chapter 2 load forecasting

112 Electrical Load Forecasting

Chapter 2Electrical Load Forecasting

Chapter content:

(1.0)System planning optimality

(2.0)Types of Patterns

(3.0)Some definitions

(4.0)Forecasting Methodology

(5.0)Method of Calculation

(6.0)Practical Example of ELF


Chapter 2Electrical Load Forecasting

Power distribution system planning is essential to assure that the growing demand of electricity can be satisfied by distribution system additions which are both technically adequate and reasonably economical the application of some type of systematic approach to generation and transmission system planning.

In the future more than the past, electric utilities will need a fast and economical planning tool to evaluate the consequences of different proposed alternatives and impact on the test of the system to provide the necessary economical, reliable and safe electric energy to consumers.

The objective of distribution system planning is to assure that the growing demand for electricity, in terms of increasing growth rates and high load densities can be satisfied in any optimum way.

Technically adequate implies the system is:

1- Reliable (voltage and service continuity)2- Supply satisfies demand.3- Equipment is not overloaded.

Total system cost is minimum, where:

Total cost = fixed cost + variable cost

Several factors affecting power distribution


Time

Load

Known unknow

Historical Future Horizonyear

?

(1.0)System planning optimality:

(1.1)Load forecasting:The load growth of the geographical area served by a utility company is the most important factor influencing the expansion of distribution system. Therefore, forecasting of load increases and system reaction to these increases is essential to the planning process. There are two common time scales of importance to load forecasting long-range, with time horizon of up to five years distant. Ideally, these forecasts would predict future load in detail, extending even to the individual customer level.

Frequently, there is a time lag between awareness of an impending event, or need, and the occurrence of that event. This time lag is the main reason for planning and forecasting. If the time lag is long, and the outcome of the final event is conditional upon identifiable factors, planning can play an important role. In such situations, forecasting is needed to determine when a need will arise so that the appropriate actions can be taken.

(1.1.1)Definition of ELF:

Electric Load Forecast is perform analysis of post or/and present data, identifying trends and patterns that exist in the data, that are then used to project load into the future.

Fig. 2.1


Load forecasts can be divided into three categories: short-term forecasts which are usually from one hour to one week, medium forecasts which are usually from a week to a year, and long-term forecasts which are longer than a year. The forecasts for different time horizons are important for different operations within a utility company. The natures of these forecasts are different as well. For example, for a particular region, it is possible to predict a next day load with an accuracy of approximately 1-3%. However, it is impossible to predict the next year peak load with the similar accuracy since accurate long-term forecasts are not available. For the next year peak forecast, it is possible to provide the probability distribution of the load based on historical weather observations. It is also possible, according to the industry practice, to predict the so-called weather normalized load, which would take place for average annual peak weather conditions or worse than average peak weather conditions for a given area.

Load forecasting has always been important for planning and operational decision conducted by utility companies. However, with the deregulation of the energy industries, load forecasting is even more important. With supply and demand --fluctuating with the changes of weather conditions and energy prices increasing by a factor of ten or more during peak situations, load forecasting is vitally important for utilities. Short-term load forecasting can help to estimate load demands and to make decisions that can prevent overloading. Timely implementations of such decisions lead to the improvement of network reliability and to the reduction occurrences of equipment failures and blackouts. Load forecasting is also important for contract evaluations and evaluations of various sophisticated financial products on energy pricing offered by the market. In the deregulated economy, decisions on capital expenditures based on long-term forecasting are also more important than in a non-deregulated economy when rate increases could be justified by capital expenditure projects.

Over the last few decades a number of forecasting methods have been introduced. Most of these methods use statistical techniques sometimes combined with artificial intelligence algorithms such as neural networks, fuzzy logic, and expert systems. Two of the methods, so-called end-use and econometric approach are broadly used for medium and long-term forecasting. A variety of methods, that include the so-called similar day approach, various regression models, time series, neural networks, statistical learning algorithms, fuzzy logic, and expert systems, have been developed for short-term forecasting.

A large variety of mathematical methods and ideas have been used for load forecasting. The development and improvements of appropriate mathematical tools will lead to the development of more accurate load forecasting techniques.


In this chapter, we are concerned to talk mainly about the long-term forecast using various regression techniques, and choosing which of them will be more accurate, to use it at the end to make calculations for an electrical load forecast for the next 10 years depending on an available historical data of the loads in the previous years.

(1.1.2)Factors Affecting ELF:Several factors affect ELF. Among them we mention:

1. Land use:

(Residential, industrial, commercial, agriculture . . . etc.). Different types of use affect the capacity of the substation, i.e. residential loads is different from industrial loads.

2. Population growth:

As the population increases more loads are needed.

3. Historical Data:

Historical data plays an important role in forecasting since they can tell how the load will behave in the future.

4. Load Densities (KVA/Km 2 ) :

Load density must be put into account during load forecasting, we must consider the range of enmities as it differs for different types of loads. Common figures for load densities are 1000 KVA/Km2 for agricultural areas; 3000 KVA/Km2 for residential areas; 5000 KVA/Km2 for city center and 10,000 KVA/Km2 for industrial areas.


(1.2)Substation expansion:

Presents some of the factors affecting the substation expansion the planner makes a decision based on tangible or intangible information. For example, the forecasted load, load density and load growth may require a substation expansion or a new substation construction. In the system expansion plan the present system configuration, capacity and the forecasted loads can play major roles.

Fig. 2.2: Factor of substation expansion

Present capacity and configuration

Physical size and

land


(1.3)Substation site selection:

The factors that affect substation site selection. The distance from the load centers and from the existing sub transmission lines as well as other limitation, such as availability of land its cost and land use regulations, are important.

Fig. 2.3: Factors affecting substation site

(1.4)Other factors:

Once the load assignments to the substation are determined then the remaining factors affecting primary voltage selection, feeder route selection, number of feeders, conductor size selection and total cost, need to be considered.

Load

availability


(2.0)Types of Patterns:

By the word pattern we mean how the load changes with time. Four basic types of patterns often exist in data series:

(2.1)Horizontal pattern: This exists when there is no trend in a data series. This can happen when there is no more expected load increase in the area. Such pattern is generally referred to as stationary. Example of this load pattern is the load pattern of Abbasia district in Cairo where there is not a single meter to build more buildings on. Figure below shows changing of load in the y-axis with time in the x-axis.

1 2 3 4 5 6 7 8 9 10 11

0

5

10

15

20

25

30

35

Fig. 2.4: Horizontal pattern

(2.2) Trend pattern:

This exists when there is an increase or decrease in the value of electric load consumption. Such pattern is generally referred to as non-stationary.

1 2 3 4 5 6

0

10

20

30

40

50

60

70

Fig. 2.5: Trend pattern


(2.3) Seasonal pattern:

This exists when data series fluctuates according to some seasonal factors.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0

5

10

15

20

25

30

Fig. 2.6: Seasonal pattern

(2.4) Cyclical pattern: This exists when data series fluctuates and doesn't repeat itself at constant time interval. One of the factors that may cause the pattern to be so is an economical crisis in the state which may cause to stop some industries in the state causing a dip in the load curve.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0

10

20

30

40

50

60

Fig. 2.7: Cyclical pattern


(3.0)Some definitions:

Load Duration Curve:

Line charts showing the percentage of time the hourly load was at or near peak values. The Y-axis is percentage of peak; the X-axis is percentage of time.

Average Annual Energy:

Average kWh consumption of energy in specific time duration.

Non-Coincident Peak Demand:

Average peak hourly demand regardless of the time of occurrence .

Maximum Demand:

The maximum diversified demand of the energy consumed for specific time periods.

Load Factor (LF):

Ratio of average energy for the year (annual kWh/8760) to peak demand .

Coincident Factor (CF):

Ratio of maximum demand to the non-coincident peak. A related measure is the diversity factor, which is the reciprocal of the coincidence factor.

Site Non-Coincident Peak Demand:

Average peak hour demand regardless of the time of occurrence .

Group Diversified Peak Demand:

The maximum, simultaneous, hourly demand of all sites for that month; it is determined by averaging demand of the sites hour by hour and then finding the maximum demand for the group.


(4.0)Forecasting Methodology:

Forecasting is simply a systematic procedure for quantitatively defining future loads. Depending on the time period of interest, a specific forecasting procedure may be classified as a short term, intermediate or long term technique.

Because system planning is our basic concern and because planning for the flow generation, transmission and distribution facilities must begin 4 - 10 year in advance of the actual in service data, we shall be concerned with the methodology of intermediate-range forecasting.

For simplicity, the word “Forecast” will usually imply an intermediate range forecast.

Forecasting techniques may be divided into 3 broad classes. Techniques may be used on extrapolation or correlation or a combination of both. Techniques may be further classified as deterministic, probabilistic, or stochastic.

(4.1)Extrapolation:

Extrapolation techniques involve fitting trend curves to basic historical data adjusted to reflect the growth. It produces reasonable results in many cases.

Such a technique is to be classified as a deterministic extrapolation, since no attempt is made to account for random errors in the data or in the analytical model. Some standard analytical functions are used in trend curves fitting, including:

1. Straight line = a + b x

2. Parabola = a + b x + cx2

The most common curve - fitting technique for finding coefficients of function in a given forecast is the method of least squares as will be discussed later


(4.2)Correlation:

Correlation techniques are used to relate system loads to various demographic and economic factors. This approach has an advantage of forcing the forecast to understand clearly the interrelationship between load growth patterns and other measurable factors. The most obvious disadvantage, however results from the need to forecast demographic and economic factors, which can be more difficult than forecasting system load. Typically, these factors may be population, employment, building permits saturation and business indicators.

(5.0)Method of Calculation:

(5.1) Simple Regression:

Regression in general is a relationship between the variable we want to forecast (dependant) and another variable (independent).

Or, we can say

Y = f (x)

If the independent variable is time, then we call it simple time-series regression, and simple refers to a single independent variable.

In a simple regression the relationship is assumed linear, i.e.

Ŷ = a + b t

The principle of regression theory is that, any function Ŷ = f (x) can be fitted to a set of data points so as to minimize the sum of errors squared at each data point and this type of fitting is called least square fit in which the objective is to:

∑i=1

n

[Y i−f ( x i) ]2=

Minimum

Where n is the number of data points


(5.2) Linear Regression:

Y=f (x )

Y¿=a+bt

¿e2=∑i=0

n

(Y−Y¿ )2

¿e2=∑i=0

n

[Y−(a+bt ) ]2

∂ e2

∂ a=2∑

i=0

n

[Y−(a+bt ) ](−1 )=0

∑i=0

n

Y=∑i=0

n

a+∑i=0

n

bt

Or

∑i=0

n

Y=na+∑i=0

n

bt

∂e2

∂b=2∑

i=0

n

[Y−(a+bt ) ](−1)=0

∑i=0

n

tY=∑i=0

n

at+∑i=0

n

bt2

(n∑i=0

n

t ¿)¿¿

¿

¿

¿


(5.3)Quadratic Regression:

Y=f (x )

Y¿=a+bt+ct2

¿e2=∑i=0

n

(Y−Y¿)2

¿e2=∑i=0

n

[Y−(a+bt+ct2 ) ]2

∂ e2

∂ a=2∑

i=0

n

[Y−(a+bt+ct2 ) ](−1)=0

∑i=0

n

Y=∑i=0

n

a+bt+ct2

∑i=0

n

Y=na+b∑i=0

n

t+c∑i=0

n

t2

∂ e2

∂ b=2∑

i=0

n

[Y−(a+bt+ct2 ) ](−t )=0

∑i=0

n

tY=a∑i=0

n

t+b∑i=0

n

t2+c∑i=0

n

t3

∂ e2

∂ c=2∑

i=0

n

[Y−(a+bt+ct2 ) ](−t2 )=0

∑i=0

n

t2Y=a∑i=0

n

t2+b∑i=0

n

t3+c∑i=0

n

t4

(n∑i=o

n

t∑i=0

n

t2 ¿)(∑i=0

n

t∑i=0

n

t2∑i=0

n

t3 ¿)¿¿

¿¿


(5.4)Polynomial Regression:

Y=f (x )

Y¿=a+bt+ct2+dt3

¿e2=∑i=0

n

(Y−Y¿)2

¿e2=∑i=0

n

[Y−(a+bt+ct2+dt 3 ) ]2

∂ e2

∂ a=2∑

i=0

n

[Y−(a+bt+ct2+dt 3 ) ](−1)=0

∑i=0

n

Y=∑i=0

n

a+bt+ct2+dt 3

∑i=0

n

Y=na+b∑i=0

n

t+c∑i=0

n

t2+d∑i=0

n

t3

∂ e2

∂ b=2∑

i=0

n

[Y−(a+bt+ct2+dt 3 ) ](−t )=0

∑i=0

n

tY=a∑i=0

n

t+b∑i=0

n

t2+c∑i=0

n

t3+d∑i=0

n

t4

∂ e2

∂ c=2∑

i=0

n

[Y−(a+bt+ct2+dt 3 ) ](−t2 )=0

∑i=0

n

t2Y=a∑i=0

n

t2+b∑i=0

n

t3+c∑i=0

n

t4+d∑i=0

n

t5

∂ e2

∂ d=2∑

i=0

n

[Y−(a+bt+ct2+dt 3 ) ](−t3 )

∑i=0

n

t3Y=a∑i=0

n

t3+b∑i=0

n

t4+c∑i=0

n

t5+d∑i=0

n

t6


(n∑i=o

n

t∑i=0

n

t2∑i=0

n

t3 ¿)(∑i=0

n

t∑i=0

n

t2∑i=0

n

t3∑i=0

n

t4 ¿)(∑i=0

n

t2∑i=0

n

t3∑i=0

n

t4∑i=0

n

t5 ¿)¿¿

¿¿


(6.0)Practical Example of ELF:

The load forecasting of a new area is to be considered. This area consists of several zones, each of a known area:

(1)Agriculture.

(2) Residential.

(3) City center.

(4) Light industrial.

(5) Heavy industrial.

The area under study is given below:

Fig. 2.8:Area under study


(6.1) Agriculture region:

year t y t2 t3 t4 y * t y *t2

1999 1 66.67 1 1 1 66.67 66.67

2000 2 66.67 4 8 16 133.34 266.68

2001 3 72.22 9 27 81 216.66 649.98

2002 4 72.22 16 64 256 288.88 1155.52

2003 5 77.78 25 125 625 388.9 1944.5

2004 6 83.33 36 216 1296 499.98 2999.88

2005 7 86.67 49 343 2401 606.69 4246.83

2006 8 88.89 64 512 4096 711.12 5688.96

2007 9 92.22 81 729 6561 829.98 7469.82

Σt Σy Σt2 Σt3 Σt4 Σ(y * t) Σ(y * t2)

45 706.67 285 2025 15333 3742.22 24488.84

Table 2.1


(6.1.1) linear model:

Where ao & a1 are computed constants as :

Therefore: ao=61.11305556, a1=3.481166667

year t y y^ e e2

1999 1 66.67 64.59422 2.075778 4.308853

2000 2 66.67 68.07539 -1.40539 1.975118

2001 3 72.22 71.55656 0.663444 0.440159

2002 4 72.22 75.03772 -2.81772 7.939559

2003 5 77.78 78.51889 -0.73889 0.545957

2004 6 83.33 82.00006 1.329944 1.768752

2005 7 86.67 85.48122 1.188778 1.413193

2006 8 88.89 88.96239 -0.07239 0.00524

2007 9 92.22 92.44356 -0.22356 0.049977

Σe2

18.44681

Table 2.2

y=a0+a1∗t

[ n ∑ t

∑ t ∑ t2 ]∗[a0

a1]=[ ∑ y

∑( y∗t )]


(6.1.2)Quadratic model:

Where

Therefore

ao=62.43785714 , a1=2.758547619 , a2=0.07226190476

year t y y^ e e2

1999 1 66.67 65.26867 1.401333 1.963735

2000 2 66.67 68.244 -1.574 2.477476

2001 3 72.22 71.36386 0.856143 0.732981

2002 4 72.22 74.62824 -2.40824 5.799611

2003 5 77.78 78.03714 -0.25714 0.066122

2004 6 83.33 81.59057 1.739429 3.025612

2005 7 86.67 85.28852 1.381476 1.908476

2006 8 88.89 89.131 -0.241 0.058081

2007 9 92.22 93.118 -0.898 0.806404

Σe2

16.8385

Table 2.3

y=a0+a1∗t+a2∗t2

[ n ∑ t ∑ t2

∑ t ∑ t2 ∑ t3

∑ t2 ∑ t3 ∑ t4 ]∗[a0

a1

a2]=[ ∑ y

∑ ( y∗t )

∑ ( y∗t2 )]


From Least Squared Error Criteria we will use Quadratic model, therefore:


year t

1999 165.2686

7

2000 2 68.244

2001 371.3638

6

2002 474.6282

4

2003 578.0371

4

2004 681.5905

7

2005 785.2885

2

2006 8 89.131

2007 9 93.118

2008 1097.2495

2

2009 11101.525

6

2010 12105.946

1

2011 13110.511

2

2012 14115.220

9

2013 15 120.075

2014 16125.073

7

2015 17130.216

9

2016 18135.504

6

2017 19140.936

8

y


Table 2.4

For 2017: Load (MVA) =140.368 MVA

(6.2)Residential region:


1999 1 118 1 1 1 118 118

2000 2 120 4 8 16 240 480

2001 3 125 9 27 81 375 1125

2002 4 125 16 64 256 500 2000

2003 5 130 25 125 625 650 3250

2004 6 140 36 216 1296 840 5040

2005 7 150 49 343 2401 1050 7350

2006 8 160 64 512 4096 1280 10240

2007 9 170 81 729 6561 1530 13770


45 1238 285 2025 15333 6583 43373

Table 2.5




Therefore: ao =104.80555556 , a1=6.55

year t y y^ e e2

1999 1 118 111.3556 6.644444 44.14864

2000 2 120 117.9056 2.094444 4.386697

2001 3 125 124.4556 0.544444 0.29642

2002 4 125 131.0056 -6.00556 36.0667

2003 5 130 137.5556 -7.55556 57.08642

2004 6 140 144.1056 -4.10556 16.85559

2005 7 150 150.6556 -0.65556 0.429753

2006 8 160 157.2056 2.794444 7.80892

2007 9 170 163.7556 6.244444 38.99309

Σe2

y=a0+a1∗t

[ n ∑ t

∑ t ∑ t2 ]∗[a0

a1]=[ ∑ y

∑( y∗t )]


206.0722

Table 2.6


Where

Therefore ao=119.0714286 , a1= -1.231385281 , a2=0.7781385281

year t y y^ e e2

1999 1 118 118.6182 -0.61818 0.382149

2000 2 120 119.7212 0.278788 0.077723

2001 3 125 122.3805 2.61948 6.861678

2002 4 125 126.5961 -1.5961 2.547548

2003 5 130 132.368 -2.36797 5.60726

2004 6 140 139.6961 0.303896 0.092353

2005 7 150 148.5805 1.41948 2.014925

2006 8 160 159.0212 0.978788 0.958026

2007 9 170 171.0182 -1.01818 1.036694

y=a0+a1∗t+a2∗t2

[ n ∑ t ∑ t2

∑ t ∑ t2 ∑ t3

∑ t2 ∑ t3 ∑ t4 ]∗[a0

a1

a2]=[ ∑ y

∑ ( y∗t )

∑ ( y∗t2 )]


Σe2

19.57835

Table 2.7

From Least Squared Error Criteria we will use Quadratic model, therefore :

year t y^

1999 1 118.6182

2000 2 119.7212

2001 3 122.3805

2002 4 126.5961

2003 5 132.368

2004 6 139.6961

2005 7 148.5805

2006 8 159.0212

2007 9 171.0182

2008 10 184.5714

2009 11 199.681

2010 12 216.3468

2011 13 234.5688

2012 14 254.3472

2013 15 275.6818

2014 16 298.5727

2015 17 323.0199

2016 18 349.0234

2017 19 376.5831


Table 2.8

For 2017: Load (MVA) =376.5831 MVA

(6.3) City Center:


1999 1 466.67 1 1 1 466.67 466.67

2000 2 488.89 4 8 16 977.78 1955.56

2001 3 555.56 9 27 81 1666.68 5000.04

2002 4 555 16 64 256 2220 8880

2003 5 588.89 25 125 625 2944.45 14722.25

2004 6 622.22 36 216 1296 3733.32 22399.92

2005 7 661.1 49 343 2401 4627.7 32393.9

2006 8 700 64 512 4096 5600 44800

2007 9 744.44 81 729 6561 6699.96 60299.64


45 5382.77 285 2025 15333 28936.56 190918

Table 2.9




Therefore: ao = 429.5263889, a1= 33.71183333

year t y y^ e e2

1999 1 466.67463.23

6 3.434278 11.79426

2000 2 488.89496.94

8 -8.05756 64.9242

2001 3 555.56530.65

9 24.90061 620.0404

2002 4 555564.37

1 -9.37122 87.81981

2003 5 588.89 598.08 -9.19306 84.51227

y=a0+a1∗t

[ n ∑ t

∑ t ∑ t2 ]∗[a0

a1]=[ ∑ y

∑( y∗t )]


3

2004 6 622.22631.79

5 -9.57489 91.6785

2005 7 661.1665.50

7 -4.40672 19.4192

2006 8 700699.21

9 0.781444 0.610655

2007 9 744.44 732.93 11.50961 132.4711

Σe2

1113.27

Table 2.10


Where

Therefore ao=443.6035714, a1= 26.03337013, a2=0.7678453203

year t y y^ e e2

1999 1 466.67470.40

5 -3.73479 13.94864

2000 2 488.89 498.74 -9.8517 97.05593

y=a0+a1∗t+a2∗t2

[ n ∑ t ∑ t2

∑ t ∑ t2 ∑ t3

∑ t2 ∑ t3 ∑ t4 ]∗[a0

a1

a2]=[ ∑ y

∑ ( y∗t )

∑ ( y∗t2 )]


2

2001 3 555.56528.61

4 26.9457 726.0708

2002 4 555560.02

3 -5.02259 25.22644

2003 5 588.89592.96

7 -4.07658 16.6185

2004 6 622.22627.44

6 -5.22626 27.31379

2005 7 661.1663.46

2 -2.36163 5.577306

2006 8 700701.01

3 -1.0127 1.025555

2007 9 744.44740.09

9 4.340545 18.84034

Σe2

931.6773

Table 2.11

From Least Squared Error Criteria we will use Quadratic model, therefore

year t y^

1999 1 470.4048

2000 2 498.7417

2001 3 528.6143

2002 4 560.0226

2003 5 592.9666

2004 6 627.4463


2005 7 663.4616

2006 8 701.0127

2007 9 740.0995

2008 10 780.7219

2009 11 822.88

2010 12 866.5739

2011 13 911.8034

2012 14 958.5686

2013 15 1006.87

2014 16 1056.706

2015 17 1108.078

2016 18 1160.986

2017 19 1215.43

Table 2.12

For 2017: Load (MVA) =1215.43 MVA

(6.4)Light Industries:


1999 1 118 1 1 1 118 118

2000 2 120 4 8 16 240 480


2001 3 125 9 27 81 375 1125

2002 4 125 16 64 256 500 2000

2003 5 130 25 125 625 650 3250

2004 6 140 36 216 1296 840 5040

2005 7 145 49 343 2401 1015 7105

2006 8 150 64 512 4096 1200 9600

2007 9 155 81 729 6561 1395 12555


45 1208 285 2025 15333 6333 41273

Table 2.13

(6.4.1)Linear model:


y=a0+a1∗t

[ n ∑ t

∑ t ∑ t2 ]∗[a0

a1]=[ ∑ y

∑( y∗t )]


Therefore: ao = 109.8055556, a1= 4.88333

year t y y^ e e2

1999 1 118 114.6889 3.311114 10.96348

2000 2 120 119.5722 0.427784 0.182999

2001 3 125 124.4555 0.544454 0.296431

2002 4 125 129.3389 -4.33888 18.82584

2003 5 130 134.2222 -4.22221 17.82702

2004 6 140 139.1055 0.894464 0.800067

2005 7 145 143.9889 1.011134 1.022393

2006 8 150 148.8722 1.127804 1.271943

2007 9 155 153.7555 1.244474 1.548717

Σe2

52.73889

Table 2.14


Where

y=a0+a1∗t+a2∗t2

[ n ∑ t ∑ t2

∑ t ∑ t2 ∑ t3

∑ t2 ∑ t3 ∑ t4 ]∗[a0

a1

a2]=[ ∑ y

∑ ( y∗t )

∑ ( y∗t2 )]


Therefore ao=115.142857 , a1= 1.972077922 , a2=0.2911255411

year t y y^ e e2

1999 1 118 117.4061 0.593939 0.352764

2000 2 120 120.2515 -0.25152 0.06326

2001 3 125 123.6792 1.320779 1.744458

2002 4 125 127.6892 -2.68918 7.231675

2003 5 130 132.2814 -2.28139 5.204719

2004 6 140 137.4558 2.544156 6.472729

2005 7 145 143.2126 1.787446 3.194963

2006 8 150 149.5515 0.448485 0.201139

2007 9 155 156.4727 -1.47273 2.168925

Σe2

26.63463

Table 2.15

From Least Squared Error Criteria we will use Quadratic model, therefore:

year t y^

1999 1 117.4061

2000 2 120.2515

2001 3 123.6792

2002 4 127.6892

2003 5 132.2814


2004 6 137.4558

2005 7 143.2126

2006 8 149.5515

2007 9 156.4727

2008 10 163.9762

2009 11 172.0619

2010 12 180.7299

2011 13 189.9801

2012 14 199.8126

2013 15 210.2273

2014 16 221.2242

2015 17 232.8035

2016 18 244.9649

2017 19 257.7087

Table 2.16

For 2017: Load (MVA) =257.7087 MVA

(6.5)Heavy Industrial:


1999 1 194.1176 1 1 1 194.1176 194.1176


2000 2 217.6471 4 8 16 435.2941 870.5882

2001 3 247.0588 9 27 81 741.1765 2223.529

2002 4 270.5882 16 64 256 1082.353 4329.412

2003 5 282.3529 25 125 625 1411.765 7058.824

2004 6 311.7647 36 216 1296 1870.588 11223.53

2005 7 335.2941 49 343 2401 2347.059 16429.41

2006 8 358.8235 64 512 4096 2870.588 22964.71

2007 9 382.3529 81 729 6561 3441.176 30970.59


45 2600 285 2025 15333 14394.12 96264.71

Table 2.17

(6.5.1)Linear model:


y=a0+a1∗t

[ n ∑ t

∑ t ∑ t2 ]∗[a0

a1]=[ ∑ y

∑( y∗t )]


Therefore: ao = 172.172, a1= 23.2353

Year t y y^ e e2

1999 1 194.1176 195.4077 -1.29005 1.664236

2000 2 217.6471 218.643 -0.99593 0.991882

2001 3 247.0588 241.8783 5.18054 26.838

2002 4 270.5882 265.1136 5.47466 29.97191

2003 5 282.3529 288.3489 -5.99593 35.95112

2004 6 311.7647 311.5842 0.180548 0.032597

2005 7 335.2941 334.8194 0.474668 0.225309

2006 8 358.8235 358.0547 0.768788 0.591035

2007 9 382.3529 381.29 1.062908 1.129773

Σe2

97.39586

Table 2.18


Where

y=a0+a1∗t+a2∗t2

[ n ∑ t ∑ t2

∑ t ∑ t2 ∑ t3

∑ t2 ∑ t3 ∑ t4 ]∗[a0

a1

a2]=[ ∑ y

∑ ( y∗t )

∑ ( y∗t2 )]


Therefore ao=172.128831 , a1= 23.5536 , a2= - 0.03183

Year t y y^ e e2

1999 1 194.1176 195.6506 -1.53296 2.349963

2000 2 217.6471 219.1087 -1.46166 2.136448

2001 3 247.0588 242.5032 4.555656 20.754

2002 4 270.5882 265.834 4.754282 22.6032

2003 5 282.3529 289.1011 -6.74814 45.53733

2004 6 311.7647 312.3045 -0.53983 0.291417

2005 7 335.2941 335.4443 -0.15022 0.022565

2006 8 358.8235 358.5205 0.303061 0.091846

2007 9 382.3529 381.5329 0.820002 0.672403

Σe2

94.45917

Table 2.19

From Least Squared Error Criteria we will use Quadratic model, therefore :

Year t

1999 1 195.6506

2000 2 219.1087

y


2001 3 242.5032

2002 4 265.834

2003 5 289.1011

2004 6 312.3045

2005 7 335.4443

2006 8 358.5205

2007 9 381.5329

2008 10 404.4817

2009 11 427.3669

2010 12 450.1884

2011 13 472.9462

2012 14 495.6404

2013 15 518.2708

2014 16 540.8377

2015 17 563.3408

2016 18 585.7803

2017 19 608.1562

Table 2.20

For 2017:Load (MVA) =608.1562 MVA

(6.6)The Final Forecasted Load Density for year 2017:


Agriculture ResidentialHeavy

IndustriesLight

IndustriesCity

Center

Load(MVA) 140.9368 376.5831 608.1562 257.7087 1215.43

Area(Km2) 125 112.5 37.5 37.5 187.5

Load Density(MVA/KM2)

1.1275 3.3474 16.2175 6.872 6.4823

Table 2.21

chapter 2 load forecasting

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