pspr

by

Dr Lalit Goel

Professor of Power Engineering & Director of Admissions

Nanyang Technological University, Singapore

IEEE PES Distinguished Lecturer Program India, Nov/Dec, 2011

Power System Reliability –

Concepts & Techniques

Outline

1.

Power System Reliability

• Generating System (HLI) Reliability Assessment

• Composite System (HLII) Reliability Assessment

• Distribution System Reliability Assessment

• Cost-Benefit Considerations

• Concluding Remarks

Outline

2. Generation

System Reliability Assessment

• Generating Unit Modeling

• Capacity Outage Probability Table

• Evaluation of Risk of Capacity Shortfall

• Comparison of Deterministic & Probabilistic Criteria

• Incorporating Load Forecast Uncertainty

An electric power system serves the basic function of supplying customers, both large and small, with electrical energy as economically and as reliably as possible. The reliability associated with a power system is a measure of its ability to provide an adequate supply of electrical energy for the period of time intended under the operating conditions encountered.

Modern society, because of its pattern of social and working habits, has come to expect the power supply to be continuously available on demand - this, however, is not physically possible in reality due to random system failures which are generally outside the control of power system engineers, operators and planners.

The probability of customers being disconnected can be reduced by

increased investment during either the planning phase, operating phase,

or both. Over-investment can lead to excessive operating costs which

must be reflected in the tariff structure. Consequently, the economic constraints can be violated even though the system may be highly reliable. On the other hand, under-investment can lead to the opposite situation. It is evident therefore that the economic and reliability constraints can be quite competitive, and this can lead to extremely difficult managerial decisions at both the planning and operating phases.

• Planning generating capacity - installed capacity equals the expected maximum demand plus a fixed percentage of the expected maximum demand;

• Operating capacity - spinning capacity equals the expected load demand plus a reserve equal to one or more largest units;

• Planning network capacity - construct a minimum number of circuits to a load group, the minimum number being dependent on the maximum demand of the group.

The criteria and techniques first used in practical applications were basically deterministic (rule-of-thumb) ones, for instance

Although the above-mentioned three and other criteria have been developed to account for randomly occurring failures, they are inherently deterministic. The essential weakness of these methods is that they do not account for the probabilistic/stochastic nature of system behavior, customer load demands and/or of component failures. Such aspects can be considered only through probabilistic criteria.

• Forced outage rate of generating units is known to be a function of unit size and therefore a fixed percentage reserve cannot ensure a consistent risk;

• Failure rates of overhead lines are functions of their lengths, design aspects, locations and environment, etc. - therefore a consistent risk of supply interruption cannot be ensured by constructing a minimum number of circuits;

• All planning and operating decisions are based on load forecasting techniques which cannot predict future loads precisely, i.e., uncertainties will always exist in the forecasts. This imposes statistical factors which should be assessed probabilistically.

Typical probabilistic aspects are as follows:

The concept of power system reliability, i.e., the overall ability of the system to satisfy the customer load requirements economically and reliably, is extremely broad. For the sake of simplicity, power system reliability can be divided into the two basic aspects of

• system adequacy, and • system security.

Adequacy relates to the existence of sufficient facilities within the system to satisfy customer load demands. These include the facilities to generate power, and the associated transmission and distribution facilities required to transport the generated energy to the load points. Adequacy, therefore, relates to static system conditions.

Power System Reliability

Security

pertains to the response of the system to the perturbations/disturbances it is subjected to. These may include conditions associated with local and widespread disturbances and loss of major generation/transmission.

Most of the techniques presently available are in the domain of adequacy assessment.

Power system functional zones Hierarchical levels

Generating capacity reliability is defined in terms of the adequacy of the installed generating capacity to meet the system load demand. Outages of generating units and/or load in excess of the estimates could result in “loss of load”, i.e., the available capacity (installed capacity - capacity

on outage) being inadequate to supply the load. In general, this condition requires emergency assistance from neighboring systems and emergency operating measures such as system voltage reduction and voluntary load curtailment. Depending on the shortage of the available capacity, load shedding may be initiated as the final measure after the emergency actions. The conventional definition of “loss of load” includes all events resulting in negative capacity margin or the available capacity being less than the load.

Generating System (HLI) Reliability

Assessment

The basic methodology for evaluating generating system reliability is to

develop probability models for capacity on outage and for load

demand, and calculate the probability of loss of load by a convolution

of the two models. This calculation can be repeated for all the periods (e.g., weeks) in a year considering the changes in the load demand, planned outages of units, and any unit additions or retirements, etc.

Generating System (HLI) Reliability

Assessment

1. loss of load probability (LOLP)

2. loss of load expectation (LOLE)

3. loss of energy expectation (LOEE)/expected energy not supplied (EENS)

4. frequency & duration (F&D) indices

5. energy index of reliability (EIR)

6. energy index of unreliability (EIU), and

7. system minutes (SM).

Probabilistic Criteria and Indices

An understanding of the probabilistic criteria and indices used in generating capacity reliability (HLI) studies is important. These include:

Generating System (HLI) Reliability

Assessment

This is the oldest and the most basic probabilistic index. It is defined as the probability that the load will exceed the available generation. Its weakness is that it defines the likelihood of encountering trouble (loss of load) but not the severity; for the same value of LOLP, the degree of trouble may be less than 1 MW or greater than 1000 MW or more. Therefore it cannot recognize the degree of capacity or energy shortage. This index has been superseded by one of the following expected values in most planning applications because LOLP has less physical significance and is difficult to interpret.

LOLP

Generating System (HLI) Reliability

Assessment

This is now the most widely used probabilistic index in deciding future generation capacity. It is generally defined as the average number of days (or hours) on which the daily peak load is expected to exceed the available capacity. It therefore indicates the expected number of days (or hours) for which a load loss or deficiency may occur. This concept implies a physical significance not forthcoming from the LOLP, although the two values are directly related.

It has the same weaknesses that exist in the LOLP.

LOLE

Generating System (HLI) Reliability

Assessment

This index is defined as the expected energy not supplied (EENS) due to those occasions when the load exceeds the available generation. It is presently less used than LOLE but is a more appealing index since it encompasses severity of the deficiencies as well as their likelihood. It therefore reflects risk more truly and is likely to gain popularity as power systems become more energy-limited due to reduced prime energy and increased environmental controls.

LOEE

Generating System (HLI) Reliability

Assessment

These are directly related to LOEE which is normalized by dividing by the total energy demanded. This basically ensures that large and small systems can be compared on an equal basis and chronological changes in a system can be tracked.

EIR and EIU

Generating System (HLI) Reliability

Assessment

The F&D criterion is an extension of LOLE and identifies expected frequencies of encountering deficiencies and their expected durations.

It therefore contains additional physical characteristics but, although widely documented, is not used in practice. This is due mainly to the need for additional data and greatly increased complexity of the analysis without having any significant effect on the planning decisions.

Frequency & Duration (F&D) Indices

Generating System (HLI) Reliability

Assessment

• All basic reliability indices can be represented by this expression, by using suitable definitions of the test function.

Objective

• Composite generating and transmission system evaluation is concerned with the total problem of assessing the ability of the generation and transmission system to supply adequate and suitable electrical energy to the major system load points (Hierarchical level II - HL II)

• The problem of calculating reliability indices is equivalent to assessing the expected value of a test function F(x), i.e., :

Composite System (HLII) Reliability

Assessment

• Expansion - selection of new generation, transmission, subtransmission configurations;

• Operation - selection of operating scenarios;

• Maintenance - scheduling of generation and transmission equipment

Applications in power system planning

Basic models

Up

_ 

Down

• G&T Equipment : Markovian or not; Two or multi-states.

Composite System (HLII) Reliability

Assessment

• System : AC or DC network representation.

• Load : Chronological or not; Markovian or not; Correlated

or not.

Basic models

Composite System (HLII) Reliability

Assessment

Reliability Measures (Conventional)

 System indices (sometimes appearing under different names) • LOLP = Loss of load probability

• LOLE = Loss of load expectation (h/year) • EPNS = Expected power not supplied (MW)

• EENS = Expected energy not supplied (MWh/year) • LOLF = Loss of load frequency (occ./year)

• LOLD = Loss of load duration (h) • LOLC = Loss of load cost ($/year)

 Load point indices

• LOLP, LOLE, etc.

Composite System (HLII) Reliability

Assessment

Reliability Measures (Well-Being)

 System indices

 Load point indices

Healthy

Marginal

At Risk

Success

• Prob {H} = Probability of healthy state • Prob {M} = Probability of marginal state

• Prob {R} = Probability of at risk state (LOLP) • Freq {H} = Frequency of healthy state (occ./year) • Freq {M} = Frequency of marginal state (occ./year)

• Freq {R} = Frequency of at risk state (LOLF) (occ./year) • Dur {H} = Duration of healthy state (h)

• Dur {M} = Duration of marginal state (h)

• Dur {R} = Duration of at risk state (LOLD) (h) • Prob {H}, Freq {H}, etc.

Composite System (HLII) Reliability

Assessment

Assessment Tools

State Selection State Analysis (adequacy)

• Enumeration • Power flow

• Monte Carlo simulation o Linear DC model

o Non-sequential o Non-linear AC model

o Sequential or chronological • Optimal power flow o Pseudo-chronological/sequential o Linear DC model

o Non-linear AC model

Composite System (HLII) Reliability

Assessment

Load Point Indices

• failure rate,





• average outage time

,

r

• average annual unavailability

,

U =



.r

• average load disconnected

,

L

• expected energy not supplied

,

E = U.L

State Space (Markov) Model

State

Probability (P)

Visiting frequency (f)

Residence time (r)

0

P

₀

= A

₁

.A

₂

f

₀

= P

₀

/r

₀

r

₀

= 1/(



₁

+



₂

)

1

P

₁

= U

₁

.A

₂

f

₁

= P

₁

/r

₁

r

₁

= 1/(



₁

+



₂

)

2

P

₂

= A

₁

.U

₂

f

₂

= P

₂

/r

₂

r

₂

= 1/(



₂

+



₁

)

3

P

₃

= U

₁

.U

₂

f

₃

= P

₃

/r

₃

r

₃

= 1/(



₁

+



₂

)

0 1 3 2 ₁ ₁ ₁ ₁ ₂ ₂ ₂ ₂

State space diagram for two-component, four-state model

Distribution System Reliability Assessment

Parallel Structure, n (independent) Components

n n i i i i=1 i=1 r 1 r  _{  } _{ }  _{ } _{ }   _{ }   



8760 



Interruption frequency f = 8760_s [Interruptions/year]

n i i=1 1 1 r



Interruption duration r =_s [hours/interruption]

n i i i=1 r       



₈₇₆₀

Annual downtime U = 8760_s [hours/year]

Us Unavailability q = _{s 8760}

Distribution System Reliability Assessment

System Oriented Reliability Indices, Number of Interruptions

• Weighting by number of customers

– System Average

Interruption Frequency Index :

f_i = number of interruptions at load point i

N_i = number of customers connected to load point i

n = number of load points interrupted

Distribution System Reliability Assessment

System Oriented Reliability Indices, Annual Interruption Time

• Weighting by number of customers

– System Average

Interruption Duration Index :

U_i = f_ir_i = annual outage time for load point i

Distribution System Reliability Assessment

System Oriented Reliability Indices, Average Interruption Duration

• Weighting by number of customers

– Customer Average Interruption Duration Index :

SAIFI



CAIDI = SAIDI

_{tot tot tot tot}

n n n n

i i i i i i i i i

i=1 i=1 i=1 i=1

n n n n

i i i i i

i=1 i=1 i=1 i=1

f N U N U N f r N = = N f N N N 

















Distribution System Reliability Assessment

System Oriented Reliability Indices, Unavailability, Energy Not Supplied • Average Service Unavailability Index

• Energy Not Supplied

• COST

of providing quality and continuity of service

< should be related to the >

• WORTH

or

BENEFIT

of having that quality and continuity

Cost-Benefit Considerations

Due to the complex and integrated nature of a power system, failures in any part of the system can cause interruptions which range from inconveniencing a small number of local residents to a major and widespread catastrophic disruption of supply.

The economic impact of these outages is not necessarily restricted to loss of revenue by the utility or loss of energy utilization by the customer but, in order to estimate the true costs, should also include indirect costs imposed on customers, society, and the environment due to the outage.

For instance, in the case of the 1977 New York blackout, 84% of the total costs of the blackout were attributed to indirect costs. In order to reduce the frequency and duration of these events, it is necessary to invest either in the design phase, the operating phase, or both. A whole series of questions come to mind:

• is it worth spending any money?

• how much should be spent?

• should the reliability be increased, maintained at existing levels, or

allowed to degrade?

• who should decide - the utility, a regulator, the customer?

• on what basis should the decision be made?

The underlying trend in all these questions is the need to determine the worth of reliability in a power system, who should contribute to this worth, and who should decide the levels of reliability and investment required to achieve them.

The basic questions that therefore need to be answered are “Is it worth it?” and “Where or on what should the next dollar be invested in the system to achieve the maximum reliability benefit?”.

The first step in answering the above questions is illustrated in the figure below, which shows how the reliability of a product/system is related to the investment cost, i.e., increased investment is required in order to improve reliability. This clearly shows the general trend that the incremental cost  C to achieve a given increase in reliability R increases as the reliability level increases. Alternatively, a given increase in investment produces a decreasing increment in reliability as the reliability is increased. In either case, high reliability is expensive to achieve.

Incremental cost of reliability

The incremental cost of reliability, C/R, is one way of deciding whether an investment in the system is worth it. However, it does not adequately reflect the benefits seen by the utility, the customer, or society in general. The two aspects of reliability and economics can be appraised more consistently by comparing reliability cost (investment cost needed to achieve a certain level of reliability) with reliability worth (benefit derived by the customer and society).

The basic concept of reliability cost/reliability worth evaluation is relatively simple and can be presented by the curves of the figure shown below. These curves show that the investment cost generally increases with higher reliability. On the other hand, the customer costs associated with failures decrease as the reliability increases.

Utility and customer costs

The total costs are the sum of these two individual costs. This total cost exhibits a minimum, and so an “optimum” or target level of reliability is achieved. Two difficulties usually arise in the total cost assessment.

Firstly, the calculated indices are usually derived only from approximate models. Secondly, there are significant problems in assessing customer perceptions of system failure costs.

The disparity between the calculated indices and the monetary costs associated with supply interruptions is shown in the figure. The left hand side of the figure shows the calculated indices at the various hierarchical levels. The right hand side indicates the interruption cost data obtained by user studies.

It can be seen that the relative disparity between the calculated indices at the three hierarchical levels and the data available for worth assessment decreases as the consumer load points are approached.

There have been many studies concerning interruption and outage costs. These studies show that, although trends are similar in virtually all cases,

the costs vary over a wide range and depend on the country of origin

and the type of customer. It is apparent therefore that considerable

research still needs to be conducted on the subject of interruption costs.

Cost-Benefit Considerations

Broadly speaking, the cost of a power interruption from the customer's perspective is dependent both on the customer and interruption characteristics. Customer characteristics include type of customer, nature of his/her activities/demand requirements. Outage costs will therefore vary substantially between customers within a class, and between classes of customers. Interruption characteristics include the parameters of frequency, duration and magnitude of outage, time of occurrence, time of year, whether partial outage or complete, etc.

The most fundamental and methodological approach that has been used

to assess direct, short-term customer outage costs is the customer survey

method. This approach appears to find favors with electric utilities for estimating the outage costs to be used for planning purposes. It is based on the premise that the customer is in the best position to assess his/her monetary losses associated with power failures. The surveys ask the monetary losses that would be sustained by them under certain specified scenarios of interruptions, and also their willingness to pay in order to avoid having those interruptions. A very important consideration in determining the interruption cost through surveys is the choice of the valuation method. Three types of approaches have been undertaken in this regard.

1. The first and the most obvious approach is a direct solicitation of the outage costs for given outage conditions. The approach provides reasonable and consistent results in situations where losses can be directly identified.

2. The second approach seeks the customers' opinions on what they would be willing to pay to avoid having the interruption(s), or conversely what amount they would be willing to accept for having to experience the outage (pay and willingness-to-accept theories). This is based on the theory that incremental willingness to pay (accept) gives the corresponding marginal increments (decrements) in service reliability.

3. The third and final approach is that of indirect worth evaluation, where customers' responses to indirect questions are used to derive a monetary figure. This approach includes the respondents' selection of interruptible/curtailable options, their predictions of what preparatory actions they might take in the event of recurring interruptions, their ranking of a set of reliability/rate alternatives and selecting an option most suitable for their needs, etc. This approach has been used in major Canadian surveys, and has also been used by many utilities and governmental agencies to estimate the costs of interruptions.

Utilization of the gathered interruption cost estimates in a practical planning context could involve converting the gathered data into a functional representation or cost model. The traditional cost model is known as a

composite customer damage function (CCDF), which defines the overall

average costs of interruptions as a function of the interruption duration in a given service area that was used in the surveys. Since the customers are asked to provide their best estimates of monetary losses for selected outage scenarios, the interruption cost data collected using the survey method are duration specific. These data can be used to create customer damage functions (CDFs) for specific customer classes (sectors). The average sector costs associated with each studied interruption scenario are used to create sector customer damage functions (SCDFs) which are then (usually) weighted using their respective energy consumptions to create a CCDF for the entire studied area.

• Probabilistic, as opposed to deterministic, indices are more popular in reliability evaluation of electric power systems.

• Fundamental reliability indices are those of probability, frequency and duration of failures, regardless of whether the system study is at HLI, HLII or HLIII system levels.

• There should be some conformity between the reliability of various parts of the power system. It is pointless to reinforce quite arbitrarily a strong part of the system where weak areas still exist. Consequently, a balance is required between generation, transmission and distribution - this does not mean that the reliability of each should be equal. The reliability of different zones will, in general, be different since HLII failures can cause widespread outages whereas distribution failures are very localized.

• There should be some benefit gained by an improvement in reliability, i.e., the incremental or marginal investment cost should be related to the customer‟s incremental or marginal valuation of the improved reliability. Reliability cost Vs reliability worth (benefit) evaluation can enable utilities to make objective decisions about investments and maintenance for enhancing supply reliability.

• Probabilistic methods are as important and critical today as they were a few decades ago, particularly in the light of increased pressure for economic justifications and the need to manage our assets effectively, efficiently and reliably.

2. Generation System Reliability Assessment

The determination of the required amount of system generating capacity to ensure an adequate supply is an important aspect of power system planning and operation.

The total problem can be divided into two conceptually different areas designated as static and operating capacity requirements.

The static capacity area relates to the long-term evaluation of this overall system requirement.

The operating capacity area relates to the short-term evaluation of the actual capacity required to meet a given load level.

Generation System Reliability Assessment

The Static requirement can be considered as the installed capacity that

must be planned and constructed in advance of the system requirements.

The static reserve must be sufficient to provide for

 the overhaul of generating equipment,

 outages that are not planned or scheduled, and

Generation System Reliability Assessment

Generating capacity reliability is defined in terms of the adequacy of the installed generating capacity to meet the system load demand. Outages of generating units and/or load in excess of the estimates could result in “loss of load”, i.e., the available capacity (installed capacity - capacity on outage) being inadequate to supply the load. In general, this condition requires emergency assistance from neighboring systems and emergency operating measures such as system voltage reduction and voluntary load curtailment. Depending on the shortage of the available capacity, load shedding may be initiated as the final measure after the emergency actions. The conventional definition of “loss of load” includes all events resulting in negative capacity margin or the available capacity being less than the load.

Generation System Reliability Assessment

The basic methodology for evaluating generating system reliability is to develop probability models for capacity on outage and for load demand, and calculate the probability of loss of load by a convolution of the two models. This calculation can be repeated for all the periods (e.g., weeks) in a year considering the changes in the load demand, planned outages of units, and any unit additions or retirements, etc. The methods available to compute generating system reliability indices are described in this module.

Generating unit parameters

Generating unit means all equipment up to the high voltage terminals

of the generator transformer and the station service transformers.

Forced outage means the occurrence of a component failure or other

condition which requires that the generating unit be removed from service immediately or up to and including the very next weekend.

The basic generating unit parameter used in static capacity evaluation is the probability of finding the unit on forced outage at some distant time in the future. This probability is defined as the unit unavailability, and historically in power system applications is known as the unit forced outage rate (FOR). It is strictly speaking not a rate in modern reliability terms, since it is a ratio of two time values.

Generating unit Modeling

Unit availability is defined as the probability of finding the unit in the operating (up) state at any future time.

 expected failure rate (f/yr)  expected repair rate (rep/yr)

m mean time to failure = MTTF = 1/ r mean time to repair = MTTR = 1/

m + r mean time between failures = MTBF = 1/f f cycle frequency = 1/T

T cycle time = 1/f

The concepts of availability and unavailability as illustrated in the above equations are associated with the simple 2-state model shown in Figure 2.1.

This model is directly applicable to a base load generating unit which is either operating or forced out of service. Scheduled outages must be considered separately.

Up Down

 

Figure 2.1 Two state model of unit

In practice, however, generating units are complex pieces of machinery and in addition to complete failures experience partial failures where they continue to operate but at reduced capacity levels. A 3- or even multi-state model is therefore required to more accurately represent the generating capacity model, as shown in Figure 2.2.

Figure 2.2 State space diagram of component with partial output state

Generation System Reliability Assessment

Full output 0 Partial output 1 Failed 2

Capacity Outage Probability Table

The purpose of the capacity model is to recognize the probabilistic nature of available generation capacity. The analytical generation model is generally in the form of discrete levels of capacity available (or unavailable) and their respective probabilities. This type of model is sufficient to calculate reliability indices in terms of probability, expected days (hours) of loss of load, and expected unserved (unsupplied) energy. There are many ways of creating and manipulating this capacity outage probability table (COPT), but the essential objectives and outcomes are the same.

As the name suggests, the COPT is a simple array of capacity levels together with their probabilities. The basic assumptions for the capacity model are as follows:

Generating capacity out of service due to forced outages is multinomially distributed with outage probabilities as parameters. The total number of available (or unavailable) capacity states in an N-unit system is 2N_{. For example, a 3-unit system (each unit can exist in 2}

states) will have 23 = 8 states of available capacity (see Figure 2.3).

1. Each generating unit exists in one of two states, operating (up) or non-operating (down). Multi-state models can also be used to represent units having derated states.

2. The failure performance of a unit is independent of the operating level, the system load and the outage pattern of other units, etc.

Capacity Outage Probability Table

 A up, B up, C up

 A up, B up, C down

 A up, B down, C up

 A down, B up, C up

 A down, B down, C up

 A down, B up, C down

 A up, B down, C down

Capacity Outage Probability Table

1U 2U 3U 1 3 2 4 6 5 7 8 1D 2U 3U 1U 2D 3U 1U 2U 3D 1D 2D 3U 1U 2D 3D 1D 2U 3D 1D 2D 3D

Capacity Outage Probability Table

The basic statistic used in developing the capacity model is the probability of a generating unit being on forced outage, i.e. the forced outage rate. If all the units in the system are identical, the COPT can be easily obtained using the Binomial distribution.

where

A = unit availability U = unit unavailability

n = number of identical units

r = number of units in the failed state, and P_r = probability of r units in the down state.

Capacity Outage Probability Table

Let there be 4 identical generating units, 25 MW, 1% FOR each.

Then

Similarly, the probability of exactly one unit being in the failed state

Thus,

P₀ = probability that zero units are in the failed state (i.e., all the 4 are up)

Example 2.1: = 4_C 0.(0.01)0.(0.99)4-0 = 0.994 = 0.960596. P₁ = 4_C 1.(0.01)1.(0.99)4-1 = 0.0388119 (A + U)4 = A4 + 4A3U + 6A2U2 + 4AU3 + U4 = P₀ +P₁ + P₂ + P₃ + P₄ = 1.000000

Capacity Outage Probability Table

The COPT can also be developed using cumulative probability, i.e., the

probability of finding a quantity of capacity on outage equal to or greater than the indicated amount.

e.g., cumulative probability of  0 MW capacity on outage is unity.

The cumulative probability values decrease as the capacity on outage increases. Although this is not completely true for the individual COPT, the same general trend is followed.

STATE (i) Capacity IN (MW) Capacity OUT (MW) State Probability (pi) Cumulative Probability (Pi) No. of UNITS DOWN 1 100 0 0.960596 1.000 NONE 2 75 25 0.0388119 0.0394038 1 3 50 50 0.000588 0.0005919 2 4 25 75 0.0000039 0.00000391 3 5 0 100 1x10-8 _1x10-8 _ALL Ʃ = 1.000

Capacity Outage Probability Table

Theoretically, the COPT incorporates all the system capacity. The table can, however, be truncated by omitting all capacity outages for which the cumulative probability is less than a specified value, e.g., 10-8. This also results in a considerable saving in computer time as the table is truncated progressively with each unit addition.

Alternatively, table rounding approach can be used by rounding the table to discrete levels after combining. The capacity rounding increment used depends upon the accuracy desired. The final rounded table contains capacity outage magnitudes that are multiples of the rounding increment. The number of capacity levels decreases as the rounding increment increases, with a corresponding decrease in accuracy.

Capacity Outage Probability Table

It is extremely unlikely, however, that all the units in a practical system will be identical, and therefore the Binomial distribution has limited application. The units can be combined, however, using basic probability concepts and this approach can be extended to a simple but powerful recursive technique in which the units are added sequentially to produce the final model.

Example 2.2: Consider a 4-generating unit system with an installed

capacity of 100 MW consisting of one 40 MW, 4% FOR unit and three 20 MW, 4% FOR units. Obtain the capacity model in the form of a COPT.

Solution: First combine the three 20 MW units (identical) using binomial concepts (Table A). Then obtain the COPT for the 40 MW unit (Table B).

Capacity Outage Probability Table

i Capacity IN (MW) Capacity OUT (MW) Probability (pi) 1 60 0 3C₀ x 0.963 _{x 0.04}0 _{= 0.884736} 2 40 20 3C₁ x 0.962 _{x 0.04}1 _{= 0.110592} 3 20 40 3C₂ x 0.961 _{x 0.04}2 _{= 0.004608} 4 0 60 3C₃ x 0.960 _{x 0.04}3 _{= 0.000064} State Capacity IN Capacity OUT Probability 1 40 0 0.96 2 0 40 0.04 Ʃ = 1.000 Table A Table B

Capacity Outage Probability Table

Now we need to combine Tables A and B, i.e., each row of Table A must be combined with each row of Table B separately, and all identical states combined together.

Rule used: If 2 events are independent, the probability of occurrence of one is not affected by the probability of occurrence of the other, i.e.,

P(A  B) = P(A). P(B)

e.g., combining state 1 of Table A with state 2 of Table B capacity in = 60 + 0 = 60 MW

capacity out = 0 + 40 = 40 MW

state probability = 0.884736 x 0.04 = 0.0353894

We thus get a total of 4 states (Table A) times 2 states (Table B) = 8 states, which can further be reduced by combining the ones with identical capacity available (or unavailable).

Capacity Outage Probability Table

STATE Capacity IN Capacity OUT How obtained? Probability a 100 0 1(A), 1(B) 0.8493465 b 60 40 1(A), 2(B) 0.0353894 c 80 20 2(A), 1(B) 0.1061683 d 40 60 2(A), 2(B) 0.0044236 e 60 40 3(A), 1(B) 0.0044236 f 20 80 3(A), 2(B) 0.0001843 g 40 60 4(A), 1(B) 0.0000614 h 0 100 4(A), 2(B) 0.0000025 Ʃ = 1.00 STATE Capacity IN Capacity OUT How obtained? Probability (state) Cumulative Prob. (i) 100 0 a 0.8493465 1.00 (ii) 80 20 c 0.1061683 0.1506534 (iii) 60 40 b + e 0.039813 0.0444851 (iv) 40 60 d + g 0.004485 0.004672 (v) 20 80 f 0.0001843 0.0001868 (vi) 0 100 h 0.0000025 0.0000025 Table C Table D

The capacity model can be created using a simple algorithm which can also be used to remove a unit from the model. This approach can also be used for a multi-state unit, i.e., a unit which can exist in one or more derated (partial output) states as well as in the fully up and fully down states. The algorithm is illustrated first using the 2-state units, and is then extended using the multi-state unit representations.

Case 1: 2-State Unit Representation

The cumulative probability of a particular capacity outage state of X MW, after a unit of capacity C MW and forced outage rate U is added, is given by

P(X) = (1 - U).P‟(X) + (U).P‟(X - C) = (A).P‟(X) + (U).P‟(X - C)

Capacity Outage Probability Table

Capacity Outage Probability Table

P‟(X) = cumulative probability of capacity outage state of X MW before a unit is added

P(X) = cumulative probability of capacity outage state of X MW after a unit is added, and

A and U are the availability and unavailability of the unit being added. The above expression is initialized by setting

P‟(X) = 1.0 for X  0, and P‟(X) = 0.0 otherwise.

Capacity Outage Probability Table

The algorithm is illustrated using the 100 MW system with the following data:

three 20 MW units,  = 0.4 f/yr,  = 9.6 rep/yr each,

and one 40 MW unit with  = 0.4 f/yr and  = 9.6 rep/yr.

Consider also that the 20 MW units are loaded first followed by the 40 MW unit.

Solution:

Given  = 0.4 failures/yr, and  = 9.6 repairs/yr.

Therefore

Capacity Outage Probability Table

Step 1:

Add the first unit. Values of X are 0 and 20 MW. P(0) = (0.96).(1.0) + (0.04).(1.0) = 1.00

P(20) = (0.96).(0.0) + (0.04).(1.0) = 0.04

Step 2:

Add the 2nd unit. Values of X: 0, 20 and 40 MW. P(0) = (0.96).(1.0) + (0.04).(1.0) = 1.00

P(20) = (0.96).(0.04) + (0.04).(1.0) = 0.0784 P(40) = (0.96).(0.0) + (0.04).(0.04) = 0.0016

Capacity Outage Probability Table

Step 3:

Add the 3rd unit. Values of X: 0, 20, 40 and 60 MW. P(0) = (0.96).(1.0) + (0.04).(1.0) = 1.00

P(20) = (0.96).(0.0784) + (0.04).(1.0) = 0.115264

P(40) = (0.96).(0.0016) + (0.04).(0.0784) = 0.004672 P(60) = (0.96).(0.0) + (0.04).(0.0016) = 0.000064

Capacity Outage Probability Table

Step 4:

Add the 4th unit. Values of X: 0, 20, 40, 60, 80 & 100 MW. P(0) = (0.96)(1.0) + (0.04)(1.0) = 1.00 P(20) = (0.96)(0.115264) + (0.04)(1.0) = 0.1506534 P(40) = (0.96)(0.004672) + (0.04)(1.0) = 0.00444851 P(60) = (0.96)(0.000064) + (0.04)(0.115264) = 0.0046718 P(80) = (0.96)(0.0) + (0.04)(0.004672) = 0.0001868 P(100) = (0.96)(0.0) + (0.04)(0.000064) = 0.0000025

It must be emphasized that for step number n, results from step (n-1) only are utilized (other than the initial conditions, if need be) - results prior to step (n-1) are of no consequence.

Capacity Outage Probability Table

Exercise 2.1:

A generating system consists of 3 generating units - two 3 MW, 2% FOR and one 5 MW, 2% FOR. Obtain the system COPT using the recursive algorithm.

Capacity Outage Probability Table

Case 2: Multi-state unit representation

The equation for 2-state representation can be modified to include multi-state unit representations.

P X p_i P X C_i i n ( )  . '(  )   1 where:

n is the no. of states of unit being added

C_i is the capacity outage (MW) of state i for the unit being added, and p_i is the probability of existence of the unit state i.

Note that for n = 2, the above equation reduces to the form for 2-state representation.

Capacity Outage Probability Table

Example 2.3 Revisited:

Consider that in the system of Example 2.3, the 4th unit (40 MW) exists in 3 states as follows: state (i) capacity out C_i (MW) state probability (p_i) 1 2 3 0 20 40 0.95 0.04 0.01 Solution:

Since units 1, 2 and 3 (the 20 MW units) are still 2-state units, the capacity models for steps 1, 2 and 3 will remain the same as before. Step 4, however, will be different due to the unit 4 which is a multi-state unit.

Capacity Outage Probability Table

Step 4: Add unit 4. X = 0, 20, 40, 60, 80, 100 MW. P(0) = (0.95)(1.0) + (0.04)(1.0) + (0.01)(1.0) = 1.0 P(20) = (0.95)(0.115264) + (0.04)(1.0) + (0.01)(1.0) = 0.1595008 P(40) = (0.95)(0.004672) + (0.04)(0.115264) + (0.01)(1.0) = 0.0190489 P(60) = (0.95)(0.000064) + (0.04)(0.004672) + (0.01)(0.115264) = 0.0014003 P(80) = (0.95)(0.0) + (0.04)(0.000064) + (0.01)(0.004672) = 0.0000492 P(100) = (0.95)(0.0) + (0.04)(0.0) + (0.01)(0.000064) = 0.0000006

Capacity Outage Probability Table

Exercise 2.2:

Obtain the COPT using the recursive algorithm for the following system:

Unit A: 10 MW, 8% FOR, 2-state Unit B: 20 MW, 8% FOR, 2-state Unit C: 30 MW, 8% FOR, 2-state

Unit D: 40 MW, exists in 3 states, with full forced outage rate of 8%, and 50% derated capacity state has a probability of 0.06.

Evaluation of Risk of Capacity Shortfall

The resultant model is known as the load duration curve (LDC) when the individual hourly load values are used, and in this case the area under the curve represents the energy required in the given period. This, however, is not the case with the DPLVC.

Typical shapes of a LDC and DPLVC are shown in Figure 2.4 - both indicate the probability that a particular load level will be exceeded.

The generation system model described in the previous chapter can be convolved with an appropriate load model to produce a system risk index.

In order to convolve the capacity model with the load model, we need to identify the various kinds of load models that can be used. The simplest load model is where each day is represented by its daily peak load. The individual daily peak loads can be arranged in descending order to form a cumulative load model known as the daily peak load variation curve

Evaluation of Risk of Capacity Shortfall

Figure 2.4 Typical shapes of a DPLVC and a LDC

time, p.u. LDC DPLVC 1.0 0.8 0.6 0.4 0.2 0 lo a d level

Evaluation of Risk of Capacity Shortfall

In the analytical approach, the applicable system COPT is combined with the system load characteristic to give an expected risk of loss of load. The units are in days if the DPLVC is used, and in hours if the LDC is used. Prior to combining the COPT it should be realized that

there is a difference between the terms “capacity outage” and “loss of

load”.

The term “capacity outage” indicates loss of generation which may or

may not result in a loss of load. This condition depends upon the generating capacity reserve margin and the system load level.

A “loss of load” will occur only when the capability of the generating capacity remaining in service is exceeded by the system load level.

Evaluation of Risk of Capacity Shortfall

The loss of load expectation (LOLE) index, using the daily peak loads, is the expected number of days in the specified period in which the daily peak load will exceed the available capacity.

LOLE = days/period...(2.1)

where:

C_i = available capacity on day i,

L_i = forecast peak load on day i, and

P_i(C_i-L_i) = probability of loss of load on day i.

Evaluation of Risk of Capacity Shortfall

Example 2.4:

Consider the generation system data and the 365-day load data shown below.

Unit no. capacity

(MW) failure rate (f/day) repair rate (rep/day) 1 25 0.01 0.49 2 25 0.01 0.49 3 50 0.01 0.49

Table 2.2: Load data daily peak load

(MW)

57 52 46 41 34

no. of occurrences 12 83 107 116 47

Evaluation of Risk of Capacity Shortfall

The cumulative COPT for the system is shown below.

Table 2.3

state (i) capacity out

(MW) cumulative probability 1 0 1.00000 2 25 0.058808 3 50 0.020392 4 75 0.000792 5 100 0.000008

Evaluation of Risk of Capacity Shortfall

Using Eqn 2.1, in conjunction with the COPT and load data, the LOLE is

obtained as follows: LOLE = 12P(100-57) + 83P(100-52) + 107P(100-46) + 116P(100-41) + 47P(100-34) LOLE = 12(0.020392) + 83(0.020392) + 107(0.0007920) + 116(0.000792) + 47(0.000792) LOLE = 2.15108 days/year

Evaluation of Risk of Capacity Shortfall

The same LOLE index can also be obtained using the DPLVC. Figure 2.5 below shows a typical system load-capacity relationship where the load model is shown as a continuous curve for a period of 365 days.

It can be seen from Figure 2.5 that any capacity outage less than the reserve will not contribute to the system LOLE.

Outages of capacity in excess of the reserve will result in varying numbers of time units during which loss of load would occur. Expressed mathematically, the contribution to the system LOLE made by capacity outage O_k is p_kt_k time units, where p_k is the individual probability of the capacity outage O_k. The total LOLE for the study interval is given by Eqn 2.2.

Evaluation of Risk of Capacity Shortfall

Figure 2.5 Relationship between load, capacity and reserve

Time load exceeds the indicated value

Da ily peak lo a d ( M W) O_k Reserve Installed capacity (MW) 0 t_k 365

Evaluation of Risk of Capacity Shortfall

where P_k is the cumulative outage prob. for capacity state O_k.

If the load characteristic in Figure 2.4 is the LDC, the value of LOLE will be in hours/period; for a DPLVC it will be in days/period.

The period of study could be a week, a month or a year. The simplest application is the use of the curve on a yearly basis. If no generating unit maintenance were performed, the COPT would be valid for the entire period.

Alternatively, the system LOLE can be obtained using the cumulative probability values from the COPT, as given by Eqn 2.3.

LOLE = time units...(2.2)

Evaluation of Risk of Capacity Shortfall

Example 2.5:

The application of Eqns 2.2 and 2.3 can be illustrated by a simple numerical example.

Consider a system containing five 40 MW units each with a FOR of 1%. The COPT of this system is shown in Table 2.4.

In the above COPT, binomial distribution concepts have been used and probability values below 10-6 have been neglected.

Table 2.4 Capacity out of service (MW) Individual probability Cumulative probability 0 0.950991 1.000000 40 0.048029 0.049009 80 0.000971 0.000980 120 0.000009 0.000009 1.000000

Evaluation of Risk of Capacity Shortfall

Let the system load model be represented by the annual DPLVC shown in Figure 2.6 - the curve is assumed to be linear for simplicity. In actual practice, however, the curve will be non-linear but the concept is still applicable.

The 100% point on the abscissa corresponds to 365 days, while that on the ordinate corresponds to the system forecast peak (160 MW in this case).

Figure 2.6 DPLVC for the Example System

100 40 100 0 D a il y pea k l o a d (% )

Percentage of days the daily peak load exceeded the indicated value

Evaluation of Risk of Capacity Shortfall

The LOLE can be evaluated using Eqn 2.2 (individual probabilities) or Eqn 2.3 (cumulative probabilities) - both the methods are illustrated here. The time periods t_k are calculated using the equation of the straight line DPLVC, and are shown in Figure 2.7. The LOLE calculations using Eqn 2.2 are shown in Table 2.5.

Figure 2.7 Time periods during which loss of load occurs

D a il y pea k l o a d (M W) 160 120 80 64 0 Installed capacity = 200 MW O₂ = 40MW O₃ = 80MW t₃ = T₃ = 41.7% T₄ = 41.7% t₄ = 83.4% Time (%) O₄ = 120MW 64 100

Evaluation of Risk of Capacity Shortfall

Table 2.5 LOLE using individual probabilities

capacity out of service (MW) capacity in service (MW) individual probability (p_k) total time (t_k) (%) LOLE (p_kt_k) (%) 0 200 0.950991 0 0 40 160 0.048029 0 0 80 120 0.000971 41.7 0.040491 120 80 0.000009 83.4 0.000751 1.00000 0.04124

The LOLE is 0.0412413% of the time base units, i.e., LOLE = 0.0412413(365/100) = 0.150410 days/yr

If the cumulative probability values are used, the time quantities used are the interval or increases in curtailed time represented by T_k in Figure 2.7. The calculations are shown in Table 2.6.

Evaluation of Risk of Capacity Shortfall

Table 2.6 LOLE using cumulative probabilities

System LOLE = 0.0412414% (365/100) = 0.15041 d/yr capacity out of service (MW) capacity in service (MW) cumulative prob. (P_k) time interval T_k (%) LOLE (P_kT_k) (%) 0 200 1.000000 0 0 40 160 0.049009 0 0 80 120 0.000980 41.7 0.040866 120 80 0.000009 41.7 0.000375 0.04124

Evaluation of Risk of Capacity Shortfall

If, however, it is assumed that instead of the load pattern being given by the DPLVC or the LDC, it is constant throughout the time period considered, then the system LOLE can be directly obtained as the

cumulative probability of the first negative margin, i.e., of the first load loss state.

For instance, if the peak load in the example system described above is assumed to be constant at 160 MW for the year, then

LOLE = Cum. Prob. of capacity out > 40 MW = (0.000980) (time period)

Evaluation of Risk of Capacity Shortfall

Note that this value is significantly higher than the 0.15041 days/yr obtained using the DPLVC - hence it provides a pessimistic appraisal of the system performance. While the load in general is not expected to remain constant over a given time period, for planning purposes a single load level may be adequate in order to evaluate alternative capacity reinforcement/expansion proposals.

Evaluation of Risk of Capacity Shortfall

Exercise 2.3

A generating system contains 120 MW capacity in six 20 MW units which are connected through step-up station transformers to a high voltage load bus, as shown in Figure 2.8. The generators have =3 failures/yr, and =97 repairs/yr each, whereas the transformers have

=0.1 failures/yr and =19.9 repairs/yr each.

For an annual forecast peak load of 95 MW, evaluate

 the system LOLE at the load bus, given that the annual DPLVC is a straight line from the 100% to the 70% points.

Evaluation of Risk of Capacity Shortfall

Figure 2.8 Single line diagram of Exercise 2.3

LOAD Generating bus

Comparison of Deterministic & Probabilistic Criteria

It was stated earlier that deterministic risk criteria such as “percentage reserve” and “loss of largest unit” do not define consistently the true risk in the system, whereas probabilistic criteria consider the actual influencing factors that govern system behaviour, unlike the deterministic ones. An example will help to illustrate this aspect.

 system 1, twenty-four 10 MW, 1% FOR units

 system 2, twelve 20 MW, 1% FOR units

 system 3, twelve 20 MW, 3% FOR units

 system 4, twenty-two 10 MW, 1% FOR units

Example 2.6:

Comparison of Deterministic & Probabilistic Criteria

All 4 systems are very similar but not identical. In each system, the units are identical and therefore the COPT can be constructed using the binomial distribution. The results are shown below in Tables 2.7 to 2.10 (arrays truncated to a cumulative probability of 10-6_).

Table 2.7 Capacity Outage Probability Table for System 1

System 1 Capacity (MW) Probability

Out In Individual Cumulative

0 240 0.785678 1.000000 10 230 0.190467 0.214322 20 220 0.022125 0.023855 30 210 0.001639 0.001730 40 200 0.000087 0.000091 50 190 0.000004 0.000004

Comparison of Deterministic & Probabilistic Criteria

Table 2.8 Capacity Outage Probability Table for System 2

System 2 Capacity (MW) Probability

Out In Individual Cumulative

0 240 0.886384 1.000000 20 220 0.107441 0.113616 40 200 0.005969 0.006175 60 180 0.000201 0.000206 80 160 0.000005 0.000005

Table 2.9 Capacity Outage Probability Table for System 3

System 3 Capacity (MW) Probability

Out In Individual Cumulative

0 240 0.693841 1.000000 20 220 0.257509 0.306159 40 200 0.043803 0.048650 60 180 0.004516 0.004847 80 160 0.000314 0.000331 100 140 0.000016 0.000017 120 120 0.000001 0.000001

Comparison of Deterministic & Probabilistic Criteria

The load level or demand on the system is assumed to be constant.

If the risk in the system is defined as the probability of not meeting the load, then the true risk in the system is given by the value of cumulative probability corresponding to the outage state one level below that which satisfies the load on the system.

Table 2.10 Capacity Outage Probability Table for System 4

System 4 Capacity (MW) Probability

Out In Individual Cumulative

0 220 0.801631 1.000000 10 210 0.178140 0.198369 20 200 0.018894 0.020229 30 190 0.001272 0.001335 40 180 0.000061 0.000063 50 170 0.000002 0.000002

Comparison of Deterministic & Probabilistic Criteria

The two deterministic criteria can now be compared with this probabilistic risk.

(a) Percentage Reserve Margin

Assume that the expected load demands in systems 1, 2, 3 and 4 are 200, 200, 200 and 183 MW respectively. The installed capacity in each system is such that there is a 20% reserve margin, i.e., a constant for all 4 systems. The probabilistic (true) risks in the 4 systems are (from the above four tables):

risk in system 1 = 0.000004

risk in system 2 = 0.000206

risk in system 3 = 0.004847

Comparison of Deterministic & Probabilistic Criteria

It can be seen that the true risk in system 3 is 1000 times greater than that in system 1. A detailed analysis of the 4 systems will show that the variation in true risk depends on forced outage rates, number of units and load demand.

The percentage reserve method cannot account for these factors and therefore, although using a “constant” risk criterion, does not provide consistent risk assessment of the system.

(b) Largest Unit Reserve

Assume now that the expected load demands in systems 1, 2, 3 and 4 are 230, 220, 220 and 210 MW respectively. The installed capacity in all four cases is such that the reserve is equal to the largest unit which again is a constant for all four systems. In this case the true (probabilistic) risks in the systems are:

Comparison of Deterministic & Probabilistic Criteria

risk in system 1 = 0.023855

risk in system 2 = 0.006175

risk in system 3 = 0.048650

risk in system 4 = 0.020229

The variation in risk is much smaller in this case, which gives some credence to the criterion. The risk merit order has changed from 3 -2 - 4 - 1 (percentage reserve criterion) to 3 - 1 - 4 - 2 (largest unit reserve criterion).

It can clearly be concluded from the above comparisons that the use of deterministic or “rule-of-thumb” criteria can lead to very divergent probabilistic risks even for systems that are very similar. The deterministic criteria are therefore inconsistent, unreliable and subjective methods for reserve margin planning.