Analysis of firm capacity calculation methods as ... · Analysis of firm capacity calculation methods as mechanisms to ensure long-term reliability in deregulated power markets 1

Analysis of firm capacity calculation methods as mechanisms to ensure long-term reliability

in deregulated power markets

Anlisis de metodologas de clculo de potencia firme

como mecanismos para asegurar la fiabilidad a largo plazo en mercados elctricos liberalizados

Proyecto fin de carrera Escuela Tcnica Superior de Ingeniera (ICAI)

Universidad Pontificia Comillas Madrid

Autor: Carla Monge Liao

Directores: Ernesto Parrilla, Carlos Batlle y Carlos Vzquez

Madrid, junio de 2005

1

Analysis of firm capacity calculation methods as mechanisms to ensure long-term reliability in deregulated power markets

Summary

The National Electric Reliability Council of the United States (NERC)

defines reliability as: the degree to which the performance of the elements

of the electrical system results in power being delivered to consumers

within accepted standards and in the amounts desired.

The concept of reliability can be broken down into short and long-

term aspects, system security the short-term ability of the system to

withstand deviations between supply and demand in real time, which is

not the object of study in this project, and system adequacy the long-

term capacity of the system to be able to meet the fluctuating energy

demand requirements throughout a broad time-span.

In the liberalized electric market framework, generation planning has

become a decentralised activity, whilst both transmission and distribution

networks remain regulated. The systems long-term reliability is now

determined by market interactions between consumers and generators.

So far no consensus has been reached on how to deal with the

adequacy problem. The debate is focused mainly on two aspects: on the

one hand, on the suitability of settling incentives to enhance investment in

order to guarantee a sufficient margin -it remains uncertain whether an

acceptable level of adequacy will be obtained through these market

mechanisms alone- and on the other, on the way to evaluate the actual

contribution to the system adequacy of the different generators.

The regulator can either choose to intervene indirectly in the market

to promote an adequate level of system reliability, or not to intervene and

to trust in market mechanisms to promote suitable system adequacy.

Interventions can range from establishing payments for available capacity

to incentive additional investments, to requiring consumers to contract a

certain level of firm power, promoting a capacity market.

2

In some of the alternatives proposed, regulators need to be able to

evaluate system generator firm capacity values -the degree to which each

generator effectively contributes to the system reliability-. This project is

going to focus on analysing, comparing and develop some of the most

commonly used firm capacity calculation methodologies.

The four methodologies evaluated in this project are: both Spanish

and US heuristic methods, the convolution model and the cooperative

game model.

The first two evaluate firm capacity through heuristic formulae.

These methods are simple to calculate, but they do not correctly reflect

each generators contribution to the systems adequacy. This is due to the

fact that the contributions do not depend solely on each individual

generators generation capacity, but also on various other factors which

depend on the rest of the generators present in the system.

The convolution model is based on a mathematical technique which

enables the probabilistic dispatch of all the units in the system. This

method carries out a marginal analysis of how each unit contributes to the

system in very specific conditions. Firm capacity values are obtained from

the results of this dispatch, according to given reliability criteria. The

advantage of this method in comparison with the previous ones is that it

takes the full power system into account when calculating individual

generator contributions. The drawback of this method however is the fact

that it does not take into account the cooperative game theory.

The cooperative game model seems to be the most complete of the

four methods analysed. Based on probabilistic production costing and on

the cooperative game theory, it focuses on the fact that the contribution to

the system reliability of the group of generators as a whole is greater than

the sum of the generators individual contributions -it implies that the

combination of all the generators has a synergic effect on the system

reliability-. The excess system reliability generated is distributed

appropriately amongst the contributing generators through linear

3

optimisation techniques. The drawback of this method is that it is too

complex to be able to simulate large-scale power systems without facing a

combinatory explosion problem.

Simulation models using Matlab have been developed in this

project to implement the previously presented methodologies. The

differences that have arisen when calculating generator firm capacities

through the different methods have been analysed and evaluated, as have

their strengths and weaknesses, the key aspects they concern, those they

have in common and those in which they differ.

The cooperative game model has been developed on the basis of an

already existing model. The efficiency of this model has been improved;

enabling it to be run entirely on Matlab, incorporating an optimisation

function and a final result ponderation using both Loss of Load

Probability (LOLP) and Non Served Energy (NSE) criteria.

In order to be able to adapt the cooperative game model to large-

scale hydrothermal power system simulations, model simplifications have

been tested in search of solutions for the combinatory explosion problem

which continue to provide accurate simulation results.

In conclusion, in order for the regulator to obtain reliable firm

capacity values for each system generator in order to be able to use these

values as the basis of a remuneration criterion with the aim to ensure

adequacy in liberalised power markets, the cooperative game model

seems to be the most complete of the four models analysed in this project,

were it not for the simulation problems it entails for large-scale power

systems. The results obtained in this project suggest that the US heuristic

method could be an interesting alternative on which approximations of

the cooperative game model could be based, due to the similarity of the

firm capacity values each of the methods calculate. Further developments

beyond the scope of this project would be to determine the precise power

system conditions under which these approximations are valid.

Analysis of firm capacity calculation methods

as mechanisms to ensure long-term reliability

in deregulated power markets

1. Introduction 5

1.1 Defining power system reliability 6

1.2 Where the adequacy problem arises: Power system

evolution 8

1.3 How to define the adequacy problem 13

1.4 Project aims 16

1.5 Document structure 17

2. State of the art 19

2.1 Regulatory measures to deal with power system reliability 19

2.2 Firm capacity calculation methods 27

3. Convolution models 31

3.1 Mathematical concept of convolution 31

3.2 Convolution applied to power systems 33

3.3 Dispatch of different types of generators 37

3.4 Problematic convolution of hydro groups 43

3.5 Traditional convolution method to determine generator

firm capacity 48

3.6 Convolution of other generators 50

4. The cooperative game model 51

4.1 Motivation 51

4.2 Aims 63

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4.3 How the cooperative game method works 65

4.4 Additional considerations 74

4.5 The advantages of the cooperative game model 78

4.6 The disadvantages of the cooperative game model 80

5. Results 84

5.1 Convolution model 84

5.2 Restricted Cooperative game model 91

5.3 Full cooperative game model 98

5.4 Comparison of the results obtained 102

6. Conclusions 107

7. Further developments 112

8. References 114


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Figure index

Fig. 1.1 Illustration of LOLP and NSE values 14

Fig. 3.1. From LDC to ILDC 33

Fig. 3.2. Inverted load duration curves illustrating the effect of

successive generators being dispatched in the system 35

Fig. 3.3. Reading data on ILDC curves 36

Fig. 3.4. Scenarios A and B with G1 active and inactive respectively 38

Fig. 3.5. Convolution of scenarios A and B to obtain a new ILDC 39

Fig. 3.6. LDC 39

Fig. 3.7. ILDC 39

Fig. 3.8. Active scenario, p = 0.85 40

Fig. 3.9. Inactive scenario, p = 0.15 40

Fig. 3.10. Convolution results 40

Fig. 3.11. Hydro plant dispatch mechanism 42

Fig. 3.12. Load-duration curve for a period of three hours 43

Fig. 3.13. Inverted load-duration curve for a period of three hours 44

Fig. 3.14. Dispatch of T1 first 45

Fig. 3.15. Dispatch of T2 first 45

Fig. 3.16. Dispatch of all units: T1, T2 and H1 47

Fig. 3.17. Convolution method algorithm 49

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Fig. 4.1. Scenario where no generators have been dispatched yet 54

Fig. 4.2. Scenario where generators have been dispatched 54

Fig. 4.3. Scenario with a high proportion of thermal units 56

Fig. 4.4. Scenario with a low proportion of thermal units 56

Fig. 4.5. Probability of scenario occurring = 0,095 57

Fig. 4.6. Probability of scenario occurring = 0,045 58

Fig. 4.7. Ideas behind the cooperative game 65

Fig. 4.8. Probabilistic dispatch algorithm. 69

Fig. 4.9. Cooperative game model 79

Fig. 5.1. Relative errors for each unit in each restricted combination 93

Fig. 5.2. Maximum percentage errors in the different restriction

combinations 93

Fig. 5.3. Medium-sized power system results for the restricted

cooperative game model 94

Fig. 5.4. Proportion of firm capacity distributions made by each model

106


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Table index

Table 3.1. Thermal and hydro unit data 43

Table 3.2. Dispatch results 45

Table 3.3. Dispatch parameters: LOLP, NSE and ERC 46

Table 3.4. Dispatch parameters for the T1, T2 and H1 dispatch 47

Table 3.5. Full dispatch results 47

Table 4.1. Generator data 57

Table 5.1. Small-sized power system input data for the convolution

model 85

Table 5.2. Small-sized power system absolute results for the

convolution model 85

Table 5.3. Small-sized power system relative results for the convolution

model 85

Table 5.4. Medium-sized power system input data for the convolution

model 89

Table 5.5 Medium-sized power system absolute results for the


Table 5.6. Medium-sized power system relative results for the


Table 5.7. Small-sized power system results for the restricted

cooperative game model 92

Table 5.8. Spanish power system input data 96

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Table 5.9.Spanish power system results for the restricted cooperative

game model 97

Table 5.10. ERC cooperative game results for a small-sized power

system 99

Table 5.11. ERC cooperative game model results for a medium-sized

power system 100

Table 5.12. Comparative table of absolute results for the small case

example 102

Table 5.13. Comparative table of relative results for the small case

example 102

Table 6.1. Table comparing model characteristics 107

Table 6.2. Firm capacity evaluation criteria for each method studied 111


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1. Introduction

Electricity is an asset which modern society has come to depend

upon entirely. The repercussions of a lack of electricity stretch deep into

the social, economic and political domains of any country. Hence it is of

primordial importance for politicians, system regulators and operators

that power systems function correctly, avoiding emergency situations

or blackouts. One of the fundamental measures of the correct

functioning of a power system is its reliability, provided through the

interaction between its principal activities: generation, transport and

distribution systems.

In the liberalized electric market framework generation planning is

decentralised, whilst both transmission and distribution networks

remain regulated. As far as electricity generation is concerned, the

search for an adequate regulatory measure to ensure long-term

reliability in liberalised power markets remains an open issue to the

present date. The regulators concern for the long-term reliability issue

is reflected through the continuous search for adequate mechanisms to

ensure the long-term availability of sufficient generation capacity in

power systems. [Prez-Arriaga 1999] The regulator can either choose to

intervene indirectly in the market to promote an adequate level of long-

term system reliability, or not to intervene and to trust in market

mechanisms alone to promote adequate long-term system reliability,

intervening only in cases of emergency.

Some of the proposed alternatives to deal with the long-term system

reliability issue require the regulator to evaluate the individual

contributions of each generator to the system reliability as a whole with

three main aims. The first aim is to be able to supervise the fulfilment of

Chapter 1. Introduction

6

these capacity obligations, in order to be able to establish an adequate

degree of short-term reliability in the system. The second aim is to be

able to make coherent predictions on the need to expand the power

systems generation capacity according to demand evolution, and hence

be able to maintain an adequate long-term level of system reliability.

The third aim is to be able to provide fair economic incentives, for each

generator in the system and for incoming generators, on the basis of

their individual contributions to the system in order that they continue

contributing optimally to cover the required demand. Not only will

these three aims enable the regulator to establish an adequate degree of

short-term reliability in the power system and to maintain it in the long-

term, but just as importantly, to make the system generation capacity

evolution and the subsequent adequate system reliability evolution a

sustainable reality.

The object of study of this project is generator firm capacity or in

other words, the individual contributions of each generator to the

system reliability as a whole. These firm capacity values constitute

one of the basic sets of data the regulator needs in order to be able to

comprehend and evaluate the power system to be dealt with. As there

is no consensus as of yet about how these firm capacity values should

be calculated, this project is going to focus on analysing and comparing

some of the most commonly used firm capacity calculation

methodologies.

1.1 Defining power system reliability

The National Electric Reliability Council of the United States

(NERC) defines reliability as: the degree to which the performance of

the elements of the electrical system results in power being delivered to

consumers within accepted standards and in the amounts desired. In


7

other words, a reliable power market is that which provides a quantity

and quality of electricity supply to satisfy all its consumers.

Furthermore, the concept of reliability can be broken down into short

and long-term aspects [Prez-Arriaga 2001].

System security is the short-term ability of the system to withstand

deviations between supply and demand in real time. It is dealt with

by the System Operator (SO) via protection devices, ancillary service

procedures and ad hoc markets for operating reserves which stabilise

the system by matching electricity production and demand

instantaneously. As a consensus has been reached regarding the

measures to provide security in deregulated power systems, and for

the time being these are functioning satisfactorily, this aspect of

reliability will not be an object of study in this project.

System adequacy is the long-term capacity of the system to be able

to meet the fluctuating energy demand requirements throughout a

broad time-span. Quantitative analysis of this variable is complex

for various reasons. Amongst others, electricity cannot be stored and

neither supply availability nor demand can be fully predicted. This

makes it complicated to always have enough generation resources

available to cover the constantly fluctuating demand requirements.

In addition, power generation plants require large investments, at

least a two to three year lead-time period to be built and operative,

and then have an average life-span of thirty years or more. Hence it

is necessary to make predictions of generation requirements well in

advance to cover future demand as accurately as possible, avoiding

capacity shortages due to underinvestment on one hand and

capacity surplus due to overinvestment on the other [Ford 2000].


8

As so far no consensus has been reached on how to deal with

adequacy, this project is going to focus on analysing methods to

calculate firm capacity as the basis of various potential regulatory

measures aimed to maintain an appropriate level of generation

investment to ensure power system adequacy.

1.2 Where the adequacy problem arises: Power system evolution

Traditionally the electric power industry did not have great

adequacy problems. Long-term guarantee of supply was ensured by

regulators through a central planning system. In view of their

predictions of future demand requirements, they made a trade-off

between generator investment costs and a socially acceptable degree of

system reliability. In order to avoid the social and political problems

that power shortages entail, they tended to overweight the reliability

factor, programming capacity reserves as a large percentage surplus of

estimated peak demand. As a result there was a high degree of system

adequacy, but average electricity prices for consumers were also higher

than they are in present day power markets.

With the restructuring of electric power systems, although the

transmission and distribution systems have remained as regulated

businesses, the power generation system has been liberalised. Central

planning has been removed and the generators themselves make their

own investment decisions. Therefore the systems adequacy level is

now determined by market interactions between consumers and


9

generators. It is not clear however that an acceptable adequacy level

will be obtained through market mechanisms alone1.

1.2.1 Power market characteristics

Relying only on market mechanisms implies that all generators

whose investment is well adapted to the existing demand and to the

existing generator collective as a whole will be able to recover their total

costs through electricity market prices alone. [Schweppe 1988], [Prez-

Arriaga 1997] This idea is based on classic economic theory, where

equilibrium is maintained between the availability of an asset, its price,

and the demands response to both these factors. In our particular case,

the unexpected outage of a generating unit or a sudden demand

increase could cause a shortage of electricity leading to price spikes.

This would cause new generation investments to rise on one hand, and

the demand to fall on the other until a new equilibrium between supply

and demand was reached once again.

In theory this would account even for peaking units with the

highest variable costs to recover their total costs. They would be able to

do so when they are the only units supplying power at prices as high as

the demand side will accept, until supply and demand equilibrium is

re-established. As it has already been described however, the long lead-

1 This new market environment not only affects adequacy by putting investment

decisions in players hands, but also affects the management of generation units.

This is the case of hydro plant reservoir management. Traditionally, the central

planning regarding these hydro plants was much more oriented to assure system

security. However, it is being observed all around the world that private utilities,

more short-term oriented, tend to diminish the weight of the system security

variable in their hydro management models.


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times necessary for capacity expansion in power systems makes the

natural economic cycles of power supply lag behind the demand, so

leading to the known boom and bust cycles [Ford 1999].

This ideal market behaviour therefore only applies to a scenario of

perfectly adapted production units, which in practice is never the case.

In addition, electricity markets have various characteristics which make

them react differently from classic economic theory.

To begin with, it is very difficult to maintain equilibrium between

supply and demand in a system where the demand side is inelastic,

passive and immature. In electricity markets most consumers pay

regulated electricity tariffs independently from power market prices.

They are unaware of real-time price fluctuations and therefore cannot

react to them. In addition, the demand side tends to assume that

electricity is an asset which is always available; it is unaware of the

existence of the adequacy problem and of the need to pay for a service

to guarantee supply availability2.

Even if the demand side did have the choice to pay for a desired

level of system adequacy, the nature of electricity distribution as a

network that supplies all standard consumers indifferently makes it

impossible to differentiate the consumers that have paid for higher

levels of adequacy from those who have not. Although it could be

physically possible to disconnect standard consumers selectively from

the network, economically it is unviable and this leads to the problem

of free riding where consumers with lower electricity tariffs, paying

2 Besides, at least in a relevant number of countries, the consumers trust on the State

providence remains fully active.


11

for a lower degree of system adequacy, could benefit from those paying

higher tariffs, accounting for a higher degree of system adequacy.

Finally there is a third problem, probably the most relevant of all,

that is: generator investment risk aversion. Due to the fact that peaking

units are only operative during very specific periods of time, their

income is very volatile and hence investors do not feel the security that

they will be able to recover their costs through electricity market prices

alone. This problem is particularly serious for peaking units which have

to recover their fixed costs in only a few hours throughout the year. As

these units lack the suitable economic incentives to expand the

production system as demand grows, they demand an additional

remuneration, independently of their expected benefits, in order to be

able to build new generation units with lower investment risks.

1.2.2 Regulatory measures

This leads us to have to determine what role the regulator should

play to ensure the correct functioning of power markets. There are

currently two main lines of action that have been adopted by power

market regulators throughout the world.

The first is an approach of minimal intervention where the regulator

only intervenes in cases where the market mechanisms diverge

significantly from the natural market equilibrium. In other words, when

the situation becomes critical, rationing is near or price spikes are

continuous [Prez-Arriaga 1997].

The second is an approach of active regulatory intervention where

the regulator establishes fixed rules and measures to guarantee the

functioning of the power market at socially acceptable standards

[Prez-Arriaga 2001].


12

The implementation of each regulatory approach depends greatly

on the characteristics of each power system and on the political

preferences of each country. In practice however, the regulator usually

implements some form of regulatory mechanism to provide an

additional remuneration for generators in order to ensure a minimum

level of system adequacy. Since the main aim of these payments is to

enhance system adequacy, the idea is that they should be based on each

generators actual contribution to the systems reliability, but in practice

they are made in very different ways.

Several regulatory mechanisms have been implemented worldwide

to enhance generation adequacy in deregulated power markets, each

with different levels of success. To the present date however, none of

them has managed to fully resolve the long-term reliability problem in

power markets.

Initially Chile and Argentina implemented different forms of

capacity payments, followed by most other Latin American countries.

Energy-only markets were initially adopted by Nordic European

countries and California, but currently only Australian power markets

rely on market mechanisms alone. Capacity markets constitute an

intermediate measure between the more interventionist capacity

payment methods and the less interventionist energy-only market

methods. This method has been adopted in places such as the pools of

PJM (Pennsylvania, New Jersey and Maryland), New England and New

York in the USA [Creti 2003]. Specific analysis of these methods escapes

the scope of this project, however, their characteristics will be outlined

in section 2.1. The different methods used to evaluate firm capacity

how each generator contributes to the systems adequacy is the main

aim of this project and will be analysed in section 2.2.


13

Definite conclusions cannot be drawn as of yet due to the limited

time these measures have been active, but recent electricity shortages

that have occurred for example in Chile, California and Brazil, or

serious problems that have arisen in the Norwegian and Australian

systems are evidence enough that supply adequacy is a variable that

should not be taken lightly in deregulated power markets.

Nevertheless, so far in regional markets such as the Internal Electricity

Market of the European Union, Mercosur in South America, the Central

American Electric Market or the Regional Transmission Groups in the

USA, the aspects regarding system adequacy have seldom been dealt

with. The issue was first addressed relatively recently be the Central

American Market and by the Spanish and Portuguese regulators when

defining the Iberian Electricity Market. Only after the California crisis

has the issue been addressed in the European Union and in the Green

Book on supply security [Prez-Arriaga 2001].

1.3 How to define the adequacy problem

Having established the precedents of how the adequacy problem

arises, we are now faced with three fundamental problems when

dealing with adequacy in deregulated power systems. Firstly, a

measure of adequacy must be defined for the system as a whole.

Secondly, according to this measure, each generators contribution to

the systems adequacy must be evaluated. Finally, a payment method

must be determined on the basis of each generators contribution in

order to maintain appropriate generator investment incentives to

ensure system adequacy.

In order to tackle the first problem of defining a measure of

adequacy for the system, different criteria can be adopted. The first


14

adequacy measure is the loss of load probability (LOLP). This

probability indicates the expected amount of hours in an established

period of time where the demand cannot be covered by the generators.

For example, US power systems establish an overall system LOLP at

one day of system failure in ten years [PJM 2004]. The second measure

is non-served energy (NSE). This is the expected amount of energy out

of a total predicted demand requirement that generators will be unable

to cover. Thirdly, the two principal methods described previously can

be combined with different weighting factors to obtain yet other

measures of the systems adequacy. The concepts of loss of load

probability and non-served energy are illustrated in Fig. 1.1

0

0,2

0,4

0,6

0,8

1

1,2

0 50 100 150 200 250 300 350 400

Probability [h]

Pow

er [M

W]

Original ILDC curve T1 T2

LOLP = 0,5

NSE = area under thelast ILDC curve

0

0,2

0,4

0,6

0,8

1

1,2

0 50 100 150 200 250 300 350 400

Probability [h]

Pow

er [M

W]

Original ILDC curve T1 T2

LOLP = 0,5

NSE = area under thelast ILDC curve

Fig. 1.1 Illustration of LOLP and NSE values

The second problem concerns the evaluation of each generators

contribution to the systems adequacy on the basis of one of the

methods described above. The individual contribution of each

generator is known as firm capacity. It is a measure of the power a

generator can assure it will be able to provide at any given moment for

the consumer. There are various possible methods to evaluate firm


15

capacity and each provides different results concerning each

generators contribution to the systems adequacy (see chapter 2).

In addition, different calculation methods must be determined for

different types of generators. For example, the relatively constant

reliability of thermal units is very different to the reliability of hydraulic

units which depends greatly on external climatic factors and moreover,

on the way the utility that owns the reservoir manages its water

reserves. This project is going to focus on how to calculate firm capacity

for the different types of generators that could appear in deregulated

power systems: thermal units and hydro units with energy limitations.

It is also going to focus on the analysis of the differences that arise when

using different methods to calculate each generators firm capacity

contributions to the systems adequacy.

Eventually these calculations could be made by the generators

themselves, deciding the capacity they are willing to offer the regulator

to support system adequacy -such as the reliability options mechanism

[Vzquez 2002]- but in all the other cases it is the regulator who

administratively determines the amount of capacity he will admit as

firm capacity.

To tackle the third problem of how to calculate the total

remuneration of the generators in order to obtain an acceptable level of

system adequacy there are also various possible solutions which

provide us with different results. One option is to distribute payments

from a fixed sum of money destined to cover capacity reserves. This

implies that the money received will depend on the number of

generators in the system and on the initial sum of money allocated.


16

Another option is to pay each unit independently, on the basis of its

contribution to the systems adequacy3.

This project is going to focus on the first and second of these issues

related to the adequacy problem.

1.4 Project aims

Parting from the basis that in order for regulators to be able to

establish, control and sustain an adequate degree of reliability in power

systems they need to be able to evaluate system generator firm capacity

values or the degree to which each generator effectively contributes

to the system reliability this project sets out with the following aims:

To evaluate the different measures which can be used to define and

determine system reliability the differences that arise when using

LOLP or NSE criteria and the differences that arise when calculating

generator firm capacities through different methods.

To analyse and compare three methodologies to evaluate firm

capacity: both Spanish and US heuristic methods, the convolution

method and the cooperative game method.

To develop simulation models using Matlab to implement the

previously presented methodologies to evaluate firm capacity in

order to be able to study their strengths and weaknesses, the aspects

they have in common and those in which they differ.

3 This option would probably imply the need to determine which is the maximum

amount of capacity the regulator is willing to pay.


17

The last of these models will be developed on the basis of an already

existing cooperative game model. The aim is to update and improve

the efficiency of this model; enabling it to be run entirely on

Matlab, incorporating an optimisation function and a final result

ponderation using both LOLP and NSE criteria.

To adapt the cooperative game model so that it is able to simulate

large-scale hydrothermal power systems. Due to the known fact that

the cooperative game model is too complex to be used for large

system simulations because of the combinatorial explosion of

variables to evaluate, the aim is to develop a simplification of the

model which continues to provide accurate simulation results.

1.5 Document structure

In the introduction of this project the concept of power system

reliability has been presented. The reasons that make it a particularly

problematic issue in the present deregulated power market framework

have been tackled. Stressing the fact that no consensus has been reached

regarding the definition of adequate regulatory measures to ensure

long-term system reliability, the need to calculate individual generator

contributions to system reliability through various possible firm

capacity calculation methodologies has been presented. Finally, the

aims of this project have been outlined.

Chapter 2 is subdivided into two sections. The first describes

international regulatory measures which have been applied to deal with

power system adequacy. The second describes the most common

methodologies which have been approached in different countries so

far to calculate firm capacity.


18

Chapter 3 describes the first of these methodologies in detail: the

convolution method. The general functioning of the method is

described together with a description of how the model has been

developed in this project, both for thermal and for hydro units. A

fundamental flaw in hydro unit dispatch is also described as an

important conclusion of this thesis.

Chapter 4 describes a second methodology which also uses

convolution techniques: the cooperative game method. This model was

initially presented in [Batlle 2000] and has been further developed in

this project.

In chapter 5, all the results obtained from the simulation models

developed throughout the project are displayed and commented,

together with a comparative analysis of all of them.

Chapter 6 finally wraps up all the conclusions and new insights

obtained from this project on the subject of firm capacity calculations

and long-term power system reliability.


19

2. State of the art Equation Chapter 2 Section 1

2.1 Regulatory measures to deal with power system reliability

Regulatory approaches to enhance generation adequacy in

deregulated power systems can be classified into eight broad categories.

[Prez-Arriaga 2001]

2.1.1 Energy only markets

In energy only markets, following classic economic theory, there is

no regulatory intervention in the market [Ford 1995]. The drawback of

this approach is that it ignores the existence of power market

characteristics such as the passivity of a significant part of the demand,

investor risk aversion or free riding which have already been

described in section 1.2.1: Power market characteristics. These

distinguishing characteristics suggest that electric power markets will

not behave in the same way as classic markets, possibly resulting in

undesirable adequacy levels with episodes of electricity shortages and

high market prices.

This approach has been followed in Nordic European countries

the NORDEL group. However, Finland and Sweden have already

begun to apply the approaches considered in sections 2.1.4: Purchases

of peaking units by the system operator and 2.1.5: Purchases of reserves

by the system operator. Norway is also beginning to consider these

options due to the growing concern that the large surplus of generation

capacity experienced in previous years has not been following the same

trend in recent years. Energy only markets have also been used in

California where they have resulted in serious generation shortages

coupled together with market design flaws and other problematic

Chapter 2. State of the art

20

circumstances [Joksow 2001]. The only country where an energy only

market is implemented at the present date is Australia, where

generation shortages have been avoided so far.

2.1.2 Long term capacity payments

Capacity payments are issued to each system generator on the basis

of an administratively determined amount of money, proportional to

the estimated contribution of each generator to the systems reliability

[Sanz 1999]. The amount of money to be received by each generator is

computed in advance annually, but the payments can be performed on

a monthly or daily basis. In practice generators only receive capacity

payments for the periods of time in which they are available.

These estimated contributions are defined as firm capacity values

and can be calculated through different methods such as mathematical

modelling, historical operation data or heuristic procedures. These

different firm capacity calculation methodologies are the object of study

of this project and will be described in section 2.2: Firm capacity

calculation methods.

The long term capacity payment approach has a theoretical

justification. It has been demonstrated that market prices alone can fully

account for the total (fixed and variable) costs of a well adapted

generation mix in perfectly competitive market conditions. In the same

way, it can be proved that any given amount of additional generation

adequacy desired by the regulator, above that provided by the market

alone, requires a capacity payment for each generator on the basis of

each of their theoretical contributions to the system reliability in order

to maintain the systems economic viability [Prez-

Arriaga&Messeguer 1996].


21

The main advantages of the long-term capacity payment approach

are the reduction of peak unit income volatility, the reduction of

investor risk aversion, the fact that it provides incentives for mature

generation units to remain in the system and the fact that it hardly

interferes with the market, allowing the demand side to mature.

The approach has its disadvantages however, as adequacy is not an

identifiable or tangible commercial product which can be clearly

defined or calculated in order that generators can be remunerated for it.

The term firm capacity attempts to provide a measure for each

generators contribution to system adequacy, but a credible and feasible

definition for firm capacity values is complex, and no consensus has

been reached as of yet as far as this concept is concerned. The

calculation methodologies for firm capacity values become even more

complicated in hydrothermal generation scenarios due to the intrinsic

differences between these two types of plants. In addition, this method

does not encourage the availability of generation units when the system

is short of reserves, and hence there are no incentives for units to

increase system reliability during short margin seasons. Finally, this

method can also induce a perverse incentive for generators to lie in

their availability status in order to receive more capacity payments.

This approach was implemented in Chile for the first time in 1981,

and with different variations has also been used in various Latin

American countries such as Argentina, Peru, Colombia, Brazil, Central

American countries and Spain.

2.1.3 Short term capacity payments

This approach, like the previous one consists in paying an

administratively determined amount of money to each generator for


22

their reliability contributions to the system. The difference in this case is

that the amount to be paid is calculated daily by overestimating the loss

of load probability for the following day [Saunders 1999].

The drawbacks of this method are basically the same as those for

the previous one. This approach however is not based on a firm

capacity criterion but on a loss of load probability criterion which can

lead to undesirable consequences. The reasons for this are that loss of

load probability can be manipulated by suppliers to increase

production plant revenues, giving way to perverse operating

incentives.

This mechanism has been implemented in the initial pool

mechanism of England and Wales, where the ex ante method of

computation of market prices was used. Due to the large interference of

this mechanism in the day-ahead market, which is extremely

undesirable to preserve system adequacy, the pool mechanism was

substituted by NETA in 2000.

2.1.4 Purchases of peaking units by the system operator

In this approach the System Operator, following the instructions of

the regulator, purchases a certain number or all of the peaking units of

the system. The incurred costs are charged on the consumers as an extra

cost associated to reliability enhancement. These peaking units are

under the System Operators absolute control, and will be used to cover

any generation deficits that may appear whilst trying to meet the

system demand.

The aim of this approach is to avoid generation units which provide

capacity at the margin from leaving the market when their revenues are


23

too volatile or insufficient to cover their total operating costs. The units

concerned would typically be peaking units, mature or inefficient units.

The drawbacks of this approach are that it is strongly

interventionist and may seriously interfere with proper market

functioning. Generators not belonging to the System Operator, as well

as potential new entrants, may judge that market prices depend too

much on the purchasing decisions dictated by the regulator. Another

inefficiency of this method is that it only remunerates part of the system

units, although all of them provide adequacy to the system. In this

sense, it doesnt give any operation incentives to non-ISO units in order

to enhance system adequacy, and hence it unfairly reduces the

remuneration for their investment.

Purchasing of peaking units is currently being used in Finland and

Italy.

2.1.5 Purchases of reserves by the system operator

This idea is based on the allocation by means of an ad hoc auction

of some capacity payments to a subset of generation units. These are

known as strategic capacity reserves. The chosen units usually have

to meet some specific requirements such as: to belong to the kind of

technology that provides capacity at the margin. These units would

typically include peaking units, mature or inefficient units as in the

previous method. In the case of the NETA pool they are the most

reliable system units. As it can be observed, this method is closely

related to the purchase of peaking units explained previously, the

difference being in this case that the regulator purchases capacity

reserves a given volume of a units production for a defined time


24

scope, typically a year or a season rather than purchasing the whole

unit.

By imposing this mechanism it is possible to make an impact on

generation adequacy. However, besides the drawbacks presented in

section 2.1.4, an additional concern that arises with this method is the

breakdown of the market into two different parts: the competitive

market and the units in control of the System Operator as strategic

reserve units. This approach may be suitable for a limited amount of

time for instance whilst some initial surplus of capacity disappears

because of natural demand growth but this market division may

result in undesirable generator behaviour patterns.

In England and Wales, under NETA, the System Operator can

purchase any amount of operating reserves in advance. Norway is also

presently securing long-term operating reserves, but only with a yearly

anticipation [Wolfgang 2004].

2.1.6 Regulatory determined competitive bidding

In this case, although free entrance of new generation is allowed

into the system, the regulator or some other administrative authority

supervises that there is no threat of insufficient generation adequacy

according to some pre-established criterion. If it considered that there

was a lack of entry of new generation, this authority could start a

competitive bidding process for the addition of the required extra

generation.

This approach was adopted by countries such as France and

Portugal in the implementation of the European Directive 96/92/EC on

the Internal Electricity Market. This was later amended by a new

European Directive presented in Stockholm by the European


25

Commission in April 2001, which explicitly forbade the tendering

procedure to acquire new generation, except when used as an

exceptional measure for reasons of supply security.

The evaluations for methods 2.1.4: Purchases of peaking units by

the system operator and 2.1.5: Purchases of reserves by the system

operator are also valid here. This approach is strongly interventionist

and may seriously interfere with the proper functioning of the market.

Market agents, as well as potential new entrants, may judge that market

prices depend too much on the purchasing decisions dictated by the

regulator. The most important drawback of this mechanism however is

that no new generation can be built outside the auction. Investors will

hence have to wait for the auction in order to obtain extra

remuneration.

2.1.7 Capacity markets

The motivation behind this approach is to guarantee a regulated

generation adequacy level for the system. This is carried out by

imposing specific purchasing commitments of firm production capacity

on all the consuming entities, usually known as Load Serving Entities

(LSEs). The approach specifies the commitment that each generation

unit must maintain in real time, i.e. the commercial product associated

to the concept of generation adequacy.

The advantage of capacity markets is that there exists an identifiable

commercial product associated to generation adequacy, as well as a

commitment by the agents to purchase the product. Despite this, the

concept of firm capacity and the conditions of delivery remain

somewhat ambiguous, particularly when hydro units are involved. The

regulator determines both the total amount of desired firm capacity and


26

the rules by which to calculate the firm capacity provided by each unit.

There is no doubt that these rules could easily be contested in the case

of coexistence of thermal and hydro units of varied reservoir capacities.

In downsides of this method are that consumers remain fully

exposed to the potential high prices in the energy market. In addition,

the generators commitment to supply power in times of need is not

precisely defined i.e. it is complicated to define the availability of a

hydro unit with limited storage capacity.

Market mechanisms determine the price of capacity, which may be

very volatile depending on the tightness of the margins of installed

capacity over the system peak load and the anticipation of the auctions

with respect to real time. The volatility may be reduced however, by

increasing the time horizon of the auctions. Otherwise, the uncertainty

in generator remuneration may not gain much in terms of stability

using this approach. Thus, investors risk aversion will not be minimised

and the system could fail in providing adequacy.

This approach has been adopted in the PJM power market and in

other regional entities of eastern USA such as New York and New

England.

2.1.8 Reliability contracts

This method was initially proposed and implemented in the

Colombian power industry by the research group of the IIT. The idea

has been further developed [Vzquez 2003, Vzquez 2002] as a way to

solve the adequacy problem in deregulated power systems.

This approach proposes to establish an organised market in which

the regulator requires the Market or System Operators to buy a

prescribed volume of reliability contracts from generators on behalf of


27

the whole demand. These reliability contracts allow consumers to

obtain a price cap on the market price in exchange for a fixed

remuneration for the generators. Additionally, consumers obtain a

satisfactory guarantee that there will be enough available generation

capacity whenever it may be needed. Otherwise the generators will be

penalised. The generators are also compensated economically for this

service; the higher the contribution to the reliability of the system, the

higher their compensation will be.

2.2 Firm capacity calculation methods

Three main categories of methods to calculate the firm capacity of a

generator can be found throughout the world: methods based on

heuristic procedures, methods based on convolution techniques and

methods based on optimisation techniques.

2.2.1 Methods based on heuristic procedures

These methods have the advantage of being simple to calculate, but

they do not accurately reflect each generators contribution to the

systems adequacy. This is mainly because each generators

contribution does not depend solely on its own generation capacity but

also on that of the rest of the system generators. There are many factors

in play when calculating each generators contribution as a function of

the rest of the system generators contributions, such as the time of

dispatch, considerations on when to dispatch hydraulic units, cost

considerations when dispatching etc.

Examples of these methods are the firm capacity in the PJM capacity

market which is defined as a function of each generators maximum

power output and its outage rate [PJM 2004]:


28

For thermal units:

max (1 )FirmCapacity P FOR= (Eq. 2.1)

For hydro units:

dispatchFirmCapacity P= (Eq. 2.2)

In Spain firm capacity is defined as a function of each generators

net power output and its average power output limited by resource

availability [OMEL 2001].

The general equation is:

12

resource limitationsnetFirmCapacity P P = + (Eq. 2.3)

For thermal units:

1 (1 )2

resource limitationsmaxFirmCapacity P FOR P = + (Eq. 2.4)

For hydro units:

[ ]1 0,35 ( )2 max max min

FirmCapacity P P P= + + (Eq. 2.5)

2.2.2 Methods based on convolution techniques

The advantage of these methods in comparison with the previous

ones is that they take the full power system into account when

calculating individual generator contributions.

In these methods the load-duration curve is convoluted, followed

by the convolution of each of the systems generators. In this manner all

the generators are dispatched depending on their nature (thermal,

hydro, etc.). Then each of the generators is removed from the system by


29

deconvolution, either individually or in different combinations with the

other generators from the system. The absence of each generator or

group of generators leaves different amounts of non-served energy in

the system. There are two possible variations of this method:

The first the standard convolution method defines firm

capacity as the power equivalent necessary to cover the non-served

energy remaining in the system after the deconvolution of each

individual generator on its own. This method was initially

implemented in Peru and has been proposed in Panama. Its main

disadvantages are that it doesnt take into account the different

dispatch variations that could occur within the system. In addition,

it doesnt take into account the fact that the sum of the generators

individual contributions to the system reliability is less than the

contribution of the group of generators as a whole, and therefore it

doesnt distribute this excess reliability amongst the contributing

generators. This important aspect is the basis of the cooperative

game theory and is taken into account in the cooperative game

method described next. It will be commented in more depth in

chapter 3.

The second the cooperative game method described in

[Batlle 2000] defines firm capacity as the solution of a cooperative

game. It has the advantage that it does take into account all possible

generator combinations in the system, unlike the previous method,

and it also takes into account the distribution of the excess

reliability provided by the group of generators as a whole.

However, it appears to be unviable for large systems where all the

possible generator combinations would be far too many and would

saturate the system.


30

The use of these methods still leaves us with problematic

considerations such as when to dispatch hydraulic units and cost

considerations when dispatching, which will have to be studied.

2.2.3 Methods based on dispatch simulations under critical

conditions

This method relies on the simulation of how all the generators are

going to be dispatched in a system under critical hydro conditions. On

the basis of the simulation results each generators firm capacity is

determined as the average power provided by each unit in every critical

hour previously defined. The disadvantages of this method are that

once again it does not take into account all possible dispatch

combinations, and that all of the groups contribute to the system

reliability. Hence, it doesnt remunerate all the system generators, and it

uses economic criteria to establish the remuneration process which can

lead to unfair remuneration mechanisms. In addition, in many cases the

simulations results are very sensitive to the input data. Examples of this

method implementation are the present and the future method in

Colombia [CREG 2000], and the current procedures used in Argentina,

Peru, Bolivia or Chile.


31

3. Convolution models Equation Chapter 3 Section 3.1

Convolution is a mathematical, probabilistic technique which has

been widely used to model power system scenarios and generator

dispatch as it enables the handling of historical data through a

probabilistic approach. Amongst the countries which have used

different convolution techniques to model their power systems are

Peru, Panama, Bolivia and Chile.

3.1 Mathematical concept of convolution

The main problem this power system modelling aims to resolve is

whether there is sufficient capacity in the system to cover predicted

demand. Clearly, the total capacity provided by the system generators

must exceed the peak demand. However, as each group has an outage

probability, a certain degree of excess capacity must be considered in

order to be able to adequately fit the system demand requirements. In

other words, a value of loss of load probability must be considered for

the system, and the aim is to minimise this value.

Each generator is therefore modelled as a random variable where

if the group is available it can contribute a given power output to the

system, and if it isnt available then its contribution will be zero.

Hence, the capacity all the generators can provide to the system is the

sum of a set of random variables.

Let us consider the set of random variables of three generators X ,

Y and Z . Each variable can be associated to a density function ( )xf x ,

( )yf y and ( )zf z . The probability that for example generator X will be

unable to supply a power load greater than x will be:

Chapter 3. Convolution models

32

[ ] ( )( )x

xP X x F x f x dx

= (Eq. 3.1)

The density function therefore indicates the probability that an

event will occur in a given interval:

[ ][ ] [ ]

[ ] ( ) ( )x dx x

x x

P x X x dx P X x dx P X x

P x X x dx f x dx f x dx+

+ = +

+ = (Eq. 3.2)

We now take two independent random variables X and Y .

Consider the sum of these variables to take on a defined value z . If in

addition Y takes on a value y , X will then have to be equal to z y .

The probability that Z is equal to z and Y is equal to y must therefore

be proportional to:

( ) ( )x yP Z z Y y f z y f y = = (Eq. 3.3)

Integrating this expression for all possible values of Y we obtain

what is known as the convolution integral:

( ) ( ) ( )z x yf z f z y f y dy

= (Eq. 3.4)

This formula expresses the density function of a random variable Z

as the convolution of the density functions of two other independent,

random variables X and Y .

z x y= (Eq. 3.5)

Convolution also has an inverse function known as deconvolution:

1x z y= (Eq. 3.6)


33

3.2 Convolution applied to power systems

For power simulations it is more common to work with the inverted

load-duration curve (ILDC) rather that with the load density function

(LD). This first function is simply the complementary function of the

latter one, and tells us the load that will go unserved in the system

rather than the load that will be served:

( ) 1 ( )ILDC LDF d F d= (Eq. 3.7)

The ILDC curve (with probability on the y -axis and unserved load

in MW on the x -axis), can be easily obtained from the usual load-

duration curve (with load in MW on the y -axis and time in hours on

the x -axis), as is shown in Fig. 3.1.

Fig. 3.1. From LDC to ILDC

As a starting point for the convolution model, the predicted ILDC

does not necessarily have to come from only one LDC. A probabilistic

ILDC can be obtained by integrating historical data from different


34

demand scenarios, with different probabilities of occurring, by

convolution.

If we define the non-served demand remaining in the system after

all the generators have been dispatched (ILDCN) as the difference

between the total system demand (ILDC) and the sum of all the system

generator capacities ( iC ):

1

N

N ii

ILDC ILDC C=

= (Eq. 3.8)

We can obtain the ILDCN of this non-served demand as the

subtraction of each generators capacity distribution functions from the

full system ILDC by convolution as follows:

1

...N NILDC ILDC C C

f f f f = (Eq. 3.9)

After the convolution of all the system generators we will be left

with a series of ILDC curves similar to the ones shown in Fig. 3.2. The

first curve is the ILDC of the full system demand when no generator

has been dispatched yet. The successive ILDCs with decreasing

probabilities are the result of dispatching each generator in the system,

so reducing the system demand to be covered each time. The last curve,

ILDCN represents the demand left uncovered once all the system

generators have been dispatched.


35

LOLP

NSE ILDCN

ILDC3

ILDC2

ILDC1

ILDC

Prob

abili

ty

Load [MW]

Fig. 3.2. Inverted load duration curves illustrating the effect of successive

generators being dispatched in the system

From the last ILDC curve obtained (ILDCN) we can extract the two

following valuable pieces of information:

The loss of load probability (LOLP) is the point where this last curve

cuts the y -axis. It represents the probability that there will still be

capacity demand left in the system after all the generators have been

dispatched.

The non-served energy (NSE) is the area beneath this last curve. It

represents the amount of energy in MWh that is left unsupplied in

the system after all the generators have been dispatched.

Convolution therefore enables us to add two probability functions,

in this particular case with probability on the y -axis and unserved load

in MW on the x -axis. In this manner, the coordinates of each point on


36

the curve represent the probability ( y -coordinate) that a load bigger

than the specified load ( x -coordinate) will go unserved in the system.

(See Fig. 3.3)

[ ][ ]

, 600 0,6

, 600 1 0,6 0,4

P UnservedCapacity x

P UnservedCapacity x

> =

= = (Eq. 3.10)

In addition, the area under the curve represents the energy which

will go unserved for power demands above or below a given point on

the curve. (See Fig. 3.3). This can be justified as we are evaluating

probability in hours.

600

0 600

( , 600) ( )

(0 , 600) ( ) ( )

NSE UnservedCapacity x ILDC x dx

NSE UnservedCapacity x ILDC x dx ILDC x dx

> =

> > =

(Eq. 3.11)

0

0,2

0,4

0,6

0,8

1

1,2

0 200 400 600 800 1000 1200

Load [MW]

Prob

abili

ty

Probability of 0,6 that a load bigger

than 600MW will go unserved

NSE due to unserved loads

greater than 600MW

NSE due to unservedloads smaller than

600MW

0

0,2

0,4

0,6

0,8

1

1,2

0 200 400 600 800 1000 1200

Load [MW]

Prob

abili

ty

Probability of 0,6 that a load bigger

than 600MW will go unserved

NSE due to unserved loads

greater than 600MW

NSE due to unservedloads smaller than

600MW

Fig. 3.3. Reading data on ILDC curves


37

3.3 Dispatch of different types of generators

When dealing with different types of generators such as thermal

and hydro units, it is important to take into account that some of them

have power and energy restrictions which condition the way in which

they can be dispatched optimally in power systems. In the following

sections different types of unit dispatch will be analysed.

3.3.1 Thermal units

In the case of thermal units, the input parameters for the model are:

maximum power output and forced outage rate.

Thermal units are dispatched using the following convolution

formula:

( ) ( ) (1 ) ( )n 1 i n i n iILDC x FOR ILDC x FOR ILDC x C+ = + + (Eq. 3.12)

Where: nILDC = inverted load-duration curve

1nILDC + = the new inverted load-duration curve after the

dispatch

iFOR = forced outage rate for generator i

x = the unserved load at each point of the curve

iC = maximum capacity for generator i

The meaning of this equation in words is that the probability of

unserved load in the system after the dispatch of generator i , ( nILDC ),

is equal to the sum of two terms. The first term: ( )i nFOR ILDC x , is the

probability that generator i is unavailable when the system demand

requires x MW of power which generator i could have been able to

supply. The second term: (1 ) ( )i iFOR ILDC x C + , is the probability that


38

generator i is available when the system demand requires x C+ MW of

power, which generator i is unable to supply as the amount required

exceeds the generators maximum capacity iC . In other words, the

equation adds the two probabilistic scenarios where a given generator

cannot cover the required demand at a given moment.

The following example illustrates this process more visually. Let us

consider a system with one unit, 1G , a scenario in which the unit is

active with a probability 1p of occurring and a scenario in which the

unit is inactive with a probability 1(1 )p of occurring. Fig. 3.4 and Fig.

3.5 show how these scenarios are integrated by convolution to obtain a

new ILDC after the dispatch of 1G .

Fig. 3.4. Scenarios A and B with G1 active and inactive respectively


39

Fig. 3.5. Convolution of scenarios A and B to obtain a new ILDC

Another discrete example can be set to illustrate more clearly how

the calculations are carried out. Let us consider the following LDC and

corresponding ILDC:

LDCILDC

MW

MW

h

h

LDCILDC

MW

MW

h

h

Fig. 3.6. LDC Fig. 3.7. ILDC

If we now add a generator to the system with a maximum power

output of 150MW and a FOR of 0.15, we are left with the two following

scenarios: Fig. 3.8 where the generator is active with a probability of


40

0.85 of occurring and Fig. 3.9 where the generator is inactive with a

probability of 0.15 of occurring.

3760

MW

Expected hours peryear

450

350

760

Group is activeprob=0.85

8760

MW


100

600

500

3760

760

Group is inactiveprob=0.15

Scenario 1 Scenario 2

3760

MW


450

350

760

Group is activeprob=0.85

8760

MW


100

600

500

3760

760


8760

MW


100

600

500

3760

760

8760

MW


100

600

500

3760

760


Scenario 1 Scenario 2

Fig. 3.8. Active scenario, p = 0.85 Fig. 3.9. Inactive scenario, p = 0.15

Integrating both scenarios by convolution can be reduced in this

discrete example to pondering each scenario by its probability of

occurring and adding them. The result obtained is shown in Fig. 3.10.

8760*0.15+3760*0.85 = 4510

3760*0.15+3760*0.85 = 3760

3760*0.15+760*0.85 = 1210

MW

Expected hours per year

450350 600

760*0.15 = 114

100

8760*0.15+3760*0.85 = 4510

3760*0.15+3760*0.85 = 3760

3760*0.15+760*0.85 = 1210

MW

Expected hours per year

450350 600

760*0.15 = 114

100

Fig. 3.10. Convolution results

The convolution technique then simply consists on repeating this

process for each of the generators in the system one by one. As units are


41

successively dispatched in the system, obviously the values of LOLP

and NSE decrease, but they never become zero as there is always a

small probability that all the generators in the system could fail at the

same time.

It is important to bear in mind that because thermal plants tend to

have the same characteristic parameters throughout the different

periods to be simulated, the convolutions of their distribution functions

are commutative:

x y y x = (Eq. 3.13)

Dispatch order is therefore irrelevant in the case of thermal units,

but we shall see in the next section how this is not the case for hydro

units.

3.3.2 Hydro units

In the case of hydro units, the input parameters for the model are:

maximum power output, minimum power output (known as run-of-

the-river capacity) and total available energy (a measure of their

reservoir capacity management).

Hydro unit dispatch is not carried out by convolution as hydro

units forced outage rate is considered to be insignificant in comparison

with their energy restrictions, and hence, in order to simplify the model,

no probability-related factor in their defining characteristics. In section

3.4 of this chapter we shall see how this causes an inherent problem in

hydro plant dispatch to arise.

Usually hydro unit dispatch is carried out in two steps:


42

Firstly the run-of-the-river capacity is dispatched by convolution as

though it where a thermal plant with 0FOR = . This is due to the fact

that it is a constant power output which is always supplied by the

hydro units.

Secondly, the rest of the hydro capacity, which is not constant, is

dispatched by moving the y -axis rightwards step by step

(1MW/step) checking in each step that either the plants energy

restriction or maximum power restriction is not violated. As soon as

either of these limiting factors is reached, the hydro plant is either

dispatched at its maximum power output if its energy restriction

was not violated before reaching it, or at a power value inferior to its

maximum power output if its energy restriction was violated before.

(See Fig. 3.11)

eH

PgE e

HgEm=

= eHg

PmU

11

MW

Prob.

1 MW 1 MW1 MW

P

p

eH

pgE

eH

PgE LOLP

eHgi

-1LOLPi

P

eH

pgE e

HgEm

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Analysis of firm capacity calculation methods as ... · Analysis of firm capacity calculation methods as mechanisms to ensure long-term reliability in deregulated power markets 1