Wydział Elektrotechniki, Elektroniki, Informatyki

i Automatyki

PRACA DYPLOMOWA MAGISTERSKA

OPERATION OF ELECTRICAL POWER SYSTEM WITH SIGNIFICANT

SHARE OF RENEWABLES

PRACA SYSTEMU ELEKTROENERGETYCZNEGO Z DUŻYM UDZIAŁEM

OZE

Autor: Jakub Przybylski

Nr albumu: 194252

Kierujący pracą: Prof. dr hab. Władysław Mielczarski

Łódź, 23 czerwca 2015

1

2

Table of Contents

Introduction .......................................................................................................................................................................... 4

Chapter 1: Electricity market in Poland..................................................................................................................... 6

1.1 Liberalization of Electrical Energy Market in Poland ............................................................................. 6

1.1.1 Unbundling ..................................................................................................................................................... 6

1.1.2 Third Party Access (TPA) .......................................................................................................................... 8

1.1.3 Energy Regulatory Office (ERO) ............................................................................................................. 9

1.2 Energy trading ..................................................................................................................................................... 10

1.2.1 Contract market ......................................................................................................................................... 10

1.2.2 Power exchange ......................................................................................................................................... 10

1.2.3 Balancing market....................................................................................................................................... 12

1.3 Power generation in Poland ........................................................................................................................... 15

1.4 Power Transmission and Distribution ....................................................................................................... 17

1.4.1 Transmission system ............................................................................................................................... 17

1.4.2 Distribution System .................................................................................................................................. 23

Chapter 2: Power system data analysis .................................................................................................................. 24

2.1 Scope of the analysis ......................................................................................................................................... 24

2.2 Review of the power system operation ..................................................................................................... 24

2.3 Selection of the worst-case days .................................................................................................................. 27

2.3.1 Motivation and methodology ............................................................................................................... 27

2.3.2 Results ............................................................................................................................................................ 28

2.3.3 Discussion..................................................................................................................................................... 32

Chapter 3: Mathematical modeling of the demand profile ............................................................................. 33

3.1 Motivation .............................................................................................................................................................. 33

3.2 Methods .................................................................................................................................................................. 33

3.2.1 Least squares method with linear regression polynomial approximation ....................... 34

3.2.2 Orthogonal polynomials ......................................................................................................................... 35

3.3 Results ..................................................................................................................................................................... 39

3.3.1 Linear regression ...................................................................................................................................... 39

3

3.3.2 Orthogonal polynomials ......................................................................................................................... 44

3.4 Discussion .............................................................................................................................................................. 51

Chapter 4: Selection of the modeled representative profiles ........................................................................ 53

4.1 Methods .................................................................................................................................................................. 53

4.2 Results ..................................................................................................................................................................... 53

4.3 Discussion .............................................................................................................................................................. 58

5 General conclusions............................................................................................................................................... 59

Appendix I ........................................................................................................................................................................... 61

Appendix II.......................................................................................................................................................................... 66

Appendix III ........................................................................................................................................................................ 70

Bibliography ....................................................................................................................................................................... 72

Acknowledgements ......................................................................................................................................................... 74

Summary .............................................................................................................................................................................. 75

Streszczenie ........................................................................................................................................................................ 76

4

INTRODUCTION

Nowadays, in the times of increasing awareness about environment of people living in

developed countries, many actions in order to minimize the negative externalities are being

done. Especially European Union’s climate policy is of the highest strictness. Establishing by the

EU more and more drastic carbon-dioxide abatement levels is meant to lead to a so called “de-

carbonization”. Reduction of share of coal-based technologies plays a key role in power industry,

which is a major consumer of coal, and so a CO2 emitter. Currently in Poland, over 85% of the

electricity is generated from either hard coal or lignite. For the reason of necessity to meet

European targets for CO2 abatement as well as share of RES, rapid increase in generation

capacity installed in renewable energy sources is observed. However, apart from the undeniable

advantages of replacing coal-fired units with wind turbines and photovoltaic panels there are

also serious issues which must have been taken into consideration.

It must be emphasized that as long as there is no effective, high-capacity electricity storage

available, energy generated by wind turbines and solar systems is uncontrollable and hardly

predictable. Thus, because of being explicitly dependent on unstable weather conditions it is

called volatile generation (VG) (Łyżwa, Przybylski, & Wierzbowski, 2015). Therefore, knowledge

of how large-scale penetration of RES affects the stability of power grid and security of supply is

one of the crucial issues in today’s power engineering, especially in the process of generation

expansion planning.

The problem of significant share of RES in conservative power systems is extremely complex

and comprises numerous aspects in various branches of science starting from strictly

engineering (power flows, relay system), through mathematics (optimization and forecasting

models) and economics (cost-effectiveness, establishing subsidies and tariffs) and ending up

with management (investment and modernization planning and execution, fundraising), law

(preparing legal regulations, executing EU directives) and social aspects (people behavior and

customs, willingness to participate in Demand Management Systems).

The aim of this thesis is to verify possible hazards in the transmission grid related to significant

presence RES and focus on effective methods of daily electricity demand modeling as well as

propose universal mathematical functions describing representative demand profiles for all

seasons.

5

Hence, the thesis is a combination of theoretical knowledge and analytical approach, and is

divided into four main parts. In the first one fundamentals of Polish electricity market and the

process of its recent liberalization are described. The second chapter is dedicated to a big-data

analysis of the Polish transmission system. It contains a review of such aspects of system

operation as demand, generation from conventional plants and wind, availability of the reserves.

Based on information elicited from the analysis the most challenging situations in the system

were selected. The objective of the third section is to find a simple and compact but accurate

method of approximation of demand profiles. The mathematical modeling was performed on

daily demand profiles based on real data. Profiles were modeled with various approximation

methods, orthogonal polynomials in particular, and the results compared. Finally, in chapter 4,

on the basis of the profiles obtained from the third part concrete functions were created and

selected as a representative profiles for each season of the year.

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CHAPTER 1: ELECTRICITY MARKET IN POLAND

1.1 Liberalization of Electrical Energy Market in Poland Introduction of Energy Law Act in 1997 is considered to be the starting point of the process of

creating energy market in Poland. Since then, electrical energy has been no longer a public good,

but a tradable commodity. The major objective was to form a competitive market, however

guaranteeing flawless security of supply. For this reason, set of instruments was necessary to be

introduced. Key elements in the process of electricity market creation are:

Unbundling

Non-discriminatory access to grid

Third Party Access (Mielczarski & Kasprzyk, CIGRE Session: C2-104, 2004)

Figure 1. Electricity market liberalization [own development].

1.1.1 Unbundling

Unbundling means separation of the market into two sectors – competitive market and

regulated market. Generation and trading are market-based and decentralized sectors, whilst

transmission and distribution are regulated, natural monopolies. Unbundling can be realized on

four levels:

Administrative unbundling – bank accounts for grid utilization and sales/generation are

separate, but organizational structure and activities are within one enterprise.

Efficient market and reduction of costs

Non-discriminatory

aceess

TPA

Unbundling

7

Management unbundling is an administrative unbundling extended by partial staff

division, i.e. allocation of employees into separate sub-entities, however centrally

managed from a head holding.

Legal unbundling – grid operation is isolated and managed independently from sales and

generation activities. However, the legally separate enterprises may function together in

a single holding company.

Ownership unbundling – the most advanced unbundling solution.

Transmission/distribution and sales and production have entirely their own proprietary

rights with neither shared activities nor dependence on central holding (Kuennekke &

Fens, 2006).

In Poland in case of Transmission System Operator (TSO) has been introduced the most far-

going model – the ownership unbundling, while for Distribution System Operators (DSOs) the

legal unbundling. It means that Distribution System Operators (DSOs) are entities both

financially and legally independent from the energy enterprises they used to belong to, whereas

Transmission System Operator (TSO) is a completely autonomous body. It should also be

mentioned that legal unbundling for DSOs concerns enterprises which supply at least 100 000

customers. Otherwise entity may operate under administrative unbundling (Mielczarski,

Development of energy systems in Poland, 2012) (Olek, 2013).

Figure 2. Electricity market transformation [own development].

Natural mononopoly

branches

Generation

Transmission

Distribution

Sales

Natural mononopoly

branches

Transmission

Distribution

Competitive market-based

branches

Generation

Trading

Sales

8

1.1.2 Third Party Access (TPA)

Third Party Access (TPA) is a market opening to the third parties based on a principle which

obliges grid infrastructure owners to allow other entities to access the network. TPA enables

active participation in the market of consumers who, in case of electricity market, are end-users

of electrical energy by allowing them to choose personally their energy retailer (other than grid

owner). Introduction of TPA is strictly related to effective market division into wholesale and

retail market. The wholesale market is a market where transactions are made mostly between

generators and distributors or sales companies within Power Exchange, bilateral contracts or

balancing market (see next paragraph). Afterwards, these entities prepare their offers in a form

of tariffs to end-users, for instance household consumers who are retail market players. Such

market structure creates room for competition at the retail level, where end-users make market-

based decisions when choosing the cheapest provider. It must be emphasized that the entity

which services can be changed on the basis of TPA is the energy seller, not the physical

electricity supplier which is a separate enterprise due to unbundling. The objective of TPA is to

provide competitive market reducing costs for end-users similarly to telecommunication market

which is a perfect example of the liberal services market (Mielczarski, The electricity market in

Poland - recent advances, 2002) (Lech, 2010).

At present, Polish market faces as buoyant growth in the number of energy supplier switches.

Initially was observed dramatic increase in provider switching activities among the largest

industrial consumers (A-tariff) followed soon by the smaller enterprises (B and C tariffs). From

the graph presented in the Fig. 3, it can be noticed that this market has been nearly saturated

within the first 3 years of operation, and then cooled down. Meanwhile, it seems there is still

great potential in the G-tariff sector. It has not loosed growth intensity yet and the number of

300,000 households which have already decided to switch their provider is not impressive

taking into account the population of 38 million people. Nevertheless, G-tariff is the only

regulated group with the determined cap price and the lowest potential for cost-savings for

consumers due to small volume of purchased energy and relatively low energy prices. As a

result, many individuals may not be sufficiently encouraged to make an effort and take activities

leading to supplier switch.

9

Figure 3. Number of supplier switches in A,B, C and G –tariff consumers. Based on (Energy Regulatory Office).

1.1.3 Energy Regulatory Office (ERO)

In order to both supervise competitiveness of the decentralized and control the centralized

sectors of the market passing the Energy Law Act lead to establishment of Energy Regulatory

Office (ERO). Currently the competences of the President of ERO include many aspects, the most

important are listed below:

1. Accepting and withdrawing concessions.

2. Approving and controlling of the tariffs in terms of compatibility with the rules

established in the Energy Law Act and executive regulations including verification of the

costs justified by the energy companies to calculate prices in tariffs.

3. Determining:

a. Correction factors specifying projected improvement in the efficient operation of

the energy companies,

b. Period of the utilization of tariffs and correction factors,

c. Level of justified rate of return of the energy companies which are claiming their

tariffs for approval,

d. Maximum share of the fixed fees in the total payments for transmission and

distribution services for particular end-users groups,

e. Substitutions fees for compulsory procurement of the energy generated from the

Renewable Energy Sources (RES),

f. Reference indicator.

4. Ensuring uniform form of development plans made by transmission and distribution

enterprises.

7 611 21 716

65 327

92 626 100 978 104 916 107 405 109 798 111 733 113 335 115 407 116 904 118 475

1 340 14 341

76 470

135 619 146 049

157 635 171 464

185 608 201 626

211 332 223 925

236 173 251 612

-

50 000

100 000

150 000

200 000

250 000

300 000

12.2010 12.2011 12.2012 12.2013 1.2014 2.2014 3.2014 4.2014 5.2014 6.2014 7.2014 8.2014 9.2014

Nu

mb

er

of

clie

nts

A, B, C - tariffs clients G - tariff clients

10

5. Controlling fulfillment of various duties of the energy enterprises regarding many

aspects of operation, for example: procurement of the energy generated in RES and

highly efficient Combined Heat and Power (CHP) plants, trading electricity via power

exchange, services quality standards etc.

6. Organizing and conducting tenders for authorizing guaranteed default suppliers as well

as for new investments in new capacity and electricity demand reduction (Energy

Regulatory Office).

1.2 Energy trading With respect to Energy Law, currently electrical energy can be traded at three segments of the

wholesale market: contract market, power exchange market and balancing market.

1.2.1 Contract market

Contract market is a market at which bilateral contracts are appointed, i.e. both sides –

generator and consumer – interested in trading particular volume at particular price negotiate

and, if agree, sign the contract. This form of trading is most common in case of large industrial

consumers who especially are in need of long-term risk hedging. Bilateral contracts are also

called Over The Counter (OTC) due to the fact that commodity is traded outside exchange and

without supervision. It should be noted that in case of OTC transaction there is no place defined

where the agreement shall be finalized – it is decided by the interested entities (Dodd, 2012).

1.2.2 Power exchange

Trading at power exchange is the most market-based method of purchasing and selling electrical

energy. Both suppliers and buyers bid their offers accordingly and the clearing price is found at

the intersection of demand and supply curve. In practice, the majority of buyers are traders and

large industrial consumers – for smaller clients access fees are too high and trading energy for

personal needs is economically inefficient. As an outcome of introduction of Energy Law Act

Poland a natural step in the process of market liberalization was creation of power exchange in

Poland. Therefore, the Ministry of Treasury announced the establishment of Polish Power

Exchange in 1999 and determined the level of obligatory trading at spot market. At the Polish

Power Exchange electricity trading can be performed at a spot market and a futures market

(Polish Power Exchange) (Szczygieł, 2005).

Futures market allows for making secure transactions based on risk hedging, accurate

forecasting and optimizing cash-flows in a long-term planning. It is a commonly used trading

medium for industrial consumers and sales companies to cover the most certain share of their or

their clients demand.

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Trading at spot market takes place at Day-Ahead Market (DAM). Both markets consist of 24

single-hour accounting periods which actually create 24 separate markets for each hour within a

single day. The scheme of transaction at DAM is based on typical for stock exchange

fundamentals of demand-supply relation. Generators bid their supply offers at their marginal

production costs which are aggregated in a price-ascending merit order (see Fig. 4). Given that

electricity demand is considered inelastic the volume of the demand determines the intersection

of the demand and supply. The intersection indicates which generators are in the market and the

clearing price is established as the price determined by the marginal cost of last generator

present in the market.

Figure 4 displays an example of price determination with the use of merit order. Let’s assume

that one bid reflects volume of 100 MWh energy and inelastic demand equals 750 MWh. Then,

demand intersects at the bid of the hard coal unit C of 180 PLN/MWh. As a result hard coal unit C

is the last unit introduced to the market (generating half of the offered volume) and clearing

price is set at the level of 180 PLN/MWh. It means that all the preceding in the merit order units

are earning profit which equals to the difference between the clearing price and their marginal

production costs, and the last unit does not make a profit.

Figure 4. Graphical explanation of the merit order with inelastic demand [own development].

It must be emphasized that marginal cost covers variable costs only – fixed costs such as

investment, amortization, loan etc. are not included. Taken into account the above, RES

technologies are in privileged position due to their zero fuel and labor costs and nearly zero

0

50

100

150

200

250

300

Wind Hydro Lignite Lignite A Lignite B Hard CoalA

Hard CoalB

Hard CoalC

Hard CoalD

CCGT

Pri

ce, P

LN/M

Wh

Energy, MWh

Clearing price

Demand curve

12

operation and maintenance costs. As a result, it can be said that increasing share of renewables

on the market pushes away conventional technologies and decreases electricity price at the

wholesale market. Nevertheless, such technologies are publically subsidized and the costs are

eventually covered by the end-users in retail market in the form of tariff surcharges.

It also should be mentioned that at the Polish Power Exchange not only electricity is traded but

also property rights, emission allowances. Polish Power Exchange is also a place of trading other

energy commodity – natural gas (Polish Power Exchange).

Figure 5. Structure of the electricity market – places of the energy trade [own development].

1.2.3 Balancing market

Balancing market is a unique market not present in any other area of trading than electricity.

Due to the fact that contracts for electrical energy are agreed at DAM the volume of energy

traded is only a short-time forecast of what will happen during the following day.

Simultaneously a physical transfer of electricity is not dependent on trading agreements and,

while being hardly storable commodity, supply must equal demand in every period of time.

Therefore traded volume does not perfectly correspond to actual flow of energy and the

imbalances need to be covered. In order to meet the mismatch between projected and real

demand transactions are carried out at balancing market. In case of excessive or insufficient

supply in the system energy is sold or purchased respectively at the balancing market.

Balancing market operates as a Day-Ahead Market under supervision of the TSO. Generators bid

their offers either for production increase or decrease in a form of bands - small volume portions

of energy with price set accordingly per each band. The offers must be submitted to the TSO for

Wholesale electricity trading

Contract market

Power Exchange market

Futures market

Spot market (Day-Ahead

market)

*Property Rights market

*Emission Allowances

market

Balancing market

13

each hour at n-1 day and the number of bands per each hour is ten and includes both increase

and reduction bands. The increase bands indicate generator’s availability to produce additional

energy at offered price, while reduction bands reflect producer’s willingness to pay the TSO a

certain price and generate less electricity than initially contracted. In Polish balancing market

the tenth band always reflects the startup price. Obviously, balancing bids offered by the

generated are technically constrained. Increase and reduction bands are limited by the

maximum output of the generating unit and technical minimum respectively (IRiESP, 2010).

When all the bids are submitted and approved by the TSO, the operator arranges them gradually

with respect to price and creates merit order for balancing market. Therefore, during intra-day

system operation the mismatch between demand and generation is covered by the balancing

bands automatically chosen as the cheapest from the merit order, as shown in Fig 6.

In scenario A the demand was overestimated and generation reduction is necessary. As a result,

the intersection of the “A” arrow and reduction bids is the band and price determinant. In this

case, the last accepted reduction bid is the 3rd band of the generator 3 at 138EUR for 200 MWh

decrease. It means that the generator 3 pays the TSO 138EUR for not generating the 200 MWh

from the initially contracted volume. The TSO, remaining neutral body in the market, transfers

the money from the producer to consumers who need to be paid for unused energy. End-users

are paid by the TSO the average-weighted value of the accepted balancing bands (Wierzbowski,

2013).

Conversely, in scenario B the demand was under contracted and additional power is needed. “B”

arrow reflects that the last balancing band accepted is the 5th band of the generator 4. Producer

4 is paid 155EUR for 40MWh of an extra power by the TSO who collects the funds from

consumers. Similarly to case A, consumers pay the average-weighted value of the accepted

balancing bands (Wierzbowski, 2013).

14

Figure 6. Balancing bands offers at Balancing Market in a merit order. Courtesy of (Wierzbowski, 2013).

Hence, both security of supply is provided and the generators are remunerated within

a competitive market for their availability to cover those imbalances. The process of cash flows

in both scenarios between generators, TSO and consumers is presented in Fig. 7.

15

Figure 7. Cash flows in scenario A and B at the Balancing Market. Courtesy of (Wierzbowski, 2013).

1.3 Power generation in Poland Fundamental in power generation is volume of installed capacity and its distribution among the

technologies. In Poland, total installed capacity is dominated by large, coal-fired power plants.

Currently - December 2014 - hard coal and lignite units constitute together three-quarters of the

total installed capacity, followed by 10% share of RES (mostly wind turbines), 7% of industrial

plants, 6% of hydropower and 3% gas-fired units. Meanwhile, analysis of the capacity structure

within last three years shows tendency of increasing role of RES and decreasing importance of

hard coal and lignite.

The structure of shares of each technology in total electrical energy generation is called energy

mix, and this expression will be frequently used in the thesis [source]. As a natural consequence

of the figures in installed capacity, majority of the electricity produced in Poland is generated in

conventional power plants based on combustion of lignite and hard coal. According to (PSE -

Raport KSE 2014) the total generation in Poland in the year 2014 reached 156 657 GWh with

the 86% of generation from the coal fired power system plants. Taking into account the fact that

fuel of most of the industrial plants is coal as well, the real share of coal in Polish energy mix is

around 90%, whilst 5% derives from RES. The reason for that is dramatically lower availability

and stochastic nature of the capacity in RES which depends on weather conditions.

16

The data concerning shares in installed capacity and energy mix in Poland is presented in the

Table 1 and 2 and Fig 8 and 9 respectively.

Table 1. Technology shares in installed capacity in Poland (PSE - Raport KSE 2014)

Installed capacity

2012 2013 2014

Hard coal 20 152 19 812 18 995 MW

Lignite 9 635 9 374 9 268 MW

Natural Gas 934 934 999 MW

Hydro 2 221 2 221 2 369 MW

Wind and other RES 2 617 3 504 3 877 MW

Industrial 2 486 2 561 2 613 MW

Total 38 045 38 406 38 121 MW

Figure 8. Diagram presenting technology shares in installed capacity in Poland. Based on (PSE - Raport KSE 2014).

50%

24%

3%

10%

6%

7%

Installed Capacity in 2014, [GWh]

Hard Coal

Lignite

Natural Gas

Wind and other RES

Hydro

Industrial

17

Table 2. Technology shares in power generation in Poland (PSE - Raport KSE 2014).

Power Generation

2012 2013 2014

Hard Coal 84 493 84 556 80 284 GWh

Lignite 55 593 56 959 54 212 GWh

Natural Gas 4 485 3 149 3 274 GWh

Wind and other RES 4 026 5 895 7 256 GWh

Hydro 2 265 2 762 2 520 GWh

Industrial 8 991 9 171 9 020 GWh

Total 159 853 162 501 156 566 GWh

Figure 9. Diagram presenting energy mix in Poland. Based on (PSE - Raport KSE 2014).

1.4 Power Transmission and Distribution

1.4.1 Transmission system

In Poland electricity generated in power plants is transported to end-users via transmission and

distribution lines. Both transmission and distribution system consist of overhead lines, cables,

transformers, substations and ancillary apparatus, however on a different voltage level.

Transmission system lines are of 750 kV (currently out of order), 400 kV, 220 kV and are owned

and managed by a single Transmission System Operator (TSO) – PSE S.A. Nowadays Polish

transmission system create:

51%

35%

2%

5% 1%

6%

Power Generation in 2014, [GWh]

Hard Coal

Lignite

Natural Gas

Wind and other RES

Hydro

Industrial

18

246 connectors of a total length of 13 519 km, including:

o 1 connector of 750 kV and 114 km length

o 77 connector of 400 kV and 5 383 km length

o 168 connector of 220 kV and 8 022 km length

103 high voltage stations

Sub-sea DC cable 450 kV cross-border connection between Poland and Sweden (PSE -

webpage) (IRiESP, 2010)

The map of the Polish transmission system with distinguished voltage-levels and indicated

cross-border connections is presented in Fig. 10.

Figure 10. Map of Polish Power Transmission System (Global Energy Network Institute)

19

1.4.1.1 TSO competences

Responsibilities of a TSO are established in Energy Law Act and the most important are:

Determining security of supply of electric energy throughout ensuring secure operation

of the power system and adequate availability of transmission capacity in transmission

grid.

Providing effective grid operation when ensuring reliability and quality standards of the

supplied energy.

Coordinating operation of 110 kV grid in cooperation with DSOs.

Maintenance, conservation and repair services (PSE - webpage)

1.4.1.2 Power system scheduling

In order to provide security of supply and balance between generation and demand accurate

scheduling methods are crucial. For this reason the daily routine in TSO is focused on

preparation of coordination plans at various time perspectives. The most long-term is Annual

Coordination Plan (ACP), followed by Monthly Coordination Plan (MCP) and series of Daily

Coordination Plans (DCPs).

1.4.1.2.1 Annual Coordination Plan - ACP

The ACP is prepared for the next 3 years and contains the following elements:

Forecasted monthly-averaged available capacity of the centrally dispatched and non-

centrally dispatched generating units.

Forecasted monthly-averaged available capacity of the generating units including

capacity losses due to submitted by the CDGU retrofit plans, capacity losses submitted by

the nCDGU as well as planned capacity losses due to grid operation conditions.

Forecasted monthly-averaged demand for typical weather conditions in daily peaks

during working days.

Forecasted maximum monthly demand.

Forecasted determined cross-border energy trade in daily peaks during working days.

Forecasted monthly-averaged nCDGU load in daily peaks during working days.

Forecasted monthly-averaged capacity reserves in power plants in daily peaks during

working days.

Plan of the closed-grid elements shut-downs.

Minimum necessary and maximum possible number of generating units in particular

nodes within the entire planned period (IRiESP, 2010).

20

1.4.1.2.2 Monthly Coordination Plan – MCP

The ACP is prepared for the next month, published no later than 25th day of the preceding month

(except for March – until 23rd of February) and contains the following elements:

Forecasted available capacity of the centrally dispatched and non-centrally dispatched

generating units.

Forecasted available capacity of the generating units including capacity losses due to

submitted by the CDGU retrofit plans, capacity losses submitted by the nCDGU as well as

planned capacity losses due to grid operation conditions.

Forecasted for the given month demand for typical weather conditions in daily peaks

during working days.

Forecasted determined cross-border energy trade in daily peaks during working days.

Forecasted nCDGU load in daily peaks during working days.

Forecasted capacity reserves in power plants in daily peaks during working days.

Plan of the closed-grid elements shut-downs.

Minimum necessary and maximum possible number of generating units in particular

nodes within the entire planned period.

Planned constraints in cross-border energy trade within entire the planned period

(IRiESP, 2010).

1.4.1.2.3 Daily Coordination Plans - DCPs

Since in short-term planning the highest accuracy of the forecast is needed the scheduling

procedure is the most complicated. Hence, there are three consecutive coordination plans:

preliminary DCP, DCP at n-1 day and current DCP at intra-day.

Preliminary DCP is published at n-2 day and provides following output data:

Capacity data for each hour of the n-day in hourly-averaged gross values:

o Demand to be covered by the national power plants.

o Sum of the generating capacity in CDGU.

o Sum of the generating capacity in nCDGU.

o Sum of the determined generation in CDGU.

o Sum of the determined generation in nCDGU.

o Required capacity upward reserve.

o Required capacity downward reserve.

Grid constraints:

o Minimum required capacity or number of units in particular nodes of the closed-

grid.

21

o Maximum required capacity or number of units in particular nodes of the closed-

grid.

Plan of the utilization of particular generating units to provide up- and down-ward

regulation services – data shared with those particular power plants (IRiESP, 2010).

Unlike all the previous plans, DCP is a solution of the economic dispatch optimization problem.

The optimization process is solved with the use of Linear Programming algorithm. The output

data comprises:

Plan of the operation of CDGU including system constraints.

Plan of the operation of CDGU divided into balancing bids bands including system

constraints.

Plan of the operation of CDGU with no system constraints included.

Plan of the operation of CDGU divided into balancing bids with no system constraints

included

List of CDGU shutdowns according to DCP.

List of CDGU startups according to DCP.

Graphical schedule of CDGU operation.

Ranking list of the CDGU load increases from the spinning reserve.

Ranking list of the CDGU startups.

Ranking list of the CDGU off-loads.

Ranking list of the CDGU shutdowns (IRiESP, 2010).

Finally, in order to adjust the operation of the power system to unexpected variations the TSO

provides Current DCP which data resolution are 15-minute periods instead of single-hours. The

basic version of the CDCP is submitted shortly after DCP, however TSO is allowed to make

numerous updates. Submission deadline for the first update is 19:00 of n-1 day, however

another updates for any 15-minute period can be made if needed either on n-1 or intra-day, but

no later than 15 minutes prior to their validity period. The updates are carried out as a result of

changes in power system operation, such as: demand variations, different cross-border trade,

available CDGU capacity, new operation constraints of the running CDGU or changes in nCDGU

generation. The output data of the CDCP covers:

State of the CDGU.

Type of losses.

Plan of the particular CDGU utilization for the primary and secondary regulation.

Available capacity in gross values.

22

Load capacity in gross values.

Minimum capacity in gross values (IRiESP, 2010).

The brief overview of the differences between the coordination plans is presented in Table 3.

Table 3. Daily Coordination Plans – overview (IRiESD, 2013).

Plan Preparation

interval

Time

planned

Submission

deadline also

Data

resolution

Updating procedure

pDCP Once a day n-day 0:00-24:00

n-2 day 16:00

1-hour Not being updated

DCP Once a day n-day 0:00-24:00

n-1 day 17:00

1-hour Not being updated

cDCP Basic version once a day, updates may be prepared several times

n-day 0:00-24:00 or other depending on time of update

Basic version – n-1 17:30, does not apply to further versions

15-minute First update – until n-1 day 19:00; Another updates made if necessary.

Figure 11. Division of Coordination Plans [own development].

Coordination plans

Technical plans

ACP MCP

Realization plans for balancing

market

DCPs

Preliminary DCP DCP Current DCP

23

1.4.2 Distribution System

Distribution system comprises: partially high- (110 kV), medium- and low-voltage infrastructure

which gives range of 0.23 kV to 110 kV. It is noteworthy that 110 kV lines, being under DSOs

ownership, are operated in cooperation with TSO on the strictly agreed terms (IRiESD, 2013).

Present shape of the distribution market is a result of reorganizations which took places in the

past 25 years. In few steps between 1989 and 2011 numerous regional distribution entities have

been merged into dominating four vertically integrated utilities: PGE, Tauron, Enea, Energa

which area of operation in terms of distribution is determined geographically (see Fig. 12).

Those utilities possess infrastructure to supply clients in the entire country except for the city of

Warsaw which is under operation of RWE. All of these 5 are enterprises supplying more than

100,000 consumers which means that their operation comes under the rule of unbundling. An

interesting DSO is PKP Energetyka which supplies railway and adjacent to railroads industrial

clients. It is the only country-wide operating DSO, however has around 50,000 clients and is not

covered by the unbundling. There are also existing several small, local DSOs servicing for

instance shopping centers, industrial complexes and others.

Figure 12. Map of areas of operation of five major energy distributors (Enea).

24

CHAPTER 2: POWER SYSTEM DATA ANALYSIS

2.1 Scope of the analysis Analysis of the Polish electric power system data was carried out for the entire year 2014, and

additionally the first quarter of 2015. The data consists of following parameters: total demand,

total CDGU (Centrally Dispatched Generating Units) generation capability, total nCDGU (non

Centrally Dispatched Generating Units) generation capability, CDGU generation, nCDGU

generation, wind generation, upward reserve available, downward reserve available. Each

parameter has its value for every hour of the year. Afterwards, based on performed analysis

selection of the worst-case days has been made.

2.2 Review of the power system operation The initial step of the analysis was to search for extreme values of the most relevant for the

system operation parameters which are demand and up- and downward reserves. The results

are presented in the Table 4, in which also can be found remarks to the given hours concerning

such issues as ambient temperature, weather conditions or day of the week. It is noteworthy to

notice that among only seven selected days, two of them are national festivities. It suggests that

alongside days of extreme weather conditions, festivity days are the most challenging ones for

secure system operation.

It can be observed that the annual difference between the maximum and minimum total demand

reaches almost 15 000 MW. In other words, in Polish system maximum load equals 250% of the

minimum. In terms of reserves, it must be noted that for both upward and downward reserve

minimum value was 0, which is highly undesirable. According to Instruction of Operation and

Maintenance of the Transmission Grid the reserves levels should equal at least 9% of projected

(day-ahead) demand for upward and 500 MW for downward reserve respectively. Whereas the

zero value for upward reserve occurred only in one hour during the entire year, zero downward

reserve was observed within 19 hours.

In order to evaluate more deeply the problem of insufficient reserves, the total time of reserves

level being below established by the TSO margin was calculated for each month and presented in

the Table 5. It can be noticed that problem of lacking upward reserves is much more frequent

than downward reserves, mainly due to different requirements (9% of the demand against

constant and low value of 500 MW). Moreover, monthly frequency of insufficient reserves

presented graphically in Fig. 13 gives impression of non-seasonally depended, stochastic

25

occurrence. However, usually when system faced shortage of one kind of reserves, the other was

within acceptable margin. The only month with significant number of inadequate both up- and

down-ward reserves was January 2015.

Furthermore, Table 5 contains information about utilization of CDGU units in percentages in

every monthly period. This parameter shows how much of the available CDGU capacity actually

contributes to the total generation. Subsequently, Fig. 14 presents distribution of hourly CDGU

utilization in July - the month with the lowest annual CDGU utilization.

Table 4. Extreme values of the key system parameters . Based on (PSE - webpage).

Date Hour Demand

value,

[MW]

Upward

reserve

value,

[MW]

Downward

reserve value,

[MW]

Day of the

week

Wind

generation,

[MW]

nCDGU

generatio

n, [MW]

Ambient

temperat

ure [0C]*

Additional

remarks

Annual min demand

21.04 6:00 10 800 3 862 -588 Monday 597 4 514 7 Easter

21.04 7:00 10 850 3 930 -520 Monday 535 4 456 8 Easter

Annual max demand

29.01 18:00 25 363 1 089 -6 677 Wednesday 2 447 8 122 -9 Evening peak,

windy, snowy and

cold

31.01 18:00 25 300 867 -6 894 Friday 2 460 8050 -4 Evening peak,

windy, snowy and

cold

Minimum upward reserve

10.09 21:00 21 563 0 -7 998 Wednesday 23 3 487 12 Only 18 529 MW

available in CDGU,

first cool day

Minimum downward reserve

16.03 2:00 –

8:00

13588-

14425

6 213-

7 339

0 Sunday 2541 – 2723 7496 –

7740

2 None

22.12 2:00 14400 4 985 0 Monday 2 950 8 212 4 None

25.12 1:00 –

9:00

11913 –

13500

4 309 –

6 645

0 Thursday 1 760 – 2 947 7103 –

8124

4 Christmas night

* weather data for central Poland - Lodz airport, from (wunderground.com)

26

Table 5. Monthly evaluation of insufficient reserves and mean CDGU utilization. Based on (PSE - webpage).

Month Number of hours with

downward reserve below

margin (500 MW)

Number of hours with

upward reserve below

margin (lower than 9% of

demand)

Mean CDGU utilization

2014

January 3 227 62%

February 4 163 62%

March 13 178 62%

April 4 268 57%

May 6 232 56%

June 1 248 49%

July 0 290 48%

August 0 205 49%

September 0 299 53%

October 1 239 59%

November 10 259 68%

December 30 237 63%

2015

January 58 238 63%

February 14 213 62%

March 10 112 62%

Figure 13. Monthly presentation of insufficient reserves. Based on (PSE - webpage).

0

50

100

150

200

250

300

350

insufficient downward reserve

insufficient upward reserve

27

Figure 14. CDGU utilization in July 2014. Based on (PSE - webpage).

2.3 Selection of the worst-case days

2.3.1 Motivation and methodology

Typically, economic dispatch and unit commitment solutions derive from representative days

(or other periods) based on statistical analysis - commonly average profiles for an entire year or

seasons. It is a very good way of shaping general tendencies in a long-term planning and

selection of representative days can be found in Chapter 4 of the thesis. However, in this part of

the research different approach is chosen – a review of dangerous situations in the transmission

system. In light of increasing penetration of VG, the hourly demand and generation profiles are

not very well corresponding with each other. Mismatch between demand and supply leads to

abnormal power system operation and challenging situations for the system operators to ensure

security of supply. As importance of studies focusing on severe conditions in power system

operation is growing, evaluation of various worst-case scenarios was decided to be done. Due to

complexity of power grid there is no exact definition of a worst-case profile. Hence, there are five

major situations studied, disregarding seasons:

1) Day with very high demand and low wind generation (Fig. 15)

2) Day with very low demand and very high wind generation (Fig. 16 and 17)

3) Day with highest ramp of CDGU generation (Fig. 18 and 19)

4) Day with largest number of hours of insufficient upward reserves (Fig. 20)

5) Day with largest number of hours of hardly sufficient downward reserves (Fig. 21)

0%

10%

20%

30%

40%

50%

60%

70%

80%1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

CDGU utilization in July 2014

CDGU utilization

Mean utilization

28

The first one is crucial in terms of high load and the generation with no support from wind

turbines. This case indicates the problem of scarcity of variable generation and the shows

necessity for backup capacity for RES. The representative day was chosen as a day with highest

difference between demand and wind generation.

The second profile presents converse situation, in which weather conditions allow for excessive

generation from RES which cannot be met by demand. Such scenario is likely to happen in a

system with large installed capacity in RES and market design providing top priority for RES

energy with no limits. The representative day was chosen as a day with lowest difference

between demand and wind generation.

The third profile states significance of flexibility needed in a power system. Large deviations in

VG must be immediately covered by CDGU. However, it must be noted that conventional units

face technical constraints such as minimum and maximum possible output, boiler inertia,

ramping time. The representative day was chosen as a day with the largest difference between

maximum and minimum CDGU generation. Graphs presenting ramps for each month of the year

are collected in Appendix II.

The following curves display days with the dangerously low reserve values. For upwards

reserves was chosen profile with largest number of hours below marginal level established by

the TSO which is 9% of projected demand. In case of downward reserves, value of 500 MW

available is required by a TSO to ensure safe operation. Since the latter margin is hardly ever

violated, level of 600 MW of available reserve was chosen as a reference value for investigation

of lacking downward reserve.

All of the above are various approaches towards distinguishing the representative days for the

most severe for power system conditions.

2.3.2 Results

The profiles are based on power system data (Daily Coordination Plan) published by Polish TSO

(PSE Operator S.A.) for 2014. Taken into account is 2014 only, since it is the first year in which

wind generation data is distinguished from the nCDGU generation. The monthly profiles for the

entire year can be found in Appendix I.

29

Figure 15. High demand with low wind generation – December 3rd, 2014. Based on (PSE - webpage).

Figure 16. Low demand with high wind generation – December 12th, 2014. Based on (PSE - webpage).

0

5 000

10 000

15 000

20 000

25 000

30 000

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

High Demand with Low Wind Generation - 3.12

NCDGU generation wind CDGU demand

0

2 000

4 000

6 000

8 000

10 000

12 000

14 000

16 000

18 000

20 000

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

Low Demand with High Wind Generation - 25.12

NCDGU generation wind CDGU demand

30

Figure 17. Low demand with low wind generation – March 16th, 2014. Based on (PSE - webpage).

Figure 18. High CDGU ramp – January 13th, 2014. Based on (PSE - webpage).

0

5 000

10 000

15 000

20 000

25 000

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

Low Demand with High Wind Generation - 16.03

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

30 000

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

High CDGU ramp case - 13.01

NCDGU generation wind CDGU demand

31

Figure 19. High CDGU ramp – January 20th, 2014. Based on (PSE - webpage).

Figure 20. Insufficient downward reserve, i.e. 10h of less than 600 MW available – March 16th , 2014. Based on (PSE - webpage).

0

5 000

10 000

15 000

20 000

25 000

30 000

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

High CDGU ramp case - 20.01

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

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

Insufficient downward reserve (10h of less than 600 MW available) - 16.03

NCDGU generation wind CDGU demand

32

Figure 21. Insufficient downward reserve, i.e. 15h of less than 9% available – July 16th, 2014. Based on (PSE - webpage).

2.3.3 Discussion

All of the profiles are equally designed. NCDGU generation, alongside with wind energy

constitute of base load due to their uncontrolled presence on the supply side. CDGU units can be

regulated, thus these are the units which balance the system correspondingly to the demand

curve. The obtained profiles vary between each other significantly, both in terms of reached

values and shape of the demand and generation pattern. It is clearly visible how intensely wind

generation affects operation of CDGU units. When high wind is unexpected it may cause

significant over generation. As today the problem is still manageable, in future when share of VG

would be few times higher might lead to severe system conditions in terms of voltage and

frequency instability.

0

5 000

10 000

15 000

20 000

25 000

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

Insufficient upward reserve (15h of less than 9% available) - 16.07

NCDGU generation wind CDGU demand

33

CHAPTER 3: MATHEMATICAL MODELING OF THE DEMAND PROFILE

3.1 Motivation The area of my scientific interest comprises power system analysis, transmission and

distribution system optimization, solving economic dispatch (ED) and unit commitment (UC)

problems, generation expansion planning (GEP), forecasting energy mix and development of

smart grids. In each of this processes repetitive and key factor is energy demand and its shape in

particular. Studies in the mentioned fields can be carried out, and typically are, with respect to

actual data given for a period of time. Nevertheless, utilization of the demand profile expressed

in a form of a function can be far more suitable in numerous investigation because of its

simplicity and universality. For this reason, this chapter of the thesis presents various

approaches to approximate a real demand with parametric functions.

3.2 Methods This part of the thesis concerns mathematical modeling of the demand profile. There are

numerous methods of creating parametric function based on given x and y coordinates and in

the thesis will be presented four of them:

1. Least squares method with linear regression polynomial approximation

Modeling with utilization of following orthogonal polynomials:

2. Chebyshev polynomial

a. Interpolation

b. Approximation

3. Hermite polynomial

a. Interpolation

b. Approximation

4. Legendre polynomial

a. Interpolation

b. Approximation

The accuracy of the investigated methods is compared and measured with a widely used in

statistical analysis Pearson correlation coefficient, also known as R-coefficient, expressed with

the following formula (Hastie, Tibshirani, & Friedman, 2009):

34

𝑅(𝑥, 𝑦) =∑(𝑥 − �̅�)(𝑦 − �̅�)

√∑(𝑥 − �̅�)2 ∑(𝑦 − �̅�)2 (1)

As presented in the listed methods in the process of function fitting apart from various

polynomials have been also used two different mathematical estimation approaches:

interpolation and approximation. In both cases, the data being estimated is a set of empirically

measured results of hourly determined electricity demand elicited from reports published by

the Polish TSO.

Approximation is a process of determining solution possibly close to dataset being

approximated. Task of an optimal approximation in polynomial base means adequate adjusting

approximating polynomial P(x) to predict accurately the value of f(x), where f(x) is a function

defined on a particular interval. The task can be solved by specifying degree of the polynomial

and quality criteria. While polynomial degree determination varies and depends on investigated

case, the latter is basically determined by errors minimization, where error of approximation is

defined as the difference f(x)- P(x) (Cichoń, 2005) (Breton & Ben-Ameur, 2005).

Interpolation is a specific example of approximation, in which at interpolation nodes

𝑥 = [𝑥0, 𝑥1, … , 𝑥𝑛] the value of interpolating function is perfectly equal to interpolated value

P(x)=f(x). Interpolation is not an area of this study.

3.2.1 Least squares method with linear regression polynomial approximation

There are many ways of fitting a model to a set of specific data, however the most popular one is

the least squares method which has been used in this study. The objective of this approach is to

minimize the residual sum of squares (RSS). The residual squares are the differences between

actual values and the values provided by the fitting model for each equation (Hastie, Tibshirani,

& Friedman, 2009).

The linear regression with least squares method was used in polynomial approximation and

performed in Microsoft Excel with the use of REGLINP function as array formula. REGLINP

function is a linear regression statistical tool useful for parameters fitting and correlation

analysis. Basically REGLINP is designed to be used for linear applications, however also allows

for more sophisticated analysis, such as polynomial or exponential when used in a form of array

formulas. In this study polynomials of the 6th, 5th, 4th and 3rd order have been investigated and

compared in terms of degree of correlation with original profile. The general formula for the

polynomial is:

35

𝑓(𝑥) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎1𝑥0 + 𝑏, (2)

where n expresses degree of the polynomial.

3.2.2 Orthogonal polynomials

The history of orthogonal polynomials dates back to second part of 19th century thanks to work

of Pafnuty Chebyshev – a Russian mathematician who built their theoretical fundamentals.

Through the years, many mathematicians developed several polynomials which fulfill given

criterion. Three of them have been chosen for this research and compared: Chebyshev, Hermite,

and Legendre polynomials (Szeg, 1939).

According to orthogonal projection theorem for any closed linear subspace of the Hilbert’s space

exists orthogonal subspace complementary to the chosen one. Orthogonality itself, means that

two objects are in such mutual relation that are perpendicular to one another. It is also common

to generalize this statement to n dimensions.

Taking the above into account orthogonal functions are such functions which inner product

equals zero (Yosida, 1980) (Chihara, 1978).

𝑓 ∘ 𝑔 = ∫ 𝑓(𝑥)𝑔(𝑥)𝑑𝑥 = 0 (3)

Similarly, orthogonal polynomials consist of sequence of polynomials that are at right angle to

each other and polynomials are orthogonal if on the set of points: 𝑥0, 𝑥1, … , 𝑥𝑛 they meet the

following criterion:

∑ Pj(xi)Pk(xi) = {0 j ≠ kconst j = k

n

i=0

(4)

where j and k indicate degree of the polynomial (Szeg, 1939) (Partington, 1986).

Unique properties of the orthogonal polynomials make them a widely used tool for

approximation. One of their most crucial feature is the fact, that their coefficient matrix is

diagonal. It means that solving complex system of equations (sometimes differential) is not

necessary and can be replaced by much simpler and more effective matrix calculations.

The fitting calculations have been performed as a matrix calculations with the utilization of the

previously mentioned property of the orthogonal polynomials.

With respect to algebraic relationships between the matrices the calculations have been carried

out in a following pattern (for exact example with actual figures see Appendix III):

36

a. Creation of matrix X sized m x 24, where m=n+1 and n is a degree of approximating

function. Columns reflect basic functions for subsequent degrees of the polynomials. In

rows are placed subsequent arguments i. In the investigated cases n varies between 3

and 6 with regard to approximating function and i indicate number of hours which is set

to 24 for each case. Below presented is general design of the matrix X for Hermite

polynomials.

X =

1 2x1 4x12-2 8x1

3-12x1 16x14-48x1

2+12 … A1x1

n- A2x1

n+…+ Am

(5) 1 2x2 4x2

2-2 8x23-12x2 16x2

4-48x22+12 …

A1x2n-

A2x2n+…+ Am

… … … … … … …

1 2xi 4xi2-2 8xi

3-12xi 16xi4-48xi

2+12 … A1xi

n- A2xin+…+

Am

b. Transposition of matrix X to XT.

c. Matrix multiplication: 𝑋𝑋𝑇 = 𝑋 ∙ 𝑋𝑇

d. Creation of matrix Z reverse matrix to XXT:

e. Creation of argument vector 𝑌 =

12…24

f. Matrix multiplication 𝑋𝑇𝑌 = 𝑋𝑇 ∙ 𝑌

g. Matrix multiplication 𝛼 = 𝑍 ∙ 𝑋𝑇𝑌

h. Obtained 𝛼 is a vector of coefficients of basic functions 𝛼 =

𝐴1

𝐴2

…𝐴𝑚

3.2.2.1 Chebyshev polynomials

Sequence developed by Chebyshev is the one which created fundamentals for further

development and modifications of the initial approach towards studies about orthogonal

polynomials.

The Chebyshev polynomials are orthogonal on the interval [-1;1] with respect to the weight

function (Paszkowski, 1975):

1

√1 − 𝑥2 (6)

Chebyshev polynomials of the first order, marked with a symbol Tn, are expressed with

following formulas (Paszkowski, 1975) (Szeg, 1939):

37

𝑇0(𝑥) = 1

𝑇1(𝑥) = 𝑥

𝑇𝑛(𝑥) = 2𝑥𝑇𝑛−1(𝑥) − 𝑇𝑛−2(𝑥) 𝑓𝑜𝑟 (𝑛 = 2, 3, … ) (7)

As a consequence of the above mentioned initial conditions and recurrence formulas the first

few consecutive Chebyshev polynomials are of a following nature (Paszkowski, 1975):

𝑇0(𝑥) = 1

𝑇1(𝑥) = 𝑥

𝑇2(𝑥) = 2𝑥2 − 1

𝑇3(𝑥) = 4𝑥3 − 3𝑥

𝑇4(𝑥) = 8𝑥4 − 8𝑥3 + 1

𝑇5(𝑥) = 16𝑥5 − 20𝑥3 + 5𝑥

𝑇6(𝑥) = 32𝑥6 − 48𝑥4 + 18𝑥2 − 1

….

(8)

Due to the limitation of the orthogonality interval to a [-1;1] for the Chebyshev polynomials, the

data must have been re-scaled to fit in this region. Therefore the initial interval of the source

data which constitute of 24 hours was recalculated using the following formula:

𝑥′ =2

24 − 1𝑥 −

1 + 24

24 − 1 , (9)

where x’ is the re-scaled value and x is the real value of time (hour).

3.2.2.2 Hermite polynomials

Hermite polynomials had been actually defined already in 1810 by Laplace, then more carefully

investigated by Chebyshev and finally published by Charles Hermite in 1864.

The Hermite sequence comprises polynomials of real coefficients and are divided into two ways

of standardization – “probabilists’ polynomials” and “physicists’ polynomials”. This study

focuses on the latter which are generated with respect to the following definition (Szeg, 1939):

𝐻𝑛(𝑥) = (−1)𝑛𝑒𝑥2 𝑑𝑛

𝑑𝑥𝑛𝑒−𝑥2

= (2𝑥 −𝑑

𝑑𝑥)

𝑛

(10)

and are solution of the following recurrence equation (Szeg, 1939):

38

𝐻𝑛+1(𝑥) = 2𝑥𝐻𝑛(𝑥) − 2𝑛𝐻𝑛−1(𝑥), (11)

at given starting conditions:

𝐻0(𝑥) = 1

𝐻1(𝑥) = 2𝑥 (12)

Therefore, first few Hermite polynomials are (Szeg, 1939):

𝐻0(𝑥) = 1

𝐻1(𝑥) = 2𝑥

𝐻2(𝑥) = 4𝑥2 − 2

𝐻3(𝑥) = 8𝑥3 − 12𝑥

𝐻4(𝑥) = 16𝑥4 − 48𝑥3 + 12

𝐻5(𝑥) = 32𝑥5 − 160𝑥3

+ 120𝑥

…

(13)

Unlike Chebyshev polynomials, Hermite sequence does not have limitation of the orthogonality

domain, thus input data does not have to be re-scaled.

3.2.2.3 Legendre polynomials

Legendre polynomials named after French mathematician Adrien Marie Legendre are defined

with Rodrigues formula (El Attar, 2009):

𝑃𝑛 =1

2𝑛𝑛!

𝑑𝑛

𝑑𝑥𝑛(𝑥2 − 1)𝑛 (𝑛 = 0,1, … ) (14)

and have following recurrence relationship (El Attar, 2009):

𝑃𝑛 =2𝑛 + 1

𝑛 + 1𝑥𝑃𝑛(𝑥) −

𝑛

𝑛 + 1𝑃𝑛−1 (𝑥) (15)

Legendre polynomials are orthogonal on the interval [-1;1] with respect to the weight function:

𝑝(𝑥) = 1 (16)

Below are listed first few Legendre polynomials (El Attar, 2009):

39

𝑃(𝑥) = 1

𝑃1(𝑥) = 𝑥

𝑃2(𝑥) =1

2(3𝑥2 − 1)

𝑃3(𝑥) =1

2(5𝑥3 − 3𝑥)

𝑃4(𝑥) =1

8(35𝑥4 − 30𝑥2 + 3)

𝑃5(𝑥) =1

8(63𝑥5 − 70𝑥3 + 15𝑥)

…

(17)

3.3 Results Major objective of the evaluation of the obtained results is comparison of the various

approximation methods and selection of the optimal degree of the polynomial. The optimal

order is here defined as the lowest order at which correlation factor R is satisfactory. In this

case, since the results were unknown before calculation have been performed, the term

“satisfactory value” cannot be exactly defined. However, it is assumed that the value of Pearson

factor should be at least 0.95, possibly consistent and shall never fall below 0.9.

3.3.1 Linear regression

As a testing demand profile initially was chosen real profile of the 1st of January 2014. In the Fig.

22 presented is chart which shows profiles obtained by polynomial approximation of a

subsequently decreasing degree, from 6th to 3rd. It can be said that there is no visible difference

between profiles shaped by the polynomials of 6th, 5th and 4th degree – the curves are almost

perfectly overlapping each other. The 3rd degree polynomial is the only one which is shaped

differently, however still following a similar pattern to the rest. Taking into account

imperfections of the drawing such as insufficient size and excessive line thickness comparison of

the Pearson coefficient is required and listed in Table 6 together with coefficients of the

polynomial in the following scheme:

6𝑡ℎ 𝑑𝑒𝑔𝑟𝑒𝑒: 𝑓(𝑥) = 𝑎6𝑥6 + 𝑎5𝑥5 + 𝑎4𝑥4 + 𝑎3𝑥3 + 𝑎2𝑥2 + 𝑎1𝑥 + 𝑎0

5𝑡ℎ 𝑑𝑒𝑔𝑟𝑒𝑒: 𝑓(𝑥) = 𝑎5𝑥5 + 𝑎4𝑥4 + 𝑎3𝑥3 + 𝑎2𝑥2 + 𝑎1𝑥 + 𝑎0

4𝑡ℎ 𝑑𝑒𝑔𝑟𝑒𝑒: 𝑓(𝑥) = 𝑎4𝑥4 + 𝑎3𝑥3 + 𝑎2𝑥2 + 𝑎1𝑥 + 𝑎0

3𝑟𝑑 𝑑𝑒𝑔𝑟𝑒𝑒: 𝑓(𝑥) = 𝑎3𝑥3 + 𝑎2𝑥2 + 𝑎1𝑥 + 𝑎0

(18)

The results confirm close alignment of the first three curves for which Pearson factor is greater

than 0.99. For the polynomial of 3rd degree larger deviation is observed and Pearson reached ca

40

0.978. Hence, according to the chosen testing conditions, the optimal solution for 1st of January

2014 could be even 3rd degree of the polynomial. Nevertheless, in spite of the relatively high

correlation 3rd degree gives significantly lower accuracy comparing to the rest, thus the

polynomial of the 4th degree should be selected as the optimal.

Figure 22. Comparison of the polynomial degrees comparison on a testing profile - January 1st, 2014

Table 6. Subsequent polynomial coefficients and R-factor of the investigated polynomial degrees.

Degree a6 a5 a4 a3 a2 a1 b R

6th 0.001207 -0.09216 2.52275 -33.6249 291.787 -1641.34 16778.9 0.9912

5th 0 -0.00164 -0.073265 1.88373 57.0501 -976.115 16220.5 0.9905

4th 0 0 -0.175941 4.19098 34.7010 -889.394 16127.6 0.9904

3rd 0 0 0 -4.60610 178.0933 -1725.12 17362.7 0.9781

Further analysis of other real profiles showed that the results do not always correspond to each

other, and that the 4th degree polynomial which was the optimal solution for randomly chosen

profile of the January 1st 2014 is not the most adequate approximation in each case. For this

reason, analysis at a larger sample of one month was performed and the Pearson factors for each

daily profile are displayed in a form of table of results in Table 7 and a chart in Fig. 23. As the

representative month was chosen January 2014.

12000

13000

14000

15000

16000

17000

18000

0 2 4 6 8 10 12 14 16 18 20 22 24

Comparison of the polynomial degrees on a testing profile - January 1st, 2014

Polynomial of 6 degree Demand Polynomial of 5 degree

Polynomial of 4 degree Polynomial of 3 degree

41

Table 7. Comparison of the R-factors between various polynomial degrees.

January 2014

Day Pearson for polynomial 6th degree

Pearson for polynomial 5th degree

Pearson for polynomial 4th degree

Pearson for polynomial 3th degree

1 0.989 0.989 0.988 0.971

2 0.988 0.988 0.969 0.966

3 0.986 0.985 0.966 0.964

4 0.985 0.985 0.980 0.980

5 0.985 0.984 0.978 0.978

6 0.989 0.989 0.984 0.984

7 0.984 0.983 0.958 0.954

8 0.982 0.981 0.952 0.950

9 0.985 0.984 0.955 0.952

10 0.982 0.981 0.953 0.950

11 0.980 0.980 0.968 0.965

12 0.981 0.980 0.971 0.971

13 0.983 0.983 0.957 0.953

14 0.981 0.980 0.949 0.947

15 0.983 0.982 0.952 0.948

16 0.980 0.978 0.949 0.946

17 0.982 0.981 0.952 0.949

18 0.980 0.979 0.966 0.962

19 0.981 0.980 0.969 0.968

20 0.985 0.984 0.958 0.953

21 0.984 0.984 0.956 0.952

22 0.983 0.983 0.955 0.950

23 0.982 0.982 0.949 0.943

24 0.981 0.980 0.946 0.939

25 0.971 0.969 0.951 0.947

26 0.973 0.970 0.951 0.951

27 0.985 0.985 0.957 0.951

28 0.980 0.980 0.946 0.940

29 0.979 0.979 0.946 0.941

30 0.982 0.982 0.949 0.941

31 0.981 0.981 0.952 0.945

42

Figure 23. Values of R-factor for various polynomial degrees within January 2014

The results show significant variations in R values among the entire month. Quite surprisingly,

results obtained from the polynomial of 4th degree correspond closely to original values actually

only on January 1st – which was initially chosen as a representative day - and on January 18th

when all four approximations were comparable. Meanwhile, among the entire period it can be

observed that 5th degree of the approximating polynomial is the last reliable fitting tool, being

very comparable to the 6th degree function. Hence, with respect to results obtained for the

January 2015 the optimal degree of the polynomial would be 5th.

Due to large variation of the obtained results and conclusions between a study over single

random day and a single random month, in order to provide higher reliability investigation of

the entire year has been carried out. The analysis was performed for each hour of the 8760

hours of the year 2014 for 4 different polynomial degrees. Then, Pearson factors were calculated

for all daily profiles. Subsequently, for easier comparison mean average and median of the R was

calculated per each month. The results are listed in Table 8 and indicated with colors in a

following pattern: green – over 0.97; yellow – between 0,97 and 0,95; red – below 0,95. Those

marked in green are the satisfactory values, yellow conditionally satisfactory, light red

unsatisfactory and bright red absolutely unacceptable. Furthermore, average and median of R

are graphically presented in a chart in Fig. 24.

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1,00

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

R

Days

R for various polynomial degrees - January 2014

R for 6th degree polynomial R for 5th degree polynomial

R for 4th degree polynomial R for 3rd degree polynomial

43

Table 8. Average and median values of R-factor.

6th degree 5th degree 4th degree 3rd degree

month mean average

median mean average

median mean average

median mean average

median

1 0.982 0.982 0.982 0.982 0.959 0.955 0.955 0.951

2 0.974 0.977 0.973 0.977 0.935 0.935 0.932 0.933

3 0.965 0.969 0.960 0.966 0.902 0.902 0.898 0.897

4 0.977 0.980 0.946 0.968 0.873 0.898 0.847 0.876

5 0.984 0.983 0.956 0.971 0.914 0.925 0.868 0.888

6 0.985 0.985 0.960 0.975 0.930 0.941 0.862 0.889

7 0.989 0.989 0.965 0.976 0.943 0.951 0.875 0.889

8 0.987 0.988 0.960 0.971 0.927 0.931 0.884 0.900

9 0.973 0.977 0.957 0.971 0.901 0.910 0.889 0.902

10 0.970 0.975 0.966 0.974 0.910 0.913 0.907 0.911

11 0.978 0.982 0.977 0.980 0.954 0.959 0.950 0.956

12 0.984 0.984 0.983 0.984 0.967 0.963 0.961 0.957

Figure 24. Graphical presentation of average and median values of R-factor throughout the year.

The acquired results follow the already observed tendency that the larger research sample the

higher differences between studied polynomial degrees may occur. Correlation factor for 3rd

0,840

0,860

0,880

0,900

0,920

0,940

0,960

0,980

1,000

1 2 3 4 5 6 7 8 9 10 11 12

R

Months

Mean average and median for each degree of the polynomial

6th mean average

6th median

5th mean average

5th median

4th mean average

4th median

3rd mean average

3rd median

44

degree are mostly unacceptable or unsatisfactory. 4th degree is dominated by unsatisfactory

results but also with presence of unacceptable values in April. Thus, these approximating

methods are considered as inaccurate and no longer discussed.

Once again, previously randomly chosen month – January, appears to be accidentally very

accurately approximated by the polynomial of the 5th degree. Such situation is repeated in three

more months: February, November and December in which both average and median are above

0.95. Nevertheless, among the remaining months performance of the 5th degree is not

dramatically, but noticeably lower. The most visible difference is observed in April, however far

more drastic in terms of mean average than median. Large deviations between average and

median are due to presence of single, extremely low values of correlation factor within these

months. It is noteworthy that only in case of 6th degree, there are no significant differences

between average and median values. It means that 6th degree is not only generally most accurate

fitting tool, but also definitely the most consistent one. Hence, 6th order is an order which is

sufficiently efficient, yet not excessively complex and should be used in demand approximation.

Besides comparison of the polynomial degrees, it is valuable to review in which periods of the

year demand is more and which less accurately estimated. The Fig. 24, with the focus on median

values, shows the general pattern which is nearly repeated in each polynomial degree. That is to

say winter and summer months are usually better fitted to real demand than spring and autumn.

The results confirm the fact that for the operator most challenging to model periods are

“transient months” – the ones in which weather substantially changes and so people’s behavior.

3.3.2 Orthogonal polynomials

The analysis of all orthogonal polynomials has been performed in the same way. The experience

learned from analysis of the linear regression method caused slightly different methodology for

evaluation of the orthogonal polynomials. High volatility of the accuracy of the results obtained

for single day and month lead to idea that the analysis shall be performed immediately for the

entire dataset, without any initial attempts at smaller samples.

Hence, set of matrix approximation calculations have been made for each day of the year 2014 in

hourly resolution, which gave population of 365 samples carried out in 8760 iterations per each

orthogonal method. As the results acquired for 3rd degree of the polynomial were mostly

considered as totally unacceptable, the investigation for orthogonal polynomials has been done

for three degrees of the polynomials: 6th, 5th and 4th.

Results obtained for Hermite, Chebyshev and Legendre polynomials are equal to each other and

same to the results derived from linear regression with least squares method analysis.

45

For this reason, in order to avoid displaying the same charts and tables obtained for each of the

investigated method, there have been additional calculation made for further analysis and

comparison of the acquired results. They are presented in a following pattern:

1. Tables 9 and 10 compare approximated with three degrees of the polynomial and real

daily demand in each hour of the chosen days: highest and lowest demand in a year.

2. Figures 25 and 26 display approximated and real daily demand profiles for the chosen

days: highest and lowest demand in a year.

3. Figures 27 - 30 present how the R factor varies in the chosen months: January, April, July,

September; in a daily resolution for each degree of the polynomial.

4. Figure 31 shows normal distribution of the R factor obtained for each day throughout

the entire year for all polynomial degrees.

46

Table 9. Comparison of the real demand and approximated with Hermite polynomials – January 29.

29th of January 2014

Actual demand Hermite polynomial 6th degree

Hermite polynomial 5th degree

Hermite polynomial 4th degree

1 18 350 18 904 18 800 17 565

2 17 738 17 084 17 170 17 546

3 17 450 16 734 16 837 17 799

4 17 413 17 308 17 354 18 267

5 17 600 18 387 18 365 18 897

6 18 425 19 666 19 595 19 638

7 20 850 20 930 20 845 20 446

8 22 675 22 047 21 978 21 279

9 23 550 22 945 22 914 22 099

10 24 100 23 603 23 618 22 875

11 24 125 24 036 24 091 23 576

12 24 300 24 285 24 362 24 178

13 24 300 24 400 24 477 24 661

14 24 363 24 438 24 493 25 008

15 23 888 24 448 24 463 25 207

16 23 563 24 465 24 434 25 249

17 24 338 24 500 24 431 25 130

18 25 363 24 539 24 453 24 851

19 25 025 24 530 24 460 24 416

20 24 850 24 388 24 365 23 833

21 24 038 23 980 24 027 23 114

22 22 550 23 135 23 238 22 276

23 21 038 21 630 21 716 21 340

24 19 688 19 199 19 095 20 330

R - 0.979 0.979 0.946

47

Table 10. Comparison of the real demand and approximated with Hermite polynomials – April 21.

21st of April 2014

Actual demand Hermite polynomial 6th degree

Hermite polynomial 5th degree

Hermite polynomial 4th degree

1 12 300 12 231 12 900 12 005

2 11 600 11 807 11 255 11 527

3 11 225 11 233 10 570 11 266

4 11 025 10 816 10 517 11 178

5 11 000 10 694 10 840 11 225

6 10 800 10 885 11 340 11 371

7 10 850 11 324 11 874 11 586

8 11 700 11 905 12 347 11 841

9 12 600 12 507 12 704 12 114

10 13 325 13 021 12 924 12 386

11 13 500 13 365 13 013 12 640

12 13 500 13 497 13 000 12 867

13 13 400 13 422 12 925 13 058

14 13 250 13 188 12 836 13 209

15 12 825 12 881 12 784 13 323

16 12 500 12 615 12 812 13 402

17 12 400 12 507 12 949 13 456

18 12 525 12 657 13 207 13 496

19 12 975 13 116 13 571 13 540

20 14 075 13 847 13 992 13 607

21 15 325 14 681 14 383 13 722

22 15 075 15 272 14 609 13 913

23 14 425 15 037 14 485 14 212

24 13 400 13 094 13 762 14 656

R - 0.978 0.910 0.816

48

Figure 25. Real and approximated demand profiles for a day with highest annual demand - January 29.

Figure 26. Real and approximated demand profiles for a day with lowest annual demand - April 21.

16 000

18 000

20 000

22 000

24 000

26 000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

January 29, 2015 - highest annual demand

Hermite polynomial 6th Hermite polynomial 5th

Hermite polynomial 4th Actual demand

10 000

11 000

12 000

13 000

14 000

15 000

16 000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

April 21, 2014 - lowest annual demand

Hermite polynomial 6th Hermite polynomial 5th

Hermite polynomial 4th Actual demand

49

Figure 27. R-factors for various polynomial degrees for representative winter month – January.

Figure 28. R-factors for various polynomial degrees for representative spring month – April.

0,94

0,945

0,95

0,955

0,96

0,965

0,97

0,975

0,98

0,985

0,99

0,995

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

January 2014

R for polynomial 6th degree R for polynomial 5th degree R for polynomial 4th degree

0,64

0,69

0,74

0,79

0,84

0,89

0,94

0,99

1,04

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

April 2014

R for polynomial 6th degree R for polynomial 5th degree

R for polynomial 4th degree

50

Figure 29. R-factors for various polynomial degrees for representative summer month – July.

Figure 30. R-factors for various polynomial degrees for representative autumn month – September.

0,9

0,91

0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

July 2014

R for polynomial 6th degree R for polynomial 5th degree

R for polynomial 4th degree

0,84

0,86

0,88

0,9

0,92

0,94

0,96

0,98

1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

September 2014

R for polynomial 6th degree R for polynomial 5th degree R for polynomial 4th degree

51

Figure 31. Gauss distribution of R-factors for various polynomial degrees.

3.4 Discussion First of all, it must be emphasized that knowledge learned from the study is that adequate scale

of research is crucial to elicit valuable and reliable information. Despite the fact that electricity

demand profile of the entire national power system may seem to be stable, volatility of load

between subsequent days introduced substantial differences in obtained coefficient factors. For

this reason, neither single day nor even a month is a sufficient dataset to evaluate and properly

select appropriate degree of the approximating polynomial. Eventually, study over an entire year

provided information about accuracy of the investigated fitting tools and utilization of basic

statistical methods such as mean average, median and normal distribution gave general

overview of the obtained results.

Secondly, it has been proved that the results vary and are highly sensitive to degree of

approximating polynomial. On the other hand, for the same degrees of polynomials results are

perfectly equal regardless of selected approximating method. It means that no matter whether

Chebyshev, Hermite, Legendre or linear regression method is used, the fitting function is

identical. Therefore, one can use any method, but it is recommended to select the fastest and the

most efficient method in calculation process. After comparison of the linear regression and

orthogonal polynomials it can be said that the latter is more favored. Calculations based on few

steps of matrix equations seem to show higher simplicity, time-efficiency and flexibility than

unclear array formulas and calculations on differential equations needed for linear regression

0

4

8

12

16

20

24

28

32

0,65 0,7 0,75 0,8 0,85 0,9 0,95 1

6th degree Gauss distribution of r 5th degree Gauss distribution of r

4th degree Gauss distribution of r 3rd degree Gauss distribution of r

52

with least squares method. However, these are subjective impressions and it is recommended to

objectively compare the methods by measuring exact computation time.

53

CHAPTER 4: SELECTION OF THE MODELED REPRESENTATIVE PROFILES

4.1 Methods The hitherto carried out study provided the optimal fitting tool which is approximation based on

polynomial of the 6th order. The general objective of the research was to model representative

electricity demand profiles. For this reason a following actions have been executed:

Selection of representative month for each season:

o Winter – January.

o Spring – April.

o Summer – July.

o Autumn – September.

Selection of representative day for period of the week:

o Working days – Wednesday.

o Weekend – Sunday.

Calculation of the average demand in each hour separately for all Wednesdays and

Sundays in a particular month. Calculations for the Wednesdays were made excluding

festivities and days free of work, eg. January 1st.

Approximation of the average Wednesday and average Sunday profiles with the

utilization of the 6th degree orthogonal polynomial.

Check of Pearson Factors.

Plot of charts of the actual and modeled profiles.

Elicitation of the final formula of the modeled functions.

4.2 Results

Table 11 compares coefficients of the polynomials. Figures 32-39 present the graphs of the

profiles for each season for working days and weekends and Equations 19-26 formulas of the

modeled function:

54

Table 11. Polynomials coefficients for representative profiles.

Season Day 𝒂𝟎 𝒂𝟏 𝒂𝟐 𝒂𝟑 𝒂𝟒 𝒂𝟓 𝒂𝟔

Winter Wednesday 22823.29 -5803.21 1612.787 -173.349 8.971688 -0.21878 0.001922

Sunday 17180.91 -1483.08 188.4996 5.663801 -1.85357 0.099615 -0.001731

Spring

Wednesday 17571.23 -1750.12 231.2214 27.95009 -5.53445 0.284816 -0.004767

Sunday 13018.91 1505.96 -946.358 191.8659 -16.5228 0.640693 -0.009224

Summer Wednesday 16966.05 -1024.21 -82.303 73.6902 -8.34512 0.357786 -0.005377

Sunday 13006.07 1751.583 -1073.64 210.174 -17.4367 0.652136 -0.009081

Autumn Wednesday 18195.54 -2765.39 511.5698 -2.16837 -3.97405 0.247271 -0.004456

Sunday 15782.46 108.1474 -501.052 137.1287 -13.2208 0.542806 -0.008097

Figure 32. January average profile for a working day

𝐹(𝑥) = 0.001922x6 + −0.21878x5 + 8.971688x4 − 173.349x3

+ 1612.787x2 − 5803.21x + 22823.29 (19)

𝑅 = 0.982

15000

16000

17000

18000

19000

20000

21000

22000

23000

24000

25000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

Polynomial 6th degree January Average Wednesday Profile

55

Figure 33. January average profile for a weekend day.

𝐹(𝑥) = − 0.001731x6 + 0.099615x5 − 1.85357x4 + 5.663801x3

+ 188.4996x2 − 1483.08x + 17180.91

(20)

𝑅 = 0.982

Figure 34. April average profile for a working day.

𝐹(𝑥) = − 0.004767x6 + 0.284816x5 − 5.53445x4 + 27.95009x3

+ 231.2214x2 − 1750.12x + 17571.23 (21)

𝑅 = 0.981

13000

14000

15000

16000

17000

18000

19000

20000

21000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

Polynomial 6th degree January Average Sunday Profile

13000

14000

15000

16000

17000

18000

19000

20000

21000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

Polynomial 6th degree April Average Wednesday Profile

56

Figure 35. April average profile for a weekend day.

𝐹(𝑥) = − 0.009224x6 + 0.640693x5 − 16.5228x4 + 191.8659x3

+ 946.358x2 − 1505.96x + 13018.91

(22)

𝑅 = 0.973

Figure 36. July average profile for a working day.

𝐹(𝑥) = − 0.005377x6 + 0.357786x5 − 8.34512x4 + 73.6902x3

− 82.303x2 − 1024.21x + 16966.05 (23)

𝑅 = 0.989

11000

12000

13000

14000

15000

16000

17000

18000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

Polynomial 6th degree April Average Sunday Profile

13000

14000

15000

16000

17000

18000

19000

20000

21000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

Polynomial 6th degree July Average Wednesday Profile

57

Figure 37. July average profile for a weekend day.

𝐹(𝑥) = − 0.009081x6 + 0.652136x5 − 17.4367x4 + 210.174x3

+ 1073.64x2 − 1751.583x + 13006.07 (24)

𝑅 = 0.992

Figure 38. September average profile for a working day.

𝐹(𝑥) = − 0.004456x6 + 0.247271x5 − 3.97405x4 − 2.16837x3

+ 511.5698x2 − 2765.39x + 18195.54 (25)

𝑅 = 0.981

11000

12000

13000

14000

15000

16000

17000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

Polynomial 6th degree July Average Sunday Profile

13000

14000

15000

16000

17000

18000

19000

20000

21000

22000

23000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

Polynomial 6th degree September Average Wednesday Profile

58

Figure 39. September average profile for a working day.

𝐹(𝑥) = − 0.008097x6 + 0.542806x5 − 13.2208x4 + 137.1287x3

− 501.052x2 − 108.1474x + 15782.46 (26)

𝑅 = 0.965

4.3 Discussion

The proposed profiles are of the satisfactory accuracy with the R factor between 0.965 for

September’s weekend and 0.992 for June’s weekend. The profiles should reflect average, normal

electricity demand within the working days or weekends for each of the season. Observation of

the profiles confirms different range of load and in some cases significantly diverse shape of the

profile between working days and weekends. Furthermore, analysis of the obtained functions

indicates that polynomials coefficients in each case are unlike. Extreme parts of the polynomial

a0 and a6 are the most similar among all the function with values of the same order. However, in

the middle parts, a2 and a3 in particular, the highest deviations are observed with values varying

between 82 and 1613, and -2 and 192 for a2 and a3 respectively.

13000

14000

15000

16000

17000

18000

19000

20000

0 4 8 12 16 20 24

De

man

d [

MW

]

Hours of the day

Polynomial 6th degree September Average Sunday Profile

59

5 General conclusions

The entire thesis consists of one theoretical and three research parts. The analysis of the Polish

Power System data dealt with operation conditions of the system with increasing share of

renewable energy sources. The system data was investigated for an entire year in a monthly

resolution, as well as selected set of worst-case daily generation-consumption profiles based on

proposed various methodology. As a result, it has been shown what are the possible jeopardies

for the system and on which aspects the improvements should be made. The results emphasize

the problem with both upward and downward reserves which may become more serious when

the penetration of RES further increases. Hence, the focus shall be placed on the development of

the generation flexibility and creation of effective financial incentives for construction of the

fast-reacting backup capacity and flexible baseload units.

The second part comprises numerical modeling of the daily profiles. Comparison of the various

methods of real data approximation was carried out with the R-factor chosen as a comparative

tool. Investigated approximation methods consist of linear regression with the least square

methods and set of orthogonal polynomials: Chebyshev, Hermite and Legendre polynomials.

Apart from various fitting method, the optimal degree of the polynomial was to be chosen. As the

objective of the study was to find possibly simple and easy-to-use function comparison of the

6th, 5th, 4th and 3rd degree was performed. Despite completely different methodology the modeled

function was identical for each method. Due to simpler calculations for further analysis was

chosen one of the orthogonal polynomials. Meanwhile, test of the polynomial degrees showed

clearly that if the satisfactory accuracy is required the only acceptable degree was 6th – the

highest one studied.

Finally, based on the approximation results, representative demand profiles were chosen.

Selection was made for each of the season for both working days and weekends. Representative

profiles were created by calculating average demand for all Wednesdays and Sundays in a

month, for working days and weekends respectively. The chosen months are the most

characteristic months for each season, i.e. January for Winter, April for Spring, July for Summer

and September for Autumn. The modeling was performed with the use of Hermite polynomial of

the 6th order. Pearson factors for the obtained profiles are of the acceptable values varying

between 0.965 and 0.992.

The research is considered to create fundaments for further study and therefore numerous

extensions are recommended. It would be useful to objectively evaluate effectiveness of linear

regression and orthogonal polynomials by comparing their exact computation time. Moreover,

for implementation to Economic Dispatch and Unit Commitment models it might be advisable to

60

rearrange representative profiles to per-unit scale with implementable peak and/or off-peak

demand. Such approach would allow for more universal applications of the modeled functions.

61

Appendix I In the Appendix I can be found supplementary data to Chapter 2.

Monthly generation-demand profiles for 2014:

Figure 40. Monthly profile for January 2014. Based on (PSE - webpage).

Figure 41. Monthly profile for February 2014. Based on (PSE - webpage).

0

5 000

10 000

15 000

20 000

25 000

30 000

1

24

47

70

93

11

6

13

9

16

2

18

5

20

8

23

1

25

4

27

7

30

0

32

3

34

6

36

9

39

2

41

5

43

8

46

1

48

4

50

7

53

0

55

3

57

6

59

9

62

2

64

5

66

8

69

1

71

4

73

7

January 2014

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

30 000

1

22

43

64

85

10

6

12

7

14

8

16

9

19

0

21

1

23

2

25

3

27

4

29

5

31

6

33

7

35

8

37

9

40

0

42

1

44

2

46

3

48

4

50

5

52

6

54

7

56

8

58

9

61

0

63

1

65

2

February 2014

NCDGU generation wind CDGU demand

62

Figure 42. Monthly profile for March 2014. Based on (PSE - webpage).

Figure 43. Monthly profile for April 2014. Based on (PSE - webpage).

Figure 44. Monthly profile for May 2014. Based on (PSE - webpage).

0

5 000

10 000

15 000

20 000

25 000

30 000

1

24

47

70

93

11

6

13

9

16

2

18

5

20

8

23

1

25

4

27

7

30

0

32

3

34

6

36

9

39

2

41

5

43

8

46

1

48

4

50

7

53

0

55

3

57

6

59

9

62

2

64

5

66

8

69

1

71

4

73

7

March 2014

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

12

34

56

78

91

11

13

31

55

17

71

99

22

12

43

26

52

87

30

93

31

35

33

75

39

74

19

44

14

63

48

55

07

52

95

51

57

35

95

61

76

39

66

16

83

70

5

April 2014

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

1

24

47

70

93

11

6

13

9

16

2

18

5

20

8

23

1

25

4

27

7

30

0

32

3

34

6

36

9

39

2

41

5

43

8

46

1

48

4

50

7

53

0

55

3

57

6

59

9

62

2

64

5

66

8

69

1

71

4

73

7May 2014

NCDGU generation wind CDGU demand

63

Figure 45. Monthly profile for June 2014. Based on (PSE - webpage).

Figure 46. Monthly profile for July 2014. Based on (PSE - webpage).

Figure 47. Monthly profile for August 2014. Based on (PSE - webpage).

0

5 000

10 000

15 000

20 000

25 000

12

34

56

78

91

11

13

31

55

17

71

99

22

12

43

26

52

87

30

93

31

35

33

75

39

74

19

44

14

63

48

55

07

52

95

51

57

35

95

61

76

39

66

16

83

70

5

June 2014

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

1

24

47

70

93

11

6

13

9

16

2

18

5

20

8

23

1

25

4

27

7

30

0

32

3

34

6

36

9

39

2

41

5

43

8

46

1

48

4

50

7

53

0

55

3

57

6

59

9

62

2

64

5

66

8

69

1

71

4

73

7

July 2014

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

1

24

47

70

93

11

6

13

9

16

2

18

5

20

8

23

1

25

4

27

7

30

0

32

3

34

6

36

9

39

2

41

5

43

8

46

1

48

4

50

7

53

0

55

3

57

6

59

9

62

2

64

5

66

8

69

1

71

4

73

7August 2014

NCDGU generation wind CDGU demand

64

Figure 48. Monthly profile for September 2014. Based on (PSE - webpage).

Figure 49. Monthly profile for October 2014. Based on (PSE - webpage).

Figure 50. Monthly profile for November 2014. Based on (PSE - webpage).

0

5 000

10 000

15 000

20 000

25 000

12

34

56

78

91

11

13

31

55

17

71

99

22

12

43

26

52

87

30

93

31

35

33

75

39

74

19

44

14

63

48

55

07

52

95

51

57

35

95

61

76

39

66

16

83

70

5

September 2014

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

30 000

1

24

47

70

93

11

6

13

9

16

2

18

5

20

8

23

1

25

4

27

7

30

0

32

3

34

6

36

9

39

2

41

5

43

8

46

1

48

4

50

7

53

0

55

3

57

6

59

9

62

2

64

5

66

8

69

1

71

4

73

7

October 2014

NCDGU generation wind CDGU demand

0

5 000

10 000

15 000

20 000

25 000

30 000

12

34

56

78

91

11

13

31

55

17

71

99

22

12

43

26

52

87

30

93

31

35

33

75

39

74

19

44

14

63

48

55

07

52

95

51

57

35

95

61

76

39

66

16

83

70

5

November 2014

NCDGU generation wind CDGU demand

65

Figure 51. Monthly profile for December 2014. Based on (PSE - webpage).

0

5 000

10 000

15 000

20 000

25 000

30 000

1

24

47

70

93

11

6

13

9

16

2

18

5

20

8

23

1

25

4

27

7

30

0

32

3

34

6

36

9

39

2

41

5

43

8

46

1

48

4

50

7

53

0

55

3

57

6

59

9

62

2

64

5

66

8

69

1

71

4

73

7

December 2014

NCDGU generation wind CDGU demand

66

Appendix II Appendix II contains column charts presenting CDGU daily ramps for each month in 2014.

Figure 52. CDGU daily ramps in January 2014. Based on (PSE - webpage).

Figure 53. CDGU daily ramps in February 2014. Based on (PSE - webpage).

Figure 54. CDGU daily ramps in March 2014. Based on (PSE - webpage).

0

2 000

4 000

6 000

8 000

10 000

12 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

January 2014

CDGU dailyramp [MW]

0

2 000

4 000

6 000

8 000

10 000

12 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

February 2014

CDGU dailyramp [MW]

0

2 000

4 000

6 000

8 000

10 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

March 2014

CDGU dailyramp [MW]

67

Figure 55. CDGU daily ramps in April 2014. Based on (PSE - webpage).

Figure 56. CDGU daily ramps in May 2014. Based on (PSE - webpage).

Figure 57. CDGU daily ramps in June 2014. Based on (PSE - webpage).

Figure 58. CDGU daily ramps in July 2014. Based on (PSE - webpage).

0

2 000

4 000

6 000

8 000

10 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

April 2014

CDGU dailyramp [MW]

0

2 000

4 000

6 000

8 000

1

33

65

97

12

9

16

1

19

3

22

5

25

7

28

9

32

1

35

3

38

5

41

7

44

9

48

1

51

3

54

5

57

7

60

9

64

1

67

3

70

5

73

7

May 2014

CDGU dailyramp [MW]

0

2 000

4 000

6 000

8 000

10 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6June 2014

CDGU dailyramp [MW]

0

2 000

4 000

6 000

8 000

10 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

July 2014

CDGU dailyramp [MW]

68

Figure 59. CDGU daily ramps in August 2014. Based on (PSE - webpage).

Figure 60. CDGU daily ramps in September 2014. Based on (PSE - webpage).

Figure 61. CDGU daily ramps in October 2014. Based on (PSE - webpage).

Figure 62. CDGU daily ramps in November 2014. Based on (PSE - webpage).

0

2 000

4 000

6 000

8 000

10 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

August 2014

CDGU dailyramp [MW]

0

2 000

4 000

6 000

8 000

10 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

September 2014

CDGU dailyramp [MW]

0

2 000

4 000

6 000

8 000

10 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6October 2014

CDGU dailyramp [MW]

0

2 000

4 000

6 000

8 000

10 000

12 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

November 2014

CDGU dailyramp [MW]

69

Figure 63. CDGU daily ramps in December 2014. Based on (PSE - webpage).

0

2 000

4 000

6 000

8 000

10 000

1

30

59

88

11

7

14

6

17

5

20

4

23

3

26

2

29

1

32

0

34

9

37

8

40

7

43

6

46

5

49

4

52

3

55

2

58

1

61

0

63

9

66

8

69

7

72

6

December 2014

CDGU dailyramp [MW]

70

Appendix III In appendix III one can find an example of matrix calculations for orthogonal polynomial

approximation. The presented matrices are for 1st January 2014, Hermite polynomial of the 3rd

order.

Hermite basic functions

T0 1

T1 2x

T2 4x^2-2

T3 8x^3-12x

X

1 2 2 -4

1 4 14 40

1 6 34 180

1 8 62 464

1 10 98 940

1 12 142 1656

1 14 194 2660

1 16 254 4000

1 18 322 5724

1 20 398 7880

1 22 482 10516

1 24 574 13680

1 26 674 17420

1 28 782 21784

1 30 898 26820

1 32 1022 32576

1 34 1154 39100

1 36 1294 46440

1 38 1442 54644

1 40 1598 63760

1 42 1762 73836

1 44 1934 84920

1 46 2114 97060

1 48 2302 110304

Xt

1 1 1 1 1 1 1 1 1 …

2 4 6 8 10 12 14 16 18 …

2 14 34 62 98 142 194 254 322 …

-4 40 180 464 940 1656 2660 4000 5724 …

Z

0,95016 -0,14417 0,005877 -7E-05

-0,14417 0,026867 -0,0012 1,5E-05

0,005877 -0,0012 5,64E-05 -7,3E-07

-7E-05 1,5E-05 -7,3E-07 9,78E-09

XtX

24 600 19552 716400

600 19600 718800 28090720

19552 718800 28130016 1,15E+09

716400 28090720 1,15E+09 4,8E+10

71

Y

15 050

14 200

13 500

13 025

12 800

12 625

12 600

12 375

12 375

12 900

13 475

13 900

14 375

14 775

14 825

15 150

16 750

17 275

17 400

17 550

17 200

16 400

15 450

14 325

XtY

350300

9150700

306242200

11362000600

alfa

17062,35

-826,875

42,05318

-0,53782

Hours Actual demand

Hermite

polynomial

1 15 050 15494,85755

2 14 200 14322,0807

3 13 500 13434,09845

4 13 025 12805,09541

5 12 800 12409,25616

6 12 625 12220,76532

7 12 600 12213,80747

8 12 375 12362,56723

9 12 375 12641,22918

10 12 900 13023,97793

11 13 475 13484,99809

12 13 900 13998,47424

13 14 375 14538,59098

14 14 775 15079,53293

15 14 825 15595,48467

16 15 150 16060,63081

17 16 750 16449,15594

18 17 275 16735,24467

19 17 400 16893,0816

20 17 550 16896,85132

21 17 200 16720,73843

22 16 400 16338,92754

23 15 450 15725,60325

24 14 325 14854,95014

72

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Mielczarski, W., & Kasprzyk, S. (2004). CIGRE Session: C2-104. The electricity market in Poland -

recent advances. Paris.

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74

Acknowledgements First and foremost, I would like to thank my head supervisor – Professor Władyslaw Mielczarski,

for helping me in selection of the area of my diploma research and placing a positive pressure on

me which made my work progress very effective.

I would also like to thank Michał Wierzbowski and Błażej Olek whose advisory and assistance on

the details of the work was very important.

I want to thank Mateusz Andrychowicz, with whom I could share ideas and collaborate on

particular subjects of our similar research areas.

Finally, I am very grateful to my beloved fiancée - Paulina Pruszkowska for being very

supportive, understanding and making every day of this busy time meaningful and enjoyable;

my loving parents on whose help I can always rely on; and my friend Aleksander Cisłak for

valuable discussions and taking care of my work-fun balance.

75

Summary There is a set of aims of this thesis. The first goal is to provide a general overview of the

liberalized electricity market in Poland, i.e. its origins, structure, members, trading schemes,

legal and operations aspects. Within the coexistence of economic, legal and technical aspects of

the market functioning the emphasis is placed on the fact that the last issue is the driving factor

of the entire concept. The crucial reason for it is a necessity of stipulating equal volume of

electricity generated and consumed in every period of time. Taking into account the

aforementioned, second part of the study focuses on the analysis of the Polish transmission grid

data, in terms of various relations between generation and consumption. Reviewing electricity

production from centrally dispatched generating units, wind turbines and the rest, together with

information about demand and available reserves allowed for determination of set of various

most challenging for the system daily profiles. The third chapter is dedicated to investigate

diverse approximation methods in order to find a compact, flexible and accurate technique of

estimating a real daily demand profile. The study comprises of comparison of both

approximating methods and degrees of fitting polynomial. Hence, investigated were polynomials

of the 6th, 5th, 4th and 3rd order in each of the following method:

Linear regression polynomials optimized with least squares method.

Various orthogonal polynomials approximation:

o Chebyshev polynomials

o Hermite polynomials

o Legendre polynomials.

In the last section of the thesis, on the basis of hitherto acquainted knowledge, concrete

parametric functions were created which represent demand profile for each of the season of the

year. For each season there are two functions – describing representative profiles for both

working days and weekends, respectively.

The most important conclusions from the research are:

Regardless of approximating method the results remain unchanged, however there are

differences in implementation and simplicity of the methods.

Among the investigated degrees of approximating polynomials 6th degree is the only

degree which met the requirements of reliability and accuracy.

76

Streszczenie Powyższa praca magisterska porusza kilka problemów. Pierwszym z celów pracy jest

przedstawienie ogólnego zarysu zliberalizowanego rynku energii w Polsce, m.in. okoliczności

jego powstawania, struktury rynku, jego uczestników, a także miejsca i sposobu prowadzenia

handlu energią oraz aspektów prawnych i operacyjnych. Choć rynek energii jest współtworzony

przez dziedziny ekonomii, prawa i myśli inżynierskiej, to kwestie techniczne, będące głównym

obiektem badań, narzucają pozostałym wiele ograniczeń determinując ich kształt. Głównym

problemem jest konieczność zapewnienia równowagi w wytwarzaniu i zużywaniu energii

elektrycznej w każdej chwili czasu. Mając to uwadze, druga część pracy obejmuje analizę danych

systemowych Krajowego Systemu Elektroenergetycznego badając zależności między generacją

i zapotrzebowaniem. Analiza produkcji energii elektrycznej z jednostek wytwórczych centralnie

dysponowanych, źródeł wiatrowych oraz pozostałych wraz z informacjami dotyczącymi

zapotrzebowania oraz dostępnością rezerw pozwoliło na określenie zbioru różnych, szczególnie

trudnych dla pracy systemu dobowych profili. Trzeci rozdział poświęcono na zbadanie różnych

metod aproksymacji w celu znalezienia prostej, elastycznej i dokładnej techniki szacowania

rzeczywistych dobowych profili zapotrzebowania. Studium dotyczy porównania zarówno

samych metod, jak i stopni wielomianów będących narzędziem dopasowującym. W rezultacie,

zbadano wielomiany 6-tego, 5-tego, 4-tego oraz 3-ego stopnia dla każdej z następujących metod

Regresja liniowa z użyciem aproksymacji metodą najmniejszych kwadratów.

Różne wielomiany ortogonalne:

o Wielomiany Czebyszewa

o Wielomiany Hermite’a

o Wielomiany Legendre’a.

Ostatnia część pracy dotyczy przedstawienia dokładnych funkcji parametrycznych, stworzonych

w oparciu o dotychczas wykonaną analizę, które odzwierciedlają reprezentatywne profile

zapotrzebowania dla każdej pory roku. Dla każdego sezonu zaproponowano dwie funkcje –

opisujące reprezentatywne profile dla dni roboczych oraz weekendów.

Najważniejsze wnioski płynące z przeprowadzonych badań są następujące:

Niezależnie od metody aproksymującej otrzymane wyniki nie ulegały zmianie.

Spośród zbadanych stopni wielomianów jedynie wielomian 6-tego stopnia spełnił

warunki dotyczące dokładności i niezawodności odwzorowania.