Master's Thesis Mateusz Andrychowicz - CIRE.pl · 1 POLITECHNIKA ŁÓDZKA WYDZIAŁ ELEKTRONIKI,...

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POLITECHNIKA ŁÓDZKA

WYDZIAŁ ELEKTRONIKI, ELEKTROTECHNIKI, INFORMATYKI I AUTOMATYKI

INSTYTUT ELEKTROENERGETYKI

MASTER’S THESIS

Praca systemów dystrybucyjnych z dużym udziałem rozproszonych źródeł energii.

(Operation of distributed systems with high penetration of distributed energy

resources.)

Author:

Mateusz Andrychowicz

Supervisor:

dr inż. Błażej Olek

Łódź, czerwiec 2015

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TABLE OF CONTENT

1 INTRODUCTION ......................................................................................................................... 5

2 GENERAL INFORMATION ABOUT POLISH POWER SYSTEM. ...................................... 6

2.1 Power system in Poland. ..................................................................................................................... 6

2.2 Energy market ................................................................................................................................... 12

2.3 Power trading .................................................................................................................................... 14

2.3.1 Balancing market ................................................................................................................................. 15

2.4 TPA and Unbundling .......................................................................................................................... 17

3 POLISH ENERGY MARKET. .................................................................................................. 19

3.1 General information about Polish energy market. ............................................................................. 19

3.2 Day Ahead Market (DAM) and Intra Day Market (IDM) ..................................................................... 21

3.3 Commodity Forward Instruments Market with Physical Delivery (CFIM), Property Right Market

(PRM) and Emission Allowances Market (EAM). ............................................................................................ 23

4 MATHEMATICAL METHODS. .............................................................................................. 25

4.1 Polynomials and orthogonal polynomials. ......................................................................................... 25

4.2 Classic orthogonal polynomials.......................................................................................................... 26

4.3 Approximation and Interpolation ...................................................................................................... 29

5 DATA ANALYSIS OF POLISH POWER SECTOR’S DEMAND ........................................ 32

5.1 Extreme demand’s profiles ................................................................................................................ 32

5.2 Reserves and use of CDGU ................................................................................................................. 35

6 RESULTS AND SIMULATIONS ............................................................................................. 37

6.1 Description of methods using in simulations ..................................................................................... 37

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6.2 Choosing the best method( interpolation, approximation) and degree of polynomial ....................... 38

6.2.1 Household in city at tariff G11 ............................................................................................................. 38

6.2.2 Final results and conclusions about the best approximation method. ............................................... 52

6.3 Selection of the best approximation method ..................................................................................... 58

6.4 Final results ....................................................................................................................................... 59

6.4.1 Household in city in tariff G11 ............................................................................................................. 60

6.4.2 Other profiles....................................................................................................................................... 65

6.5 Summation ........................................................................................................................................ 70

7 APPLICATIONS ........................................................................................................................ 71

8 APPENDIX ................................................................................................................................. 72

9 REFERENCES ............................................................................................................................ 89

10 STRESZCZENIE .................................................................................................................... 91

11 SUMMARY ............................................................................................................................. 92

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1 Introduction

The aim of the thesis is to analyse the distribution network in terms of demand’s profiles.

The analysis consists of creating representative profiles for particular customers who

operate in different energy tariffs. The representative profiles are created by using various

types of mathematical tools: interpolation and approximation. Customers are divided into

three tariff groups: G11, G12 and C11. Groups G11 and G12 are also divided into

households in a city and in a village.

The scope of the thesis includes:

• description of the Polish Power System,

• description of the Polish Energy Market,

• definition of the basic terms of mathematical tools related to approximation and

interpolation,

• analysis of the Polish power sector according to demand’s data,

• analysis of the most convenient methods of approximation and interpolation,

• description of demand’s profile for each study group,

• creation of the representative profiles for each group,

• preparation of the final conclusion.

The analyzed topic is important to ensure the security of supply. In view of the nature of

electricity which cannot be stored for large scale production, and the demand has to be

balanced in each period of a day. Nowadays, when renewable energy sources have in each

year higher share of energy production it is important to have the best available tools to

create demand’s profile for each day.

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2 General information about Polish power system.

2.1 Power system in Poland.

Polish power sector is based on coal. In 2013 49,6 % of electricity was generated from hard

coal and 34,1 % from lignite. Another source which is important in polish power system

there are renewable energy sources which generated 10,4 % of electricity in 2013. Below

there is a graph which present structure of electricity production in Poland in 2013.

Figure 1. Structure of electricity production in Poland in 2013 (Own development based on [1]).

Energy production in 2012 was 162,139 TWh and the installed capacity was 38 046 MW.

Largest producer of energy was PGE S.A. which produced 63,2 TWh they also had the

largest amount of installed capacity – 12 990 MW. Second biggest company was TAURON

Polska Energia S.A. which produced 22,1 TWh and had 5 509 MW in installed capacity.

Company Capacity [MW] Generation [TWh]

PGE S.A. 12 990 63,2

TAURON Polska

Energia S.A.

5 509 22,1

Enea S.A. 3 109 13,0

Energa S.A. 1 150 4,6

PAK S.A. 2 462 11,7 Table 1. Characteristic of biggest power company in Poland Own development based on [2], [12])

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Installed capacity is the maximum amount of energy which can be generated by power

plant. In Poland installed capacity rise from 34 595 in 2000 to 38 406 in 2013. Available

capacity is different than installed capacity due to power’s loss which last longer than one

year.

Year Installed capacity [MW] Available capacity [MW]

2000 34 595 33 418

2001 34 722 33 628

2002 34 945 33 621

2003 34 805 34 209

2004 34 715 34 114

2005 34 673 34 544

2006 34 864 34 718

2007 35 096 34 872

2008 35 342 34 922

2009 35 954 35 243

2010 35 756 35 509

2011 37 367 37 010

2012 38 046 37 720

2013 38 406 38 112 Table 2. Installed and available capacity in years 2000-2013 (Own development based on [3], [5])

Daily maximum demand is a value of power that power system has to assure to stable work

of a power system in conditions of maximum demand. Daily maximum demand rise from

22 050 MW in 1994 to 24 761 MW in 2013.

Year Daily maximum demand [MW]

1994 22 050

1995 23 056

1996 23 439

1997 24 337

1998 23 384

1999 22 821

2000 22 289

2001 22 868

2002 23 207

2003 23 288

2004 23 108

2005 23 477

2006 24 640

2007 24 611

2008 25 120

2009 24 593

2010 25 449

2011 24 780

2012 25 845

2013 24 761 Table 3. Average daily maximum demand and daily maximum demand in years 1994-2013 (Own development based on [3])

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In Polish power system there are various level of power lines. Level of the voltage depends

on the distance for which energy has to be sent. There are two type of power line: power

transmission line and power distribution line. For the power transmission line belongs line

of voltage:

• 750 kV,

• 400 kV,

• 220 kV.

For the power distribution lines belong lines of voltage:

• 110 kV,

• 230 V to 30 kV.

Lines of the voltage 750 kV, 400 kV and 220 kV are used to transmission energy on the long

distance. Lines of the voltage 110 kV are used to transmission energy on the distance less

than one hundred kilometers. 230 V to 30 kV lines are used in local distribution system.

Figure 2. Map of Polish transmission grid. (Own development based on [25])

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In Poland there is one TSO – PSE. Aim of PSE is to provide the security of the Polish Power

System operations. Another aims of TSO in Poland is transmission the energy in compliance

with all criteria and balancing demand, generation and management the system

constraints in case of ensure the security of operation the powers system. Main activities of

PSE there are:

• assurance the Polish Power System’s secure and economical operation as a part of

European Power System,

• assurance the necessary development and maintenance of the cross-border

interconnections and national transmission grid,

• assurance the cross-border exchange’s transmission capacity on market based

methods,

• assurance the necessary technical infrastructure for operations on the domestic

wholesale electricity market.

TSO is also responsible for preparing the coordination plans. They contain plans for

operation the Centrally Dispatched Generating Unit in different time interval. Coordination

plans are divided into:

• daily coordination plan,

• monthly coordination plan,

• yearly coordination plan,

• daily preliminary coordination plan,

• daily current coordination plan,

• daily technical-trading balance.

Daily coordination plan is created by TSO in day n-1 for each hour of day n. It is made up

based on reported transaction for physical power transmission and trading and technical

data from balancing bids taking into account system restrictions and required reserve. Daily

plan contains operation plan for Centrally Dispatched Generating Unit for each hour of day

taking into account forecast for daily demand, reserve and system constraints.

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Time Action

Initiating actions.

Day n-1

9.00-14.30

Update of generation units data .

Bidding data are collected.

Day n-1

9.00-14.30

Trading and technical data for particular generation units are update.

Data from DSOs about generation forecast from particular units are collected.

Creation of daily coordination plan.

Day n-1

12.00-15.00

Data of power system balance are updated.

Data of actual condition of power system are collected.

Day n-1

12.00-15.00

Power system’s restrictions are updated.

Day n-1

14.30-15.00

Power flows are appointed.

Table 4. Schedule of creating daily coordination plan (Own development based on [16)

Monthly coordination plan is created by TSO for one month. Plans for each month except

march are made up to 25th day of month before. For march plans are made up to 23th of

February. Monthly coordination plan for particular day of the month contain:

• forecast of installed capacity in CDGU and nCDGU

• forecast of available capacity in CDGU and nCDGU including repairs which are

reported by producers and loss caused by work condition of system in daily peak,

• demand for each day in normal conditions and in peaks,

• forecast of cross-border electricity exchange,

• forecast of load for nCDGU in daily peaks,

• forecast of power reserves in national generation units,

• forecast of power reserves for TSO,

• forecast of restrictions for cross -border electricity exchange.

Yearly coordination plan is created by TSO for three years. They are made up to 30th of

November in each year. Yearly coordination plan for particular month contain:

• forecast of average monthly installed capacity in CDGU and nCDGU,

• forecast of average monthly available capacity in CDGU and nCDGU including repairs

which are reported of by producers and loss caused by work condition of system in

daily peak from working day,

• forecast of demand for each month in normal conditions and in peaks including

weather conditions from working day,

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• forecast of maximum demand for each month ,

• forecast of average monthly cross-border electricity exchange from working day,

• forecast of average monthly load for nCDGU in daily peaks from working day,

• forecast of average monthly power reserves in national generation units from

working day,

• forecast of restrictions for cross -border electricity exchange in all period.

Daily preliminary coordination plan is created by TSO for one day. It is made based on daily

technical-trading balance and system data.

Time Action

Actions which initiate daily preliminary coordination plan creation.

Day n-2

8.00-11.00

Update the system data and the power balance is national power system: demand in power system, level of reserves, cross-border electricity exchange.

Update the technical data of generation units.

Update the available capacity of generation units.

Analysis of system’s restrictions.

Day n-2

8.00-13.00

Preliminary analysis of system’s restrictions

Update the transmission network’s availability.

Preliminary plans creation using system services.

Day n-2

8.00-13.00

Plan of using CDGU to regulation is created.

Levels of power reserves are created.

Actions which close the creation of plan.

Day n-2

13.00-

16.00

Daily preliminary coordination plan is created.

Day n-2

to 16.00

Daily preliminary coordination plan is available.

Table 5. Schedule of creating daily preliminary coordination plan. (Own development based on [16])

Daily current coordination plan created by TSO in day n-1 for each 15 minutes period of

day n. It is created directly after creation of daily coordination plan. Daily current

coordination plan can be modified if special circumstances occur. Special circumstances are

connected with:

• power’s demand,

• cross -border electricity exchange,

• CDGU and nCDGU generation,

• available CDGU capacity.

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Daily technical-trading balance is created by TSO every day for seven days. It is made up

based on actual and forecast situation in power system. Daily technical-trading balance

contain values:

• power demand,

• available capacity in CDGU,

• available capacity in nCDGU,

• generation from CDGU,

• generation from nCDGU,

• upward reserve,

• downward reserve,

• system restrictions.

President of Energy Regulatory Office was appointed by virtue of Energy Law from 10 April

1997 and he is responsible for regulation in energy sector. His competences and duties are

connected with the state energy policy. The main activities of the President of ERO there

are connected with:

• creation of sustainable economic growth in the country,

• ensuring energy security,

• rational and economical use of energy and fuels,

• counteracting negative effects of natural monopolies,

• development of competition,

• environmental protection.

2.2 Energy market

Power sector developed in the middle of XX century. In that time power sector was

centralized and run by county’s mechanisms. In Poland energy market was being

introduced since April 1997 when the Energy Law came into force.

The aim of introducing electricity market was appropriate power prices for consumers

while:

• security of supply,

• high quality of energy,

• increase efficiency of energy using,

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• assurance the energy sector money for developed power sector,

• low level of gases emission.

Participants of energy market are divided into 4 groups:

• power producers,

• companies which deal with energy trade,

• companies which deal with energy transmission,

• customers.

For the power producers belong system power plants. In Poland there are 19 power plants

this type and only coal power plants belong to system power plants. Another energy

sources which belong to power producers there are heat and power plants and renewable

energy sources.

Companies which deal with power trade buy power from producers and sell it to final

consumers. Prices and terms of bargain are set individually between companies which buy

and sell power or result of power buying rules. A special group make a companies which

came into being after section in half of the 2007. This companies sell energy to households

located at the former regional power companies area. Prices of the power in this group

there are specified in tariffs which were approved by Energy Regulatory Office.

According to the technical aspects power is transmitted as high voltage as possible. Power

is transmitted by two type of power line: power transmission line and power distribution

line. According to this there are two type of companies which deal with energy

transmission. First of its is transmission system operator which is PSE, 100 % of its shared

belong to State Treasury. Second type there are distribution system operators. Five biggest

distribution companies are: PGE, Tauron, Energa, Enea, RWE.

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Figure 3. Area of operating the biggest distribution system operators. (Own development based on [2])

Customers divide in two groups. First group there are households. In first group prices of

the energy are dependent of the tariff for which customers belong. Second group there are

other customers which convent agreements with the power companies individually. [7]

2.3 Power trading

Electricity market consist of three part which operate as an complementary whole.

Electricity market is divided for:

• technical market,

• financial market,

• active power market.

Technical market consist of trading the power’s reserves, ancillary control services and the

trading of the power generated in specific power units to ensure technical parameters of

power system. Such services were introduced to ensure appropriate standards of working

polish power system, which are particularly:

• stable and secure work of power system,

• assurance the appropriate level of power’s reserves and ancillary control services,

• assurance the reliability and appropriate quality of the power.

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Financial market is a market where the products is energy with specified volumes and

prices. Trading concern energy but not the physical supply of energy. Aim of this market is

to secure participants from changing the energy prices in future.

Active power market is a market where product of trading is energy with specified volumes,

prices, time and place of delivery. Active power market consists of the parts. This parts are:

• the bilateral transactions segment,

• the exchange-based segment,

• the balancing segment.

Bilateral contracts are covenanted for energy trading between market’s participants.

Conditions of the agreements like energy prices, volumes or date of delivery depend on

individual trade agreements. Trade agreements are known only for the sides which

covenanted the agreements, only energy volumes, time, place of delivery and a technical

aspects of the transmission have to be sent to the transmission system operator. [29]

Exchange-based segment have a form of conventional exchange bargains or contracts. This

type of trading is carried out by Polish Power Exchange. which operate based on law of

commodity exchange and a internal rules of exchange. Exchange contracts are covenanted

in form of day ahead and futures contracts. Energy in day ahead market is contracted for

each hour in the next day since the bargain. Energy in futures energy market is contracted

for period of month or longer. Participant of the exchange-based segment can be natural

person or legal person which have a company operate on the energy field or customers

eligible to use a transmission services. [29]

2.3.1 Balancing market

Balancing market is a part where transmission system operator compare generation and

demand of the energy taking into account all bargains and contracts for energy. PSE S.A.

buy or sell energy to balance the energy in power system. Operating the balancing market

is possible thanks to quick increasing or reduction the energy production by power units. In

this reason prices of the energy on the balancing market are appointed based on contracts

between transmission system operator and energy producers. Participants on the balancing

market can be:

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• trade companies,

• generators,

• customers,

• power exchanges.

Each of participants have to sign Transmission Agreements with the PSE S.A. [17]

Figure 4. Structure of the balancing market in the decentralized electricity market. [30], [22]

Producers have to present bids to TSO for each hour of a Day Ahead Market. Bids are

arranged in ten bands.

Figure 5. Example of the balancing bid. [30]

Bids of Day Ahead Market are trimmed in merit order. If contracted energy is higher than

forecast demand then R bands are accepted. If contracted energy is lower than forecast

demand then TSO has to bay energy to cover the differences by accepting P bands.

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Figure 6. Comparison of the “P” and “R” offers acceptance. [30], [18], [19]

2.4 TPA and Unbundling

TPA(Third Party Access) is a condition and warranty to exist energy free market. In 2007 in

Poland power companies had to divided production and transmission of energy for two

different activities. From that date customers are obliged to buy energy and transmission

services separately. Customer has a possibility to chose energy seller but transmission is

connected with company which responsible for area where customer is located. There are

two forms of buying energy:

• energy from seller who responsible is for area where customer is located,

• energy from any seller which is on the market. [8]

Energy prices in first form are specified in seller’s tariffs, which are approved by Energy

Regulatory Office. In second form customer can buy energy from all kind of seller:

producer, energy trading company, on power exchange or through the internet energy

trading platform.

Unbundling is one of the most important action which was taken to develop

competitiveness in power sector. Before unbundling large power companies were integrate

vertically. It cause that the power sector was monopolized and large power companies

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blocked potential competition. Unbundling is a action which separate transmission and

distribution services from production and supply the energy. There are three type of

unbundling:

• functional unbundling,

• accounting unbundling,

• legal unbundling.

Main rule of functional unbundling that system operator is independent from any kind of

activities which are not associated with transmission or distribution. In this type of

unbundling operators of transmission or distribution system should have possibility to take

action related with assets which are needed to maintenance and repairs, operation and

development the power grids. In this type of unbundling people who are responsible for

management the transmission or distribution system operator cannot be responsible for

issues connected with generation and supply.

Accounting unbundling require from power companies separated accountancy for

particular business area like transmission, generation and supply. Energy Regulatory Office

is a subject which oversee this type of unbundling.

Legal unbundling is a type of unbundling where activities of transmission and distribution

energy are performed by companies which are legally separated from companies which

produce energy. [24]

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3 Polish energy market.

3.1 General information about Polish energy market.

After 1997 when the Energy Law came into force, energy market was being introduced. 7th

December 1999 was registered a Electrim S.A. which was responsible for created and

launched an energy market. Half year after registered the Electrim S.A. spot market was

introduced. Prices on spot market were reference to prices on bilateral contracts. In 2003

POLPX (Polish Power Exchange) received a license from Polish Securities and Exchange

Commission for operation of a commodity market. Supervision of the Financial Supervision

Commission and existing legal regulations are guarantee of suitable functioning of the

market and prevent from any kind of speculation.

In 2005 POLPX was responsible for creation and manage a register of certificates of origin

the electrical power generated from RES. In 2007 it was extended to power generated in

co-generation. In that same time it was launched a Property Rights Market, where

producers of co-generation and renewable energy and entities obliged to acquire

certificates of origin can trade property rights of this certificates.

In 2006 POLPX cooperated with the National Administrator of the Emission Allowance

Certificates Trading System launched a spot market for CO2 emission certificates. On this

market participants can trade European Union Allowance units.

• In 2008 POLPX launched an Electrical Power Derivatives Market and in December

2012 Day-Ahead Market

• Commodity Forward Instruments Market with Physical Delivery

• Intra Day Market

• Property Rights Market

• Emission Allowances Market.

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Figure 7. Shareholders of Polish Power Exchange. (Own development based on [4])

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3.2 Day Ahead Market (DAM) and Intra Day Market (IDM)

Day Ahead Market has operated since 30th June, 2000. It is a physical spot market for

electrical power. Roles of this market are:

• establish a electrical power prices for other contracts which are covenanted on

wholesale market,

• make participants possible to initially balance their contracts,

• make possible to preliminary valuation of power company in view of valuation of

electrical power they produce,

• giving a significant signals for building new generation capacity.

This market in terms of is located between balancing offers on the balancing market and

term contracts with physical power delivery on the intra day market. Transaction on Day

ahead Market are dealt based on:

• hourly demand level from the previous day,

• hourly profile consumption,

• short-term weather forecast,

• expected fluctuation in demand and production for electrical power.

Day Ahead Market on POLPX consist of 24-hour markets, where on each one type of hourly

contract is recorded. Additionally on the Day Ahead Market 3 three block contracts are

recorded. Type of block contracts [23]:

• BASE where contracts are covenanted for delivery of 1 MWh of power in each hour of

a day,

• PEAK where contracts are covenanted for delivery of 1 MWh of power in the demand

peak hour between 7.00 and 22.00,

• Offpeak where contracts are covenanted for delivery of 1 MWh of power in the

demand low hour in two time interval: 0.00 to 7.00 and 22.00 to 24.00.

Day Ahead Market operates every day even in the day off. Accuracy on DAM is PLN

0,01/MWh and minimum order volume is 0,1 MWh.

Participants of DAM sent bids for demand and sales for particular hours. From demand bids

there is created a supply curve and from sales bids demand curve. Transaction price for

each hour is a price where supply and demand curve intersect each other. Rules of

accepting the bids on market are:

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• all demand bids with prices higher than transaction price for particular hour are fully

accepting,

• all supply bids with prices lower than transaction price for particular hour are fully

accepting,

• all demand and supply bids with the same price as transaction price for particular are

fully accepting or reduced in the same proportion for all bids.

Figure 8. Way of creating transaction price. (Own development based on [1])

P- price

Q- quantity

Intra Day Market next to Day Ahead Market is a spot market. This market was launched to

allow adjustment of contractual positions by members of Exchange after closing the trading

on the DAM on the day before delivery the energy and within the energy supply delivery

day.

Overall information about IDM:

• hourly contracts,

• physical delivery of electrical power,

• trading: everyday form 7.15 to 14.30,

• trading is based on continuous and fixing trading systems .

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Figure 9. Schedule of how Day Ahead Market and Intra Day Market operate. (Own development based on[4])

Day n-1 - Exchange members can place orders in during continuous trading and the fixing

system. Orders can be bided for any day n-1 hour.

Day n - Exchange members can bid orders during continuous trading. In next hours of the

exchange session, there is decrease the number of hours for biding the orders .

3.3 Commodity Forward Instruments Market with Physical Delivery

(CFIM), Property Right Market (PRM) and Emission Allowances

Market (EAM).

November 19th, 2008 was the date when Commodity Forward Instruments Market with

Physical Delivery was established. This type of market is intended for long horizon contract

for energy with specified date of delivery in future and price. [26]

Forward Instruments Market with Physical Delivery sessions are available from Monday to

Friday from 8.00 to 14.00. Market operate in compliance with the rules which are specified

in the Exchange Regulations. There are four types of instruments on CFIM:

• Weekly,

• Monthly,

• Quarterly,

• Yearly.

December 28, 2005 Property Rights Market was established. Firstly only certificates of

origin for energy generated in Renewable Energy Sources ware available on PRM. In the

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next years other certificates of origin ware introduced. Now on PRM four type of

certificates of origin are available. These certificates are:

• certificates of origin for energy generated in Renewable Energy Sources,

• certificates of origin for energy generated in high-efficient CHP units fired with gas

obtained during processing of biomass or with methane,

• certificates of origin for energy generated in high-efficient CHP units fired with the

total installed power lower than 1 MW or with gaseous fuels,

• certificates of origin for energy generated in other high-efficient CHP units.

For each company which sell energy there is imposed duty to obtain the certificates of

origin and to submit them to the ERO President for redemption. Another possibility to fulfill

this duty is paid substitution payment in account of The National Fund of Environment

Protection and Water management. In most cases prices of certificates of origin are lower

then substitution payment.

The Emission Allowances Market has been in place since September 19, 2006 and it is an

integral part of the national and European CO2 Emission Allowances Trading System.

The Emission Allowances Market is characterized by:

• Trading is available twice a week on Tuesdays and Thursday,

• On market there are continuous, fixing trading and OTC systems.

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4 Mathematical methods.

4.1 Polynomials and orthogonal polynomials.

Polynomial is an algebraic expression which contain of variable and constant factors. They

are connected by addition, subtraction and powers signs. Polynomial due to their simplicity

and good knowledge of property are used in many branches of mathematics. A polynomial

with one term is called a monomial, with two terms is called a binomial and with three

terms is called a trinomial. [13]

A polynomial of one variable can be presented as:

�� ∗ �� + �� ∗ �� +⋯+ � ∗ � + �� ∗ �� + � (4.1)

where:

� …�� are constant factors,

is integer,

� is variable factor.

Example of polynomial:

−�� + 2 ∗ �� + 2 ∗ � − 5 ∗ � + 6 (4.2)

Figure 10. Example of fourth degree polynomial. (Own development)

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Orthogonal polynomials are type of polynomials where two of them are in the sequence in

of orthogonal to each other under some inner product. This type of polynomials are classes

of polynomials {pn(x)} defined on the interval characteristic to them. The most common

used orthogonal polynomials are classic orthogonal polynomials which are:

• Chebyshev polynomials,

• Hermite polynomials,

• Laguarre polynomials,

• Legendre polynomials.

Orthogonal polynomials are very useful to solve mathematical and physical problems. [60],

[10]

4.2 Classic orthogonal polynomials

Chebyshev polynomials are named after Pafnut Chebyshev. There are two kind of

Chebyshev polynomials first(Tn) and second(Un) kind. Polynomial of n degree consist of

sequence of n Chebyshev polynomials. Subject of the Tn and Un are at [-1;1] interval. [28]

Recurrence relation of Chebyshev polynomials first kind:

�� = 1 (4.3)

�� = � (4.4)

�� = 2� − 1 (4.5)

�� = 4�� − 3� (4.6)

…

�� = 2� ∗ �� − �� (4.7)

Weighting function of Tn:

�� = 1√1 − �

(4.8)

Trigonometric definition of Tn:

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�� = cos� � !!"#�� (4.9)

Recurrence relation of Chebyshev polynomials first kind:

$�� = 1 (4.10)

$�� = 2� (4.11)

$�� = 4� − 1 (4.12)

$�� = 8�� − 4� (4.13)

…

$�� = 2� ∗ $�� − $�� (4.14)

Weighting function of Un:

�� = &1 − � (4.15)

Trigonometric definition of Un:

$�� = cos� � !!"#�� (4.16)

Using Chebyshev polynomials in interpolation minimizes the problem of Runge’s

phenomenon and gives the better approximation results. [14], [20]

Hermite polynomials(Hn) are named after Charles Hermite. Subject of the Hn is at [-∞;∞]

interval. [15]

Recurrence relation of Hermite polynomials:

'�� = 1 (4.17)

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'�� = 2� (4.18)

'�� = 4� − 2 (4.19)

'�� = 8�� − 12� (4.20)

…

�� = 2� ∗ �� − 2� − 1� ∗ �� (4.21)

Weighting function of Hn:

�� = (�)* (4.22)

Laguarre polynomials(Ln) are named after Edmond Laguarre. Subject of the Ln is at [0;∞]

interval.

Recurrence relation of Laguarre polynomials:

+�� = 1 (4.23)

+�� = � − 1 (4.24)

+�� = 12 �� − 4� + 2�

(4.25)

+�� = 16 �−�� + 9� − 18� + 6�

(4.26)

…

+�-�� = �2 + 1 − �� ∗ +�� − ∗ +�� + 1

(4.27)

Weighting function of Ln:

�� = (�) (4.28)

29

Legendre polynomials(Pn) are named after Edmond Laguarre. Subject of the Ln is at [-1;1]

interval.

Recurrence relation of Legendre polynomials:

.�� = 1 (4.29)

.�� = � (4.30)

.�� = 12 �3� − 1�

(4.31)

.�� = 12 �5�� − 3��

(4.32)

�� = (�) (4.33)

…

.�-�� = 2 + 1 + 1 � ∗ .�� −

+ 1 ∗ .�� (4.34)

Weighting function of Ln:

�� = 1 (4.35)

Polynomial interval w(x)

Chebyshev - first

kind

[-1;1] 1√1 − �

Chebyshev - second

kind

[-1;1] &1 − �

Hermite [-∞;∞] (�)*

Laguarre [0;∞] (�)

Legendre [-1;1] 1

Table 6. Interval and weighting function for various type of orthogonal polynomials. (Own development based on

4.3 Approximation and Interpolation

Approximation is a process which determine solutions which are close to real solutions.

Approximation is a method which is used very often in mathematical problems. In many

30

cases approximation based on creation a function according to data set. Function should be

created in that way to ensure the least error between value from data set and values lied

on the function curve. It leads to minimization the error function. The most popular way to

measure that error is mean square error.

There are a lot of approximation methods. The most popular there are:

• least squares,

• uniform approximation,

• linear approximation.

In a lot of methods there are used various type of mathematical functions. Approximation

function could be presented in many ways but the most popular are:

• polynomials,

• regression models,

• spline functions,

• artificial biological neural networks. [9]

Least squares is an approximation which aim is to minimize the error on [a;b] interval. Level

of error in particular points is defined by weighted function(w(x)). Mathematical

interpretation of minimization the error:

/ = 0 �� ∗ 12�� − 3��45�6

7

(4.36)

where:

g(x) – approximation function,

f(x) – function which is approximated.

Interpolation is a mathematical method which aim is created a interpolation function which

taking a specified values for specified data points. Interpolation is a special type of

approximation. The most popular method of interpolation there are:

• linear interpolation,

• polynomial interpolation ,

• spline interpolation.

31

One of the most popular polynomial interpolation is interpolation which use Lagrange

polynomials. In Lagrange polynomials for polynomial n power there are chosen n+1 points:

x0, x1, … xn and polynomial has shape:

+8�� = 93��:� ; � − �<�: − �<

�

<=7�><?:

�

:=

(4.37)

In each xi polynomial has the same value as function f. [11]

32

5 Data analysis of Polish power sector’s demand

5.1 Extreme demand’s profiles

Balancing the Power system’s demand and generation is crucial issue to ensure security of

supply. Now operators use normal system to present representative day’s profile which is

used to prepare the plan for each day. Nowadays the increasing penetration of RES

especially wind generators cause difficulties when generation from wind power plant is too

high or too small. It cause mismatch between demand and supply. According to this there

are days when a powers system operates under abnormal conditions, that cause

challenging situations for operators. Due to all challenging situations we face the problem,

how to create a profile of representative day which include CDGU (Centrally Dispatched

Generating Unit), nCDGU (Non-Centrally Dispatched Generating Unit) and demand. It will

help to optimize work of power system and increase the security of supply. The thesis refer

to finding appropriate profiles for worst-case scenarios. Below there are four cases which

refer to demand.

• Day with the highest demand.

• Day with the lowest demand.

• Day with the largest difference between off-peak and peak demand.

• MIN_MAX profile.

Date Hour Demand

value,

[MW]

Upward

reserve

value,

[MW]

Downward

reserve

value, [MW]

Day of the

week

nCDGU

generation,

[MW]

Ambient

temperature

[0C]*

Additional

remarks

Annual min demand

4.05 6:00 10 900 5 063 -201 Saturday 3 430 10 Long weekend

5.05 6:00 10 900 4 481 -538 Sunday 3 467 9 Long weekend

3.05 6:00 11 013 4 192 - 538 Friday 4 333 11 Long weekend

Annual max demand

25.01 18:00 24 875 1 530 -7 245 Tuesday 6 386 -7 Evening peak, and cold

22.01 18:00 24 813 1 884 -6 687 Saturday 6 832 -8 Evening peak, snowy and cold

Minimum downward reserve

25.12 3:00 - 9:00

12 000 - 12 625

4 811 - 6 486

0 Wednesday 7 406 - 7 471

3-6 Christmas night

1.01 3:00 - 10:00

12 225 - 13 325

4 904 - 5 645

0 Tuesday 7 192 - 7 280

3-4 New year

2.01 2:00 - 4:00

12 550 - 12 725

4 696 - 5 186

0 Wednesday 6 675 - 6 839

2 Day after New year

Table 7. Extreme values for power system in 2013. (Own development)

33

Figure 11. Highest demand in year 2013. (Own development)

Profile appointed based on maximum demand in the day. (25.01 at 18).

Figure 12. Lowest demand in year 2013. (Own development)

Profile appointed based on minimum demand in the day (4.05 at 6).

34

Figure 13. Largest difference between demands. (Own development)

Profile appointed as difference between highest and lowest demand in the day (2.01).

Figure 14. Min-Max profile – type 1. (Own development)

Figure 15. Min-Max profile – type 2. (Own development)

35

Figure 14 and 15 present the Min-Max profile. This profile was created based on lowest

value in off-peak of demand and largest value in peaks of demand. Each daily profile has

similar shape. It contains of night and day off-peak and morning and evening peak. To draw

a profile there were used values for different days: night volley – 4.05 at 6; day volley –

31.03 at 17; morning peak – 23.01 at 14; evening peak – 24.01 at 18. In figure 15 evening

peak was moved from 18 to 20 to made profile more similar to average profile.

5.2 Reserves and use of CDGU

PSE use two types of reserves. First is upward reserve and the second is downward reserve.

In accordance with the Transmission Network Code upward reserve has to be higher than

9% of demand and downward reserve has to be higher than 500 MW.

Month Upward reserve less

than 9%

[hours per month]

Downward reserve

less than 500 MW

[hours per month]

Use of CDGU

[%]

January 227 17 66

February 241 0 67

March 210 13 66

April 178 3 67

May 206 2 69

June 178 0 69

July 210 1 68

August 257 2 68

September 178 3 72

October 261 0 70

November 178 9 68

December 161 28 62

Table 8. Reserves and use of CDGU in 2013. (Own development)

In accordance with table 8 upward reserve were not covered by 2275 hour per year which

is 26 %. Downward reserve were not covered by 78 hour per year which is less than 1%.

Regarding to this values only upward reserve is taken into account when creating a

representative day’s profile in this thesis.

36

Figure 16. Demand’s profile for a day with highest insufficient upward reserve. (Own development)

Figure 16 shows a profile which was drawn based on the day with 15 hours with insufficient

upward reserve. 15 hours per day of insufficient upward reserve is the highest value for day

in all 2013.

Figure 17. Use of CDGU in 2013, 2014 and three month of 2015. (Own development)

In all month of 2013 and first three month of 2015 use of CDGU was between 62% and 72%.

In 2014 situation was different and in April, May, September and October use of CGDU was

lower than 60% and in June, July and August even lower than 50%.

0%

10%

20%

30%

40%

50%

60%

70%

80%

1.2

01

3

2.2

01

3

3.2

01

3

4.2

01

3

5.2

01

3

6.2

01

3

7.2

01

3

8.2

01

3

9.2

01

3

10

.20

13

11

.20

13

12

.20

13

1.2

01

4

2.2

01

4

3.2

01

4

4.2

01

4

5.2

01

4

6.2

01

4

7.2

01

4

8.2

01

4

9.2

01

4

10

.20

14

11

.20

14

12

.20

14

1.2

01

5

2.2

01

5

3.2

01

5

US

E o

f C

DG

U [

%]

Date

Use of CDGU

37

6 Results and simulations

6.1 Description of methods using in simulations

Simulations are divided in two parts. At first there are used two mathematical methods to

interpolate data: Lagrange polynomials and Chebyshev polynomials. At second part there

are used Chebyshev polynomials and regression model to approximate data. In all methods

there is used Microsoft Office Excel to simulations.

At Lagrange polynomials for each degree of polynomial there are set various hours as

reference nodes. Identified hours for various polynomial:

• third degree polynomial – hours: 1,9,16,24,

• fourth degree polynomial – hours: 1,7,13,18,24,

• fifth degree polynomial – hours: 1,6,10,15,19,24,

• sixth degree polynomial – hours: 1,5,9,13,16,20,24,

At Chebyshev polynomials in interpolation for each degree of polynomial there are set

various hours as reference nodes. Identified hours for various polynomial:

• third degree polynomial – hours: 1,9,16,24,

• fourth degree polynomial – hours: 1,7,13,18,24,

• fifth degree polynomial – hours: 1,6,10,15,19,24,

• sixth degree polynomial – hours: 1,5,9,13,16,20,24,

From Chebyshev recurrence relation for each reference nodes and data’s value there is

built a matrix. Next step is to calculate a determinants of the matrix. Determinants of the

matrix are then used to designate a value for each hour.

At Chebyshev polynomials in approximation there are built two matrix. First consist of

Chebyshev recurrence relation for all 24 hours(X) and second matrix is its transposition(XT).

Next step is multiplication these two matrix (XTX). Next there is multiplication of XT and

data’s value for all hours(Y). Final step it to calculate a determinants of the matrix from XTX

matrix and XTY which is used to designate a value for each hour. [21]

To create profile at regression model there is used Microsoft Excel. REGLIMP function allow

to approximate real date. Final results of this approximation there are four different

polynomials of degree: 3rd, 4th, 5th and 6th.

38

6.2 Choosing the best method( interpolation, approximation) and

degree of polynomial

Five groups of data was analysis to chose the best method and degree of polynomial. Types

of analyzed groups:

• household in city at tariff G11,


• household in village at tariff G11,


• commercial premises at tariff C11.

Analysis for all groups was for 1-7 January and 1-7 July. Criterion of the choice is the lowest

approximation error. [27]

6.2.1 Household in city at tariff G11

Figure 18. Chebyshev polynomials interpolation – 1.1.2008. (Own development)

39

Figure 19 . Lagrange polynomials interpolation – 1.1.2008. (Own development)

Figure 20 . Chebyshev polynomials approximation – 1.1.2008. (Own development)

Figure 21 . Regression model approximation – 1.1.2008. (Own development)

Profile for household in city at tariff G11 has a one morning valley and and evening peak.

40

Hour Approximation

error of third

degree

polynomial

Approximation

error of fourth

degree

polynomial

Approximation

error of fifth

degree

polynomial

Approximation

error of sixth

degree

polynomial

1 0,0% 0,0% 0,0% 0,0%

2 4,9% 5,6% 6,2% 28,4%

3 11,2% 11,8% 5,6% 29,2%

4 15,9% 15,5% 1,8% 14,9%

5 15,1% 12,7% 0,6% 0,0%

6 11,2% 5,7% 0,0% 11,4%

7 8,9% 0,0% 3,1% 17,3%

8 7,7% 4,2% 5,9% 14,9%

9 0,0% 15,0% 0,2% 0,0%

10 0,9% 15,2% 0,0% 8,0%

11 8,5% 6,6% 5,8% 5,3%

12 15,0% 1,2% 10,2% 0,3%

13 13,4% 0,0% 6,3% 0,0%

14 11,2% 1,7% 2,2% 1,5%

15 10,0% 2,0% 0,0% 5,3%

16 0,0% 12,2% 11,2% 0,0%

17 6,0% 4,3% 3,6% 9,1%

18 8,7% 0,0% 1,2% 11,8%

19 5,3% 2,5% 0,0% 4,8%

20 7,7% 1,4% 5,5% 0,0%

21 7,4% 2,5% 8,3% 10,0%

22 11,3% 8,0% 14,5% 13,8%

23 5,5% 3,5% 9,5% 21,5%

24 0,0% 0,0% 0,0% 0,0%

Average

approximation

error

7,7% 5,5% 4,2% 8,6%

Maximum

approximation

error

15,9% 15,5% 14,5% 29,2%

Median of

approximation

error

8,1% 3,9% 3,3% 6,7%

Table 9. Approximation error for Chebyshev polynomial in interpolation 1.1.2008. (Own development)

41

Hour Approximation

error of third

degree

polynomial

Approximation

error of fourth

degree

polynomial

Approximation

error of fifth

degree

polynomial

Approximation

error of sixth

degree

polynomial

1 0,0% 0,0% 0,0% 0,0%

2 4,9% 5,6% 6,2% 28,4%

3 11,2% 11,8% 5,6% 29,2%

4 15,9% 15,5% 1,8% 14,9%

5 15,1% 12,7% 0,6% 0,0%

6 11,2% 5,7% 0,0% 11,4%

7 8,9% 0,0% 3,1% 17,3%

8 7,7% 4,2% 5,9% 14,9%

9 0,0% 15,0% 0,2% 0,0%

10 0,9% 15,2% 0,0% 8,0%

11 8,5% 6,6% 5,8% 5,3%

12 15,0% 1,2% 10,2% 0,3%

13 13,4% 0,0% 6,3% 0,0%

14 11,2% 1,7% 2,2% 1,5%

15 10,0% 2,0% 0,0% 5,3%

16 0,0% 12,2% 11,2% 0,0%

17 6,0% 4,3% 3,6% 9,1%

18 8,7% 0,0% 1,2% 11,8%

19 5,3% 2,5% 0,0% 4,8%

20 7,7% 1,4% 5,5% 0,0%

21 7,4% 2,5% 8,3% 10,0%

22 11,3% 8,0% 14,5% 13,8%

23 5,5% 3,5% 9,5% 21,5%

24 0,0% 0,0% 0,0% 0,0%

Average

approximation

error

7,7% 5,5% 4,2% 8,6%

Maximum

approximation

error

15,9% 15,5% 14,5% 29,2%

Median of

approximation

error

8,1% 3,9% 3,3% 6,7%

Table 10. Approximation error for Lagrange polynomials in interpolation 1.1.2008. (Own development)

42

Hour Approximation

error of third

degree

polynomial

Approximation

error of fourth

degree

polynomial

Approximation

error of fifth

degree

polynomial

Approximation

error of sixth

degree

polynomial

1 5,3% 2,3% 1,3% 2,2%

2 1,0% 0,5% 0,9% 4,2%

3 4,7% 3,2% 2,2% 2,3%

4 8,7% 5,9% 4,8% 2,6%

5 6,7% 3,4% 2,7% 3,9%

6 1,4% 1,7% 1,7% 2,6%

7 1,9% 3,9% 3,3% 2,2%

8 3,5% 4,1% 3,0% 1,2%

9 12,0% 11,2% 10,0% 8,1%

10 10,6% 8,9% 7,9% 8,7%

11 1,6% 0,5% 1,1% 1,4%

12 6,1% 8,1% 8,3% 5,4%

13 4,8% 6,6% 6,5% 3,8%

14 2,8% 4,3% 3,9% 2,1%

15 2,0% 2,9% 2,4% 1,9%

16 8,5% 8,2% 8,8% 7,8%

17 1,7% 1,9% 2,4% 0,5%

18 1,6% 0,8% 0,6% 2,8%

19 1,8% 3,1% 3,0% 1,2%

20 1,1% 0,4% 0,1% 0,5%

21 1,1% 0,3% 0,2% 0,9%

22 5,6% 4,7% 5,3% 2,6%

23 0,3% 0,1% 0,4% 2,2%

24 5,6% 2,2% 3,3% 0,5%

Average

approximation

error

4,2% 3,7% 3,5% 3,0%

Maximum

approximation

error

12,0% 11,2% 10,0% 8,7%

Median of

approximation

error

3,2% 3,1% 2,8% 2,3%

Table 11. Approximation error for Chebyshev polynomial in approximation 1.1.2008. (Own development)

43

Hour Approximation

error of third

degree

polynomial

Approximation

error of fourth

degree

polynomial

Approximation

error of fifth

degree

polynomial

Approximation

error of sixth

degree

polynomial

1 5,3% 2,3% 1,3% 2,2%

2 1,0% 0,5% 0,9% 4,2%

3 4,7% 3,2% 2,2% 2,3%

4 8,7% 5,9% 4,8% 2,6%

5 6,7% 3,4% 2,7% 3,9%

6 1,4% 1,7% 1,7% 2,6%

7 1,9% 3,9% 3,3% 2,2%

8 3,5% 4,1% 3,0% 1,2%

9 12,0% 11,2% 10,0% 8,1%

10 10,6% 8,9% 7,9% 8,7%

11 1,6% 0,5% 1,1% 1,4%

12 6,1% 8,1% 8,3% 5,4%

13 4,8% 6,6% 6,5% 3,8%

14 2,8% 4,3% 3,9% 2,1%

15 2,0% 2,9% 2,4% 1,9%

16 8,5% 8,2% 8,8% 7,8%

17 1,7% 1,9% 2,4% 0,5%

18 1,6% 0,8% 0,6% 2,8%

19 1,8% 3,1% 3,0% 1,2%

20 1,1% 0,4% 0,1% 0,5%

21 1,1% 0,3% 0,2% 0,9%

22 5,6% 4,7% 5,3% 2,6%

23 0,3% 0,1% 0,4% 2,2%

24 5,6% 2,2% 3,3% 0,5%

Average

approximation

error

4,2% 3,7% 3,5% 3,0%

Maximum

approximation

error

12,0% 11,2% 10,0% 8,7%

Median of

approximation

error

3,2% 3,1% 2,8% 2,3%

Table 12. Approximation error for Regression model in approximation 1.1.2008. (Own development)

In household in city at tariff G11 the best results of approximation according with figures 18

to 21 are for Chebyshev polynomials for approximation and for regression model. This

statement is confirmed by results in tables 9 to 12. The best (the lowest) median of

approximation error is for Chebyshev polynomials of sixth degree for approximation and for

regression model of sixth degree polynomials and it is 2,3 %. The best average

44

approximation error is also for Chebyshev polynomials of sixth degree for approximation

and for regression model of sixth degree polynomials and it is 3 %. The worst (the highest)

median of approximation error is for Chebyshev and Lagrange polynomials of third degree

for interpolation and it is 8,1 %. The worst average approximation error is also Chebyshev

and Lagrange polynomials of third degree for interpolation and it is 7,7 %.

Date Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

1.1.2008 8,1% 3,9% 3,3% 6,7%

2.1.2008 7,8% 7,6% 2,9% 1,9%

3.1.2008 10,7% 9,8% 2,1% 5,4%

4.1.2008 11,0% 9,2% 4,7% 6,0%

5.1.2008 7,9% 3,9% 8,6% 4,5%

6.1.2008 6,1% 2,3% 3,5% 3,5%

7.1.2008 9,3% 10,9% 2,8% 4,4%

1.7.2008 6,7% 7,6% 6,1% 4,4%

2.7.2008 5,5% 5,1% 7,5% 5,5%

3.7.2008 12,4% 9,2% 6,2% 6,4%

4.7.2008 16,5% 7,8% 9,1% 4,6%

5.7.2008 8,2% 7,4% 4,4% 2,3%

6.7.2008 7,9% 6,7% 9,0% 2,1%

7.7.2008 9,8% 7,1% 5,0% 3,7%

Table 13. Median of approximation error of Chebyshev polynomial interpolation. (Own development)

Figure 22. Chebyshev interpolation – winter period. (Own development)

Polynomial of third degree lowest value of median of approximation error has for 6 of

January and the highest for 4 of January, all its value are between 11% and 6,1%.

45

Polynomial of fourth degree lowest value of median of approximation error has for 6 of

January and the highest for 3 of January, all its value are between 9,8% and 2,3%.

Polynomial of fifth degree lowest value of median of approximation error has for 3 of


Polynomial of sixth degree lowest value of median of approximation error has for 2 of


Figure 23. Chebyshev interpolation – summer period. (Own development)

Polynomial of third degree lowest value of median of approximation error has for 2 of July

and the highest for 4 of July, all its value are between 16,5% and 5,5%. Polynomial of fourth

degree lowest value of median of approximation error has for 2 of July and the highest for 3

of July, all its value are between 9,2% and 5,1%. Polynomial of fifth degree lowest value of

median of approximation error has for 5 of July and the highest for 4 of July, all its value are

between 9,1% and 4,4%. Polynomial of sixth degree lowest value of median of

approximation error has for 6 of January and the highest for 3 of July, all its value are

between 6,4% and 2,1%.

46

Date Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

1.1.2008 8,1% 3,9% 3,3% 6,7%

2.1.2008 7,8% 7,6% 2,9% 1,9%

3.1.2008 10,7% 9,8% 2,1% 5,4%

4.1.2008 11,0% 9,2% 4,7% 6,0%

5.1.2008 7,9% 3,9% 8,6% 4,5%

6.1.2008 6,1% 2,3% 3,5% 3,5%

7.1.2008 9,3% 10,9% 2,8% 4,4%

1.7.2008 6,7% 7,6% 6,1% 4,4%

2.7.2008 5,5% 5,1% 7,5% 5,5%

3.7.2008 12,4% 9,2% 6,2% 6,4%

4.7.2008 16,5% 7,8% 9,1% 4,6%

5.7.2008 8,2% 7,4% 4,4% 2,3%

6.7.2008 7,9% 6,7% 9,0% 2,1%

7.7.2008 9,8% 7,1% 5,0% 3,7%

Table 14. Median of approximation error of Lagrange polynomial interpolation. (Own development)

Figure 24. Lagrange interpolation – winter period. (Own development)







Polynomial of sixth degree lowest value of median of approximation error has for 2 of


47

Figure 25 . Lagrange interpolation – summer period. (Own development)




of July, all its value are between 9,2% and 5,1%. Polynomial of fifth degree lowest value of





Date Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

1.1.2008 3,2% 3,1% 2,8% 2,3%

2.1.2008 7,8% 8,8% 3,4% 3,0%

3.1.2008 10,1% 11,0% 6,0% 6,3%

4.1.2008 8,5% 9,6% 4,9% 6,3%

5.1.2008 3,8% 3,3% 4,0% 3,6%

6.1.2008 4,8% 3,5% 4,1% 3,8%

7.1.2008 8,2% 10,3% 5,2% 5,9%

1.7.2008 6,7% 6,1% 5,5% 5,3%

2.7.2008 6,6% 6,5% 7,5% 5,0%

3.7.2008 7,9% 7,7% 4,5% 3,6%

4.7.2008 7,5% 7,1% 4,4% 3,3%

5.7.2008 7,6% 6,7% 4,3% 3,1%

6.7.2008 10,5% 7,5% 8,2% 4,1%

7.7.2008 9,7% 10,0% 7,2% 3,7%

Table 15. Median of approximation error of Chebyshev polynomial approximation. (Own development)

48

Figure 26. Chebyshev approximation– winter period. (Own development)






January and the highest for 3 of January, all its value are between 6% and 2,8%. Polynomial

of sixth degree lowest value of median of approximation error has for 1 of January and the

highest for 3 and 4 of January, all its value are between 6,3% and 2,3%.

Figure 27 . Chebyshev approximation– winter period. (Own development)



49


of July, all its value are between 10% and 6,1%. Polynomial of fifth degree lowest value of





Date Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

1.1.2008 3,2% 3,1% 2,8% 2,3%

2.1.2008 7,8% 8,8% 3,4% 3,0%

3.1.2008 10,1% 11,0% 6,0% 6,3%

4.1.2008 8,5% 9,6% 4,9% 6,3%

5.1.2008 3,8% 3,3% 4,0% 3,6%

6.1.2008 4,8% 3,5% 4,1% 3,8%

7.1.2008 8,2% 10,3% 5,2% 5,9%

1.7.2008 6,7% 6,1% 5,5% 5,3%

2.7.2008 6,6% 6,5% 7,5% 5,0%

3.7.2008 7,9% 7,7% 4,5% 3,6%

4.7.2008 7,5% 7,1% 4,4% 3,3%

5.7.2008 7,6% 6,7% 4,3% 3,1%

6.7.2008 10,5% 7,5% 8,2% 4,1%

7.7.2008 9,7% 10,0% 7,2% 3,7%

Table 16. Median of approximation error of Regression model approximation. (Own development)

Figure 28. Regression model approximation – winter period. (Own development)



50




January and the highest for 3 of January, all its value are between 6% and 2,8%. Polynomial

of sixth degree lowest value of median of approximation error has for 1 of January and the

highest for 3 and 4 of January, all its value are between 6,3% and 2,3%.

Figure 29. Regression model approximation – winter period. (Own development)




of July, all its value are between 10% and 6,1%. Polynomial of fifth degree lowest value of





In figures from 22 to 29 and tables from 13 to 16 are presented data of Median of

approximation error of various level of degree and type of approximation. Data are for

winter period 1-7 January and for summer period 1-7 July. The lowest difference between

highest and lowest value of median of approximation error is for summer period for

Chebyshev polynomials of sixth degree for approximation and for regression model of sixth

degree polynomials and it is 2,7 %. The highest difference between highest and lowest

value of median of approximation error is for summer period for Chebyshev and Lagrange

polynomials of third degree for interpolation and it is 11 %.

51

Type of

approximation

Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

Chebyshev

polynomial

interpolation

8,2% 7,5% 4,9% 4,6%

Lagrange

polynomial

interpolation

8,2% 7,5% 4,9% 4,6%

Chebyshev

polynomial

approximation

7,7% 7,3% 4,7% 3,7%

Regression

model

7,7% 7,3% 4,7% 3,7%

Table 17. Median of approximation error of various polynomial degree for various type of approximation. (Own development)

Type of

approximation

Average

approximation

error of third

degree

polynomial

Average

approximation

error of fourth

degree

polynomial

Average

approximation

error of fifth

degree

polynomial

Average

approximation

error of sixth

degree

polynomial

Chebyshev

polynomial

interpolation

9,0% 5,5% 4,2% 5,4%

Lagrange

polynomial

interpolation

9,0% 5,5% 4,2% 5,4%

Chebyshev

polynomial

approximation

6,5% 6,6% 5,0% 2,9%

Regression

model

6,5% 6,6% 5,0% 2,9%

Table 18. Average approximation error of various polynomial degree for various type of approximation. (Own development)

For both criteria median of approximation error and average approximation error the

lowest value are for Chebyshev polynomials of sixth degree for approximation and for

regression model of sixth degree polynomials and there are 3,7 % for median of

approximation error and 2,9 % for average approximation error. The highest values of

approximation error and average approximation error are for Chebyshev and Lagrange

52

polynomials of third degree for interpolation and there are 8,2 % for median of

approximation error and 9 % for average approximation error

6.2.2 Final results and conclusions about the best approximation

method.

At rest analyzed groups:




• commercial premises at tariff C11,

there are presented graphs for 7th of January to compare their shape and final results.

Figure 30. Profile for household in village at tariff G11. (Own development)

Profile for household in village at tariff G11 has a one morning valley and evening peak.

53

Figure 31. Profile for household in city at tariff G12. (Own development)

Profile for household in city at tariff G12 has a two valley: morning and evening it also has

two peaks: afternoon night.

Figure 32. Profile for household in village at tariff G12. (Own development)

Profile for household in village at tariff G12 has a two valley: morning and evening it also

has two peaks: afternoon night.

54

Figure 33. Profile for commercial premises at tariff C11. (Own development)

Profile for commercial premises at tariff C11has a one morning valley and and evening

peak.

These profiles are for winter period for week’s days. They are shown to demonstrate that

profiles for each tariffs are different. Research are carried out for rest periods and also for

weekend’s days.

Profile Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

Household

in city at

tariff G11

8,2% 7,5% 4,9% 4,6%

Household

in city at

tariff G12

11,8% 7,6% 5,7% 5,5%

Household

in village at

tariff G11

12,7% 8,9% 10,6% 7,0%

Household

in village at

tariff G12

6,2% 5,1% 9,7% 3,7%

Commercial

premises at

tariff C11.

8,2% 4,4% 4,0% 3,0%

Table 19. Median of approximation error for various polynomial’s degree and various type of profiles for Chebyshev

interpolation for winter and summer periods. (Own development)

55

Profile Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

Household

in city at

tariff G11

8,2% 7,5% 4,9% 4,6%

household

in city at

tariff G12

11,8% 7,6% 5,7% 5,5%

Household

in village at

tariff G11

12,7% 8,9% 10,6% 7,0%

Household

in village at

tariff G12

6,2% 5,1% 9,7% 3,7%

Commercial

premises at

tariff C11.

8,2% 4,4% 4,0% 3,0%

Table 20. Median of approximation error for various polynomial’s degree and various type of profiles for Lagrange

interpolation for winter and summer periods. (Own development)

Profile Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

Household

in city at

tariff G11

7,7% 7,3% 4,7% 3,7%

Household

in city at

tariff G12

5,7% 6,0% 5,0% 4,4%

Household

in village at

tariff G11

11,0% 6,7% 6,5% 6,6%

Household

in village at

tariff G12

6,1% 5,9% 5,7% 4,9%

Commercial

premises at

tariff C11.

7,3% 4,1% 3,7% 3,6%

Table 21. Median of approximation error for various polynomial’s degree and various type of profiles for Chebyshev

approximation for winter and summer periods. (Own development)

56

Profile Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

Household

in city at

tariff G11

7,7% 7,3% 4,7% 3,7%

Household

in city at

tariff G12

5,7% 6,0% 5,0% 4,4%

Household

in village at

tariff G11

11,0% 6,7% 6,5% 6,6%

Household

in village at

tariff G12

6,1% 5,9% 5,7% 4,9%

Commercial

premises at

tariff C11.

7,3% 4,1% 3,7% 3,6%

Table 22. Median of approximation error for various polynomial’s degree and various type of profiles for Regression model


Profile Average

approximation

error of third

degree

polynomial

Average

approximation

error of fourth

degree

polynomial

Average

approximation

error of fifth

degree

polynomial

Average

approximation

error of sixth

degree

polynomial

Household

in city at

tariff G11

9,0% 5,5% 4,2% 5,3%

Household

in city at

tariff G12

7,9% 6,7% 7,9% 5,4%

Household

in village at

tariff G11

15,4% 5,6% 8,8% 7,1%

Household

in village at

tariff G12

7,0% 4,5% 12,0% 2,5%

Commercial

premises at

tariff C11.

9,5% 4,4% 7,0% 2,9%

Table 23. Average approximation error for various polynomial’s degree and various type of profiles for Chebyshev interpolation

for winter and summer periods. (Own development)

57

Profile Average

approximation

error of third

degree

polynomial

Average

approximation

error of fourth

degree

polynomial

Average

approximation

error of fifth

degree

polynomial

Average

approximation

error of sixth

degree

polynomial

Household

in city at

tariff G11

9,0% 5,5% 4,2% 5,3%

Household

in city at

tariff G12

7,9% 6,7% 7,9% 5,4%

Household

in village at

tariff G11

15,4% 5,6% 8,8% 7,1%

Household

in village at

tariff G12

7,0% 4,5% 12,0% 2,5%

Commercial

premises at

tariff C11.

9,5% 4,4% 7,0% 2,9%

Table 24. Average approximation error for various polynomial’s degree and various type of profiles for Lagrange interpolation

for winter and summer periods. (Own development)

Profile Average

approximation

error of third

degree

polynomial

Average

approximation

error of fourth

degree

polynomial

Average

approximation

error of fifth

degree

polynomial

Average

approximation

error of sixth

degree

polynomial

Household

in city at

tariff G11

6,5% 6,6% 5,0% 2,9%

Household

in city at

tariff G12

7,8% 6,2% 5,8% 5,2%

Household

in village at

tariff G11

10,5% 8,2% 7,6% 7,7%

Household

in village at

tariff G12

5,5% 4,7% 4,7% 4,3%

Commercial

premises at

tariff C11.

8,7% 4,8% 3,3% 3,1%

Table 24. Average approximation error for various polynomial’s degree and various type of profiles for Chebyshev


58

Profile Average

approximation

error of third

degree

polynomial

Average

approximation

error of fourth

degree

polynomial

Average

approximation

error of fifth

degree

polynomial

Average

approximation

error of sixth

degree

polynomial

Household

in city at

tariff G11

6,5% 6,6% 5,0% 2,9%

Household

in city at

tariff G12

7,8% 6,2% 5,8% 5,2%

Household

in village at

tariff G11

10,5% 8,2% 7,6% 7,7%

Household

in village at

tariff G12

5,5% 4,7% 4,7% 4,3%

Commercial

premises at

tariff C11.

8,7% 4,8% 3,3% 3,1%

Table 26. Average approximation error for various polynomial’s degree and various type of profiles for Regression model


6.3 Selection of the best approximation method

Analysis which compare methods of approximation was carried out on 1680 data for

various profiles and periods using various types of approximation. After analysis it can be

seen that all results for Chebyshev interpolation and Lagrange interpolation are the same,

the same situation is for Chebyshev approximation and Regression model approximation. It

is the reason why in further comparisons there will be taken into consideration Chebyshev

interpolation and approximation. Chebyshev methods are chooses because data entry and

obtaining the results are simpler than for others methods.

Methods Median of

approximation

error of third

degree

polynomial

Median of

approximation

error of fourth

degree

polynomial

Median of

approximation

error of fifth

degree

polynomial

Median of

approximation

error of sixth

degree

polynomial

Chebyshev

interpolation 8,4% 5,8% 6,9% 4,6%

Chebyshev

approximation 7,6% 6,0% 5,1% 4,4%

Table 27. Median of approximation error for various polynomial’s degree for data to all profiles.

59

Analysis of data show that better results of median of approximation error are better for

Chebyshev approximation. What is more the highest polynomial’s degree the lower median

of approximation error.

Methods Average

approximation

error of third

degree

polynomial

Average

approximation

error of fourth

degree

polynomial

Average

approximation

error of fifth

degree

polynomial

Average

approximation

error of sixth

degree

polynomial

Chebyshev

interpolation 9,2% 6,4% 7,5% 4,9%

Chebyshev

approximation 7,8% 6,1% 5,3% 4,8%

Table 28. Average approximation error for various polynomial’s degree for data to all profiles. (Own development)

Analysis of data show that better results of average approximation error are better for

Chebyshev approximation. What is more the highest polynomial’s degree the lower average

approximation error.

For polynomial’s sixth degree difference between Chebyshev approximation and

interpolation for median of median of approximation error is 0,2% and for average

approximation error is 0,1%. So little differences are caused by number of data for each

day. Errors are appointed for 24 data which are equal the hours of the day, for Chebyshev

interpolation of six degree the seven errors are 0% which has high influence for results. If

time interval is more frequent than one hour than results for Chebyshev interpolation are

worse and results for Chebyshev approximation are better.

Summing up all results the best methods to further analysis will be the Chebyshev

approximation of sixth degree polynomial.

6.4 Final results

Aim of the research is to create a characteristic demand profile for each period of a year

and divided week into week’s days and weekend’s days. Below is the proceeding algorithm

which describe how final profile is created.

Figure 34. Proceeding algorithm which describe how final profile is created.

6.4.1 Household in city in tariff G11

Figure 35. Profile for winter period for week’s days for household in city in tariff G11.

Equation which describe profile for winter per

tariff G11:

3�@� = 3,98 ∗ 10�

�0,00356

Data approximation using Chebyshev approximation of polynomial of 6th degree and appointment design feature based on approximated data

Gathering all design feature in one sheet

Appointment new data for each hour of a each day in 2008 and appointment 24 period of a year using median

Characteristicapproximation of polynomial of 6characteristic design feature based on approximated data

0,15

0,2

0,25

0,3

0,35

0,4

0,45

1 2 3 4 5 6

De

ma

nd

[M

W]

thm which describe how final profile is created. (Own development)

Household in city in tariff G11

. Profile for winter period for week’s days for household in city in tariff G11. (Own development

Equation which describe profile for winter period for week’s days for household in city in

�C ∗ @D � 4,86 ∗ 10�D ∗ @E � 1,95 ∗ 10�� ∗ @� �

00356 ∗ @� � 0,0325 ∗ @ � 0,131 ∗ @ � 0,346

Data approximation using Chebyshev approximation of polynomial degree and appointment design feature based on

Gathering all design feature in one sheet

Appointment new data for each hour of a each day in 2008 and appointment 24 characteristic data for each hour of a day for each period of a year using median

haracteristic date data approximation using Chebyshev approximation of polynomial of 6th degree and appointment characteristic design feature based on approximated data

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hours of the day

G11_city_winter_week

60

Own development)

iod for week’s days for household in city in

(6.1.)

data for each hour of a day for each

degree and appointment

24

Czybyszew function

Approximated data

61

This profile has one night valley and one evening peak. Demand rise from 4 to 20 and then

is rapid decrease.

Figure 36. Profile for winter period for weekend’s days for household in city in tariff G11. (Own development)

Equation which describe profile for winter period for weekend’s days for household in city

in tariff G11:

3�@� = −1,79 ∗ 10�G ∗ @D + 1,23 ∗ 10�E ∗ @E − 3,08 ∗ 10�� ∗ @� +

0,00318 ∗ @� − 0,00741 ∗ @ − 0,0457 ∗ @ + 0,319

(6.2.)


is rapid decrease.

Figure 37. Profile for spring period for week’s days for household in city in tariff G11. (Own development)

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_city_winter_weekend

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_city_spring_week

Czybyszew function

Approximated data

62

Equation which describe profile for spring period for week’s days for household in city in

tariff G11:

3�@� = −1,67 ∗ 10�G ∗ @D + 9,84 ∗ 10�D ∗ @E − 1,93 ∗ 10�� ∗ @� +

0,00117 ∗ @� + 0,00497 ∗ @ − 0,0571 ∗ @ + 0,266

(6.3.)

This profile has one night valley and one evening peak. Demand rise rapidly from 4 to 11

than rise to 17 is slowly than between 17 and 22 is quick rise after that hour there is rapid

decrease.

Figure 38. Profile for spring period for weekend’s days for household in city in tariff G11. (Own development)

Equation which describe profile for spring period for weekend’s days for household in city

in tariff G11:

3�@� = −2,67 ∗ 10�G ∗ @D + 1,79 ∗ 10�E ∗ @E − 4,36 ∗ 10�� ∗ @� +

0,00451 ∗ @� − 0,0155 ∗ @ − 0,0105 ∗ @ + 0,239

(6.4.)


than between 13 and 17 demand is on the same level than to 22 is quick rise after that hour

there is rapid decrease.

0,1

0,15

0,2

0,25

0,3

0,35

0,4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_city_spring_weekend

Czybyszew function

Approximated data

63

Figure 39. Profile for summer period for week’s days for household in city in tariff G11. (Own development)

Equation which describe profile for summer period for week’s days for household in city in

tariff G11:

3�@� = −1,39 ∗ 10�G ∗ @D + 8,99 ∗ 10�D ∗ @E − 2,09 ∗ 10�� ∗ @� +

0,00201 ∗ @� − 0,00537 ∗ @ − 0,0137 ∗ @ + 0,186

(6.5.)


than rise to 17 is slowly than between 17 and 22 is quick rise after that hour there is rapid

decrease.

Figure 40. Profile for summer period for weekend’s days for household in city in tariff G11. (Own development)

Equation which describe profile for summer period for week’s days for household in city in

tariff G11:

0,1

0,15

0,2

0,25

0,3

0,35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_city_summer_week

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_city_summer_weekend

Czybyszew function

Approximated data

64

3�@� = −2,90 ∗ 10�G ∗ @D + 2,09 ∗ 10�E ∗ @E − 5,61 ∗ 10�� ∗ @� +

0,00681 ∗ @� − 0,0351 ∗ @ + 0,0580 ∗ @ + 0,143

(6.6.)

This profile has two valley: night and evening and two peak: afternoon and evening.

Demand rise rapidly from 5 to 14 than demand fluctuate between 14 and 22 and then there

is rapid decrease.

Figure 41. Profile for autum period for week’s days for household in city in tariff G11. (Own development)

Equation which describe profile for autumn period for week’s days for household in city in

tariff G11:

3�@� = −2,36 ∗ 10�C ∗ @D − 5,52 ∗ 10�G ∗ @E + 9,16 ∗ 10�E ∗ @� +

−0,00240 ∗ @� + 0,0254 ∗ @ − 0,104 ∗ @ + 0,282

(6.7.)


is rapid decrease.

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_city_autumn_week

Czybyszew function

Approximated data

65

Figure 42. Profile for autumn period for weekend’s days for household in city in tariff G11. (Own development)

Equation which describe profile for autumn period for weekend’s days for household in city

in tariff G11:

3�@� = −2,35 ∗ 10�G ∗ @D + 1,61 ∗ 10�E ∗ @E − 4,09 ∗ 10�� ∗ @� +

+0,00451 ∗ @� − 0,0174 ∗ @ − 0,00360 ∗ @ + 0,236

(6.8.)

Where:

t – hour of the day


is rapid decrease.

6.4.2 Other profiles

For profiles:




• commercial premises at tariff C11,

• PV

there are presented only equations for particular period of a year.

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day


Czybyszew function

Approximated data

66

Profile G11 for household in village

Period Equation

Week’s days in winter 3�@� = −1,18 ∗ 10�G ∗ @D + 6,58 ∗ 10�D ∗ @E +

−1,08 ∗ 10�� ∗ @� − 3,07 ∗ 10�� ∗ @� +

+0,0156 ∗ @ − 0,106 ∗ @ + 0,400

(6.9.)

Weekend’s days in

winter

3�@� = −1,53 ∗ 10�G ∗ @D + 1,02 ∗ 10�E ∗ @E +

−2,49 ∗ 10�� ∗ @� + 0,00243 ∗ @� +

−0,00393 ∗ @ − 0,0451 ∗ @ + 0,361

(6.10.)

Week’s days in spring 3�@� = −1,53 ∗ 10�G ∗ @D + 8,83 ∗ 10�D ∗ @E +

−1,57 ∗ 10�� ∗ @� + 4,45 ∗ 10�� ∗ @� +

+0,0123 ∗ @ − 0,0890 ∗ @ + 0,368

(6.11.)

Weekend’s days in

spring

3�@� = −2,63 ∗ 10�G ∗ @D + 1,71 ∗ 10�E ∗ @E +

−3,96 ∗ 10�� ∗ @� + 0,00367 ∗ @� +

−0,00771 ∗ @ − 0,0409 ∗ @ + 0,343

(6.12.)

Week’s days in

summer

3�@� = −1,38 ∗ 10�G ∗ @D + 8,40 ∗ 10�D ∗ @E +

−1,72 ∗ 10�� ∗ @� + 0,00106 ∗ @� +

+0,00566 ∗ @ − 0,0653 ∗ @ + 0,297

(6.13.)

Weekend’s days in

summer

3�@� = −1,78 ∗ 10�G ∗ @D + 1,18 ∗ 10�E ∗ @E +

−2,80 ∗ 10�� ∗ @� + 0,00258 ∗ @� +

−0,00354 ∗ @ − 0,0472 ∗ @ + 0,298

(6.14.)

Week’s days in

autumn

3�@� = −5,39 ∗ 10�C ∗ @D + 1,45 ∗ 10�D ∗ @E +

+4,29 ∗ 10�E ∗ @� − 0,00197 ∗ @� +

+0,0252 ∗ @ − 0,115 ∗ @ + 0,347

(6.15.)

Weekend’s days in

autumn

3�@� = −1,07 ∗ 10�G ∗ @D + 5,72 ∗ 10�D ∗ @E +

−8,62 ∗ 10�E ∗ @� − 2,39 ∗ 10�� ∗ @� +

−0,0157 ∗ @ − 0,102 ∗ @ + 0,365

(6.16.)

Table 29. Equations for particular period of a year for tariff G11 for household in village. (Own development)

67

Profile G12 for household in city

Period Equation

Week’s days in winter 3�@� = −3,35 ∗ 10�C ∗ @D + 5,03 ∗ 10�D ∗ @E +

−2,13 ∗ 10�� ∗ @� + 0,00362 ∗ @� +

−0,0222 ∗ @ + 0,00287 ∗ @ + 0,628

(6.17.)

Weekend’s days in

winter

3�@� = −2,30 ∗ 10�G ∗ @D + 2,06 ∗ 10�E ∗ @E +

−6,79 ∗ 10�� ∗ @� + 0,0101 ∗ @� +

−0,0633 ∗ @ − 0,0996 ∗ @ + 0,612

(6.18.)

Week’s days in spring 3�@� = −1,44 ∗ 10�G ∗ @D + 1,18 ∗ 10�E ∗ @E +

−3,50 ∗ 10�� ∗ @� + 0,00460 ∗ @� +

−0,0232 ∗ @ − 0,00308 ∗ @ + 0,517

(6.19.)

Weekend’s days in

spring

3�@� = −4,29 ∗ 10�G ∗ @D + 3,43 ∗ 10�E ∗ @E +

−0,00104 ∗ @� + 0,0144 ∗ @� +

−0,0899 ∗ @ − 0,194 ∗ @ + 0,369

(6.20.)

Week’s days in

summer

3�@� = −2,02 ∗ 10�G ∗ @D + 1,49 ∗ 10�E ∗ @E +

−3,98 ∗ 10�� ∗ @� + 0,00452 ∗ @� +

−0,0174 ∗ @ − 0,0153 ∗ @ + 0,303

(6.21.)

Weekend’s days in

summer

3�@� = −3,50 ∗ 10�G ∗ @D + 2,63 ∗ 10�E ∗ @E +

−7,34 ∗ 10�� ∗ @� + 0,00914 ∗ @� +

−0,0464 ∗ @ + 0,0549 ∗ @ + 0,267

(6.22.)

Week’s days in

autumn

3�@� = −3,10 ∗ 10�H ∗ @D − 5,46 ∗ 10�G ∗ @E +

+4,84 ∗ 10�E ∗ @� − 0,00138 ∗ @� +

+0,0185 ∗ @ − 0,109 ∗ @ + 0,481

(6.23.)

Weekend’s days in

autumn

3�@� = −2,42 ∗ 10�G ∗ @D + 1,92 ∗ 10�E ∗ @E +

−5,63 ∗ 10�E ∗ @� + 0,00734 ∗ @� +

−0,0379 ∗ @ − 0,0267 ∗ @ + 0,443

(6.24.)

Table 30. Equations for particular period of a year for tariff G12 for household in city. (Own development)

68

Profile G12 for household in village

Period Equation

Week’s days in winter 3�@� = −4,29 ∗ 10�C ∗ @D − 1,93 ∗ 10�D ∗ @E +

+1,88 ∗ 10�E ∗ @� + 2,42 ∗ 10�E ∗ @� +

+0,0381 ∗ @ − 0,0617 ∗ @ + 0,574

(6.25.)

Weekend’s days in

winter

3�@� = −2,65 ∗ 10�C ∗ @D + 3,40 ∗ 10�D ∗ @E +

−1,36 ∗ 10�� ∗ @� + 0,00208 ∗ @� +

−0,00799 ∗ @ − 0,0407 ∗ @ + 0,601

(6.26.)


−3,56 ∗ 10�� ∗ @� + 0,00469 ∗ @� +

−0,0237 ∗ @ + 0,00850 ∗ @ + 0,476

(6.27.)

Weekend’s days in

spring

3�@� = −2,30 ∗ 10�G ∗ @D + 1,84 ∗ 10�E ∗ @E +

−5,49 ∗ 10�� ∗ @� + 0,0732 ∗ @� +

−0,0395 ∗ @ − 0,0405 ∗ @ + 0,492

(6.28.)

Week’s days in

summer

3�@� = −2,69 ∗ 10�G ∗ @D + 2,03 ∗ 10�E ∗ @E +

−5,80 ∗ 10�� ∗ @� + 0,00759 ∗ @� +

−0,0427 ∗ @ + 0,0693 ∗ @ + 0,346

(6.29.)

Weekend’s days in

summer

3�@� = −2,89 ∗ 10�G ∗ @D + 2,27 ∗ 10�E ∗ @E +

−6,71 ∗ 10�� ∗ @� + 0,00904 ∗ @� +

−0,0522 ∗ @ + 0,0903 ∗ @ + 0,348

(6.30.)

Week’s days in

autumn

3�@� = −9,29 ∗ 10�C ∗ @D + 7,41 ∗ 10�D ∗ @E +

−2,24 ∗ 10�E ∗ @� − 0,00308 ∗ @� +

−0,0164 ∗ @ − 0,00652 ∗ @ + 0,435

(6.31.)

Weekend’s days in

autumn

3�@� = −1,82 ∗ 10�G ∗ @D + 1,48 ∗ 10�E ∗ @E +

−4,54 ∗ 10�E ∗ @� + 0,00635 ∗ @� +

−0,0369 ∗ @ + 0,0501 ∗ @ + 0,454

(6.32.)

Table 31. Equations for particular period of a year for tariff G12 for household in village. (Own development)

69

Profile C11 commercial premises

Period Equation

Week’s days in winter 3�@� = 2,89 ∗ 10�G ∗ @D − 1,62 ∗ 10�E ∗ @E +

+4,87 ∗ 10�� ∗ @� − 0,00892 ∗ @� +

+0,0982 ∗ @ − 0,419 ∗ @ + 1,27

(6.33.)

Weekend’s days in

winter

3�@� = −8,35 ∗ 10�G ∗ @D + 6,02 ∗ 10�E ∗ @E +

−0,00161 ∗ @� + 0,0192 ∗ @� +

−0,0946 ∗ @ − 0,130 ∗ @ + 0,944

(6.34.)


−3,98 ∗ 10�� ∗ @� + 0,00335 ∗ @� +

+0,0147 ∗ @ − 0,171 ∗ @ + 1,01

(6.35.)

Weekend’s days in

spring

3�@� = −8,19 ∗ 10�G ∗ @D + 6,07 ∗ 10�E ∗ @E +

−0,00167 ∗ @� + 0,0205 ∗ @� +

−0,106 ∗ @ + 0,171 ∗ @ + 0,896

(6.36.)

Week’s days in

summer

3�@� = −3,48 ∗ 10�G ∗ @D + 2,90 ∗ 10�E ∗ @E +

−8,52 ∗ 10�� ∗ @� + 0,00962 ∗ @� +

−0,0224 ∗ @ − 0,0993 ∗ @ + 1,06

(6.37.)

Weekend’s days in

summer

3�@� = −1,01 ∗ 10�D ∗ @D + 7,79 ∗ 10�E ∗ @E +

−0,00223 ∗ @� + 0,0290 ∗ @� +

−0,160 ∗ @ + 0,286 ∗ @ + 1,01

(6.38.)

Week’s days in

autumn

3�@� = −2,66 ∗ 10�G ∗ @D + 1,91 ∗ 10�E ∗ @E +

−4,44 ∗ 10�E ∗ @� + 0,00233 ∗ @� +

+0,0365 ∗ @ − 0,306 ∗ @ + 1,22

(6.39.)

Weekend’s days in

autumn

3�@� = −6,17 ∗ 10�G ∗ @D + 4,72 ∗ 10�E ∗ @E +

−0,00135 ∗ @� + 0,0172 ∗ @� +

−0,0915 ∗ @ + 0,119 ∗ @ + 1,03

(6.40.)

Table 32. Equations for particular period of a year for tariff G11 for commercial premises. (Own development)

70

PV profile which is approximated is a profile for PV installation for a sunny day without

clouds. Equation for this profile:

3�@� = −3,24 ∗ 10�G ∗ @D + 2,78 ∗ 10�E ∗ @E − 8,76 ∗ 10�E ∗ @� +

+0,0122 ∗ @� − 0,0722 ∗ @ + 0,170 ∗ @ − 0,119

(6.41.)

Where:

t – hour of the day

6.5 Summation

Research was conducted for 6 profiles. Four of them ware household, one of them was

commercial premises and one PV installation. For each customer profile was divided into

eight different profiles in view of period of the year and period of the week. There are some

similarities between profiles but profiles for particular customers are different. There are

also similarities between profiles for the same period for customers in the same tariff.

71

7 Applications

The thesis is divided into three theoretical and two research parts. The analysis of the

Polish Power System is made in view of demand and possible threat of security of supply in

case of renewable energy sources higher share in energy production. Special days were

chosen for the analysis. The criterion for selection of these days were maximum and

minimum demand in the year, maximum daily difference between demand’s peak and

valley, and minimum downward reserve in the year. Another factors which were analyzed

there were hours per month for upward reserve less than 9% and downward reserve less

than 500 MW. According to the analysis upward reserve in 2013 was not covered by 2275

hour per year, which is 26 %, and downward reserve in 2013 was not covered by 78 hour

per year, which was less than 1%. The last issue which was taken into consideration was use

of CGDU expressed as a percentage of all available capacity. According to data average level

was between 62% and 72% exceptions were June, July and August in 2014 when the level of

use the CDGU was lower than 50%.

Choosing the best approximation method was made based on the median of approximation

error and the average approximation error. Four methods were compared, two of them

were interpolation and two approximation. Interpolation methods based on orthogonal

polynomials: Chebyshev and Lagrange. Approximation methods based on Chebyshev

polynomials and linear regression with the least squares. Each method was analyzed for 3rd,

4th, 5th and 6th degree of polynomial. The results for Chebyshev interpolation and Lagrange

interpolation were the same, the same situation is for Chebyshev approximation and linear

regression. The best results were for 6th degree of Chebyshev approximation and 6th

degree of linear regression. The median of approximation error for this method were 4,4%

and average approximation error were 4,8%.

Finally, using the 6th degree of Chebyshev approximation the representative profiles for

each season at year for both working days and weekends were created. They are created

based on the median value of demand for each hour for each season. The analysis of

demand’s data let to create representative profiles for each considered period of the year.

Each profile was different from others and their shape suits the data for particular period of

the year. Each profile was also described by equation.

72

8 Appendix

Figure 43. Profile for winter period for week’s days for household in village in tariff G11. (Own development)

Figure 44. Profile for winter period for weekend’s days for household in village in tariff G11. (Own development)

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

G11_village_winter_week

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0,55

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

G11_village_winter_weekend

Czybyszew function

Approximated data

73

Figure 45. Profile for spring period for week’s days for household in village in tariff G11. (Own development)

Figure 46. Profile for spring period for weekend’s days for household in village in tariff G11. (Own development)

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_village_spring_week

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_village_spring_weekend

Czybyszew function

Approximated data

74

Figure 47. Profile for summer period for week’s days for household in village in tariff G11. (Own development)

Figure 48. Profile for summer period for weekend’s days for household in village in tariff G11. (Own development)

0,1

0,15

0,2

0,25

0,3

0,35

0,4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_village_summer_week

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_village_summer_weekend

Czybyszew function

Approximated data

75

Figure 49. Profile for autumn period for week’s days for household in village in tariff G11. (Own development)

Figure 50. Profile for autumn period for weekend’s days for household in village in tariff G11. (Own development)

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G11_village_autumn_week

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day


Czybyszew function

Approximated data

76

Figure 51. Profile for winter period for week’s days for household in city in tariff G12. (Own development)

Figure 52. Profile for winter period for weekend’s days for household in city in tariff G12. (Own development)

0,15

0,25

0,35

0,45

0,55

0,65

0,75

0,85

1 2 3 4 5 6 7 8 9 1011 1213 1415 1617 1819 2021 222324

De

ma

nd

[M

W]

Hours of the day

G12_city_winter_week

Czybyszew function

Approximated data

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G12_city_winter_weekend

Czybyszew function

Approximated data

77

Figure 53. Profile for spring period for week’s days for household in city in tariff G12. (Own development)

Figure 54. Profile for spring period for weekend’s days for household in city in tariff G12. (Own development)

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G12_city_spring_week

Czybyszew function

Approximated data

0,1

0,2

0,3

0,4

0,5

0,6

0,7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G12_city_spring_weekend

Czybyszew function

Approximated data

78

Figure 55. Profile for summer period for week’s days for household in city in tariff G12. (Own development)

Figure 56. Profile for summer period for weekend’s days for household in city in tariff G12. (Own development)

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day

G12_city_summer_week

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

G12_city_summer_weekend

Czybyszew function

Approximated data

79

Figure 57. Profile for autumn period for week’s days for household in city in tariff G12. (Own development)

Figure 58. Profile for autumn period for weekend’s days for household in city in tariff G12. (Own development)

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0,55

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day


Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0,55

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day


Czybyszew function

Approximated data

80

Figure 59. Profile for winter period for week’s days for household in village in tariff G12. (Own development)

Figure 60. Profile for winter period for weekend’s days for household in village in tariff G12. (Own development)

0,15

0,25

0,35

0,45

0,55

0,65

0,75

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day

G12_village_winter_week

Czybyszew function

Approximated data

0,1

0,2

0,3

0,4

0,5

0,6

0,7

1 2 3 4 5 6 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G12_village_winter_weekend

Czybyszew function

Approximated data

81

Figure 61. Profile for spring period for week’s days for household in village in tariff G12. (Own development)

Figure 62. Profile for spring period for weekend’s days for household in village in tariff G12. (Own development)

0,1

0,2

0,3

0,4

0,5

0,6

0,7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

G12_village_spring_week

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0,55

0,6

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

G12_village_spring_weekend

Czybyszew function

Approximated data

82

Figure 63. Profile for summer period for week’s days for household in village in tariff G12. (Own development)

Figure 64. Profile for summer period for weekend’s days for household in village in tariff G12. (Own development)

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

G12_village_summer_week

Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

G12_village_summer_weekend

Czybyszew function

Approximated data

83

Figure 65. Profile for autumn period for week’s days for household in village in tariff G12. (Own development)

Figure 66. Profile for autumn period for weekend’s days for household in village in tariff G12. (Own development)

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0,55

0,6

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day


Czybyszew function

Approximated data

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

0,55

0,6

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day


Czybyszew function

Approximated data

84

Figure 67. Profile for winter period for week’s days for commercial premises in tariff C11. (Own development)

Figure 68. Profile for winter period for weekend’s days for commercial premises in tariff C11. (Own development)

0,15

0,35

0,55

0,75

0,95

1,15

1,35

1,55

1,75

1,95

2,15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

C11_winter_week

Czybyszew function

Approximated data

0,15

0,35

0,55

0,75

0,95

1,15

1,35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

De

ma

nd

[M

W]

Hours of the day

C11_winter_weekend

Czybyszew function

Approximated data

85

Figure 69. Profile for spring period for week’s days for commercial premises in tariff C11. (Own development)

Figure 70. Profile for spring period for weekend’s days for commercial premises in tariff C11. (Own development)

0,15

0,35

0,55

0,75

0,95

1,15

1,35

1,55

1,75

1,95

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

C11_spring_week

Czybyszew function

Approximated data

0,15

0,35

0,55

0,75

0,95

1,15

1,35

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

C11_spring_weekend

Czybyszew function

Approximated data

86

Figure 71. Profile for summer period for week’s days for commercial premises in tariff C11. (Own development)

Figure 72. Profile for summer period for weekend’s days for commercial premises in tariff C11. (Own development)

0,15

0,35

0,55

0,75

0,95

1,15

1,35

1,55

1,75

1,95

2,15

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

C11_summer_week

Czybyszew function

Approximated data

0,15

0,35

0,55

0,75

0,95

1,15

1,35

1,55

1,75

1 2 3 4 5 6 7 8 9 10111213141516 1718192021222324

De

ma

nd

[M

W]

Hours of the day

C11_summer_weekend

Czybyszew function

Approximated data

87

Figure 73. Profile for autumn period for week’s days for commercial premises in tariff C11. (Own development)

Figure 74. Profile for autumn period for weekend’s days for commercial premises in tariff C11. (Own development)

0,15

0,35

0,55

0,75

0,95

1,15

1,35

1,55

1,75

1,95

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day

C11_autumn_week

Czybyszew function

Approximated data

0,15

0,35

0,55

0,75

0,95

1,15

1,35

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

De

ma

nd

[M

W]

Hours of the day

C11_autumn_week

Czybyszew function

Approximated data

88

Figure 75. PV profile for sunny day. (Own development)

-0,05

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

1 6 11 16 21

De

ma

nd

[p

.u.]

Hours of the day

PV

Czybyszew function

Approximated data

89

9 References

1. http://www.rynek-energii-elektrycznej.cire.pl/

2. http://www.ure.gov.pl/

3. http://www.pse.pl/

4. http://tge.pl/

5. http://www.are.waw.pl/

6. Boas M.L. (2006) “Mathematical methods in the physical sciences”

7. Borgosz-Koczwara, M., Trück, S., Kozłowski, M. (2003) - Polski rynek energii

elektrycznej na tle rynków Unii Europejskiej, REE’2003.

8. Brzeziński S., Pietrasieński P., Zborowski K. (2013) - The effect of the liberalization

of electricity market in Poland. Polish Journal of Management Studies volume 7

9. Cheney E.W.(1980) - approximation theory volume III

10. Chilhara T.S. (1978) - An Introduction to Orthogonal Polynomials

11. Davis J. (1975) - Interpolation and Approximation

12. Energy sector in Poland (2012), Polish information and foreign investment agency

13. Fortuna Z., Macukow B., W¡sowski J. (1982) - Metody numeryczne, Warszawa

14. Fox L., Parker I. (1968) - Chebyshev polynomials in numerical

15. Grad H. (2006) - Note on N-dimensional Hermite polynomials

16. Instruction of Transmission System Operation and Maintenance (2001)– Polskie

Sieci Elektroenergetyczne S.A., Warsaw, Poland

17. Kasprzyk S, Rakowski JI, Zerta M, Poręba S. (2001) - Electricity balancing market in

Poland – Polskie Sieci Elektroenergetyczne S.A., Warsaw, Poland

18. Kasprzyk, S. (2005). Rozdział obciążeń na rynku bilansującym w Polsce. (in English

Economic dispatch in balancing market in Poland). In: Anders, G. J., Mielczarski, W,

(2005). Development of electricity markets, Edition: The European power supply

industry, European Electricity Market 2005 – EEM 05’. Institute of Electrical Power

Engineering: April 2005

90

19. Kasprzyk, S., Mielczarski, W. (2005). Modern Commitment and Dispatch in the

Balancing Market. In: Anders, G. J., Mielczarski, W, (2005). Development of

electricity markets, Edition: The European power supply industry, European

Electricity Market 2005 – EEM 05’. Institute of Electrical Power Engineering: April

2005

20. Mason J.C., Davis C., (2002) - Chebyshev Polynomials

21. Meinardus G. (1968) - Aproksymacja funkcji i jej metody numeryczne, Warszawa

22. Mielczarski, W. (2000) - Rynki energii elektrycznej: Wybrane aspekty techniczne i

ekonomiczne. Warszawa: Agencja Rynku Energii

23. Mielczarski W. (2002) - The electricity market in Poland – recent advances, Power

Economics February 2002.

24. Nowak B. (2007), - Rozdział przedsiębiorstw zintegrowanych pionowo w sektorze

energii elektrycznej i gazu na podstawie dyrektyw. Mit czy rzeczywistość.

25. Poland energy Report (June 2012), Enerdata

26. Popczyk, J. (2000) - Giełda energii elektrycznej i wybrane zagadnienia związane,

REE’2000.

27. Smale S, (2003) “Estimating the approximation error in learning theory”

28. Szeg G. (2010) - Orthogonal Polynomials

29. Weron, A. (2003) - Kontrakty giełdowe oraz pozagiełdowe a zarządzanie ryzykiem.

Ocena rozwiązań polskich i zagranicznych, REE’2003.

30. Wierzbowski M. (2013) - Optimization of distribution electrical networks in the

modern power systems”– doctoral dissertation.

91

10 Streszczenie

W pierwszej części pracy została przedstawiona elektroenergetyka w Polsce. W części tej

opisano między innymi produkcję energii elektrycznej w kraju z podziałem na nośnik

energii. Rozdział ten zawiera też informacje o zainstalowanej mocy w Polsce w

poszczególnych latach, liniach najwyższego oraz wysokiego napięcia, wstępny opis rynku

energii oraz opis TPA oraz Unbandling.

Kolejny rozdział zawiera dokładny opis rynku energii w Polsce. Zawiera on w pierwszej

części wstępny opis rynku energii w kojonej części opisane są: Rynek Dnia Następnego,

Rynek Dnia Bieżącego, Rynek Terminowy Towarowy, Rynek Praw Majątkowych oraz Rynek

Uprawnień do Emisji CO2.

Czwarty rozdział zawiera opis metod matematycznych. Do metod matematycznych, które

zostały opisane w tym rozdziale zalicza się: interpolacja, aproksymacja oraz wielomiany

ortogonalne. Do wielomianów ortogonalnych, które są opisane w tym rozdziale zaliczają się

wielomiany: Czebyszewa, Hermite’a, Legengre’a oraz Laguarre’a. W ostatniej części owego

rozdziału opisana jest metoda interpolacyjna wykorzystująca wielomiany Lagrange’a.

Następny rozdział dotyczy analizy danych odnośnie zapotrzebowania na energię przez

Polski system elektroenergetyczny. Analiza polega na przeanalizowaniu rzeczywistych

danych ze względu na zagrożenia mogące wystąpić podczas pracy systemu. Zagrożenia jakie

są brane pod uwagę to: dni z najwyższym zapotrzebowaniem na energię, dni z

najmniejszym zapotrzebowaniem na energię, dni z najmniejsza rezerwą ponad stan oraz

dni, w których występuję największa różnica w zapotrzebowaniu pomiędzy szczytem a

doliną.

Szósty rozdział jest to rozdział poświęcony na wybór najlepszej metody, która pozwoli na

stworzenie profili reprezentatywnych. Metody które są porównywane to: interpolacja

wykorzystująca wielomiany Czebyszewa oraz Lagrange’a oraz aproksymacja wykorzystująca

wielomiany Czebyszewa oraz regresje liniową. W kolejnej części tego rozdziału tworzone są

profile reprezentatywne dla użytkowników korzystających z taryf: G11, G12, C11 oraz profil

PV. Profile tworzone są dla każdej pory roku z odróżnieniem na dni robocze oraz weekendy.

Każdy wykres jest również opisany przez równanie matematyczne.

Na zakończenie sformułowano wnioski, które stanowią podsumowanie przeprowadzonych

analiz oraz uzyskanych wniosków.

92

11 Summary

In the first part of the thesis the situation in the Polish Power System is presented. This part

describes the energy production divided into energy sources . Also, it includes the

information about capacity in particular years, transmission power-lines, basic information

about energy market and information about TPA and Unbunbling.

The next chapter includes detailed description of the Polish energy market. In the first part

of this chapter there is some basic information about the energy market and the next part

describes: Day Ahead Market, Intra Day Market, Commodity Forward Instruments Market

with Physical Delivery, Property Right Market and Emission Allowances Market.

The fourth chapter includes a description of the mathematical tools like: interpolation,

approximation and orthogonal polynomials. The orthogonal polynomials which are

described here comprise polynomials like: Chebyshev, Hermite, Legengre and Laguarre. In

the last part of this chapter Lagrange interpolation is described.

The next chapter concerns the analysis of Polish power sector’s demand. The aim of this

chapter is to analyse the real data in view of possible threats which can occur during power

system operation. The threat which are taken into consideration are: the maximum and

minimum demand in the year, the minimum downward reserve in the year and the

maximum daily difference between demand’s peak and valley.

The sixth chapter is devoted to choosing the best approximation method. The methods

which are compared: imethods based on orthogonal polynomials: Chebyshev and Lagrange

and approximation methods based on Chebyshev polynomials and linear regression with

the least squares. In the next part of this chapter there are representatives for customer

from tariffs: G11, G12, C11 and PV profile. Profiles are created for each season of the year

for both working days and weekends. Each profile is also described by equation.

At the end there are conclusions which form a summary to the analysis and results.

Master's Thesis Mateusz Andrychowicz - CIRE.pl · 1 POLITECHNIKA ŁÓDZKA WYDZIAŁ ELEKTRONIKI,...

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