovich and Vladimir M. Shiljkut, The Peak load forecasting

afterwards its intensive reductionIEEE Transactions on power delivery, vol, 24, N0.3,July 2009 pp 1552-1559

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