2 Probabilistic power system reliability analysis1

2.1 Introduction

The reliability of power systems has been studied for many years. Throughout the years, many different methods, concepts and techniques have emerged. This chapter gives an overview of the current state of probabilistic reliability analysis.

This chapter begins with a description of the basic concepts and methods of probabilistic reliability analysis in sections 2.2 and 2.3. Then, section 2.4 describes the different applications of probabilistic reliability analysis.

2.2 Basic concepts

For the reliability analysis of the power system, both deterministic and probabilistic methods can be used. In deterministic methods, criteria are given for the occurrence of certain events. For example, the n-1 criterion states that in system operation, a single failure must not develop into a significant failure of the system. Such a criterion does not say anything about the probability of occurrence of the individual events. In probabilistic reliability analysis, these probabilities are taken into account. In this way, situations with low probability are no longer overestimated and complex situations that were not considered in the deterministic methods can be revealed.

The concept of system reliability is a general concept that is used to indicate the

ability of a system to fulfil its function. In probabilistic power system reliability, reliability is divided into two main fields as shown in figure 2.1 [1]. The system security refers to the ability of the system to respond to disturbances arising within that system. System adequacy is the existence of sufficient facilities within the system to fulfil its function. System security is therefore more related to system dynamics while system adequacy is more related to the steady state of the system.

Figure 2.1: Subdivision of system reliability [2].

As the power system is a complex system, it is divided into several hierarchical

levels for reliability studies, illustrated in figure 2.2. The first level consists of the generation system, the second level is the transmission network and the third level is the distribution system. Most studies are mainly devoted to one hierarchical level,

1 B.W.Tuinema (PhD-researcher: Power System Reliability, b.w.tuinema@tudelft.nl)

while assumptions are made for the other hierarchical levels. In some research, an additional hierarchical level, i.e. HL0, the availability of energy resources, is defined.

Figure 2.2: Hierarchical levels [2].

2.3 Methods for probabilistic reliability analysis

Different methods have been developed for the reliability analysis of power systems. Some of these methods resulted from general reliability analysis or mathematics, while other methods have been developed with the application for power systems in mind. This section describes some of the reliability methods and indices that are used for probabilistic reliability analysis of power systems. A more detailed description of the different methods is given in power system reliability literature like [1] and [3].

2.3.1 Methods for components and small parts of the system

A basic reliability model is the stress-strength model as shown in figure 2.3. It shows the probabilistic distribution of the strength of a component and the stress on that component. The area where the stress on the component is higher than the strength of the component is the area where failures occur. In the power system, this model can represent the reliability of the generation system. The available capacity of the generation system has a certain probabilistic distribution (system strength), while the load has another distribution (system stress). When the load is higher than the available generation capacity, system failures occur. The probability of a system failure can be calculated with a mathematical convolution.

800 1000 1200 1400 1600 18000

1

2

3

4

5

x 10-3 Stress-Strength Model

Stress Strength

Failures

Figure 2.3: Stress-Strength model.

A second model that is often used in reliability analysis is the Markov model. In its

simplest form, the Markov model consists of an UP and a DOWN state, like in figure 2.4. The Markov model gives information about the probability of the system states (PUP and PDOWN), de failure rate (λ) and the repair rate (µ). Mostly, this simple Markov model is used to represent the failure behaviour of single components within the system.

Figure 2.4: Simple two-state model [1].

The Markov model can be expanded with additional states like derated or

maintenance states. The Markov models of different components can be combined into a Markov model for a small part of the system. Figure 2.5 gives an example of a Markov model for a system that consists of two components and one additional spare. As Markov models become complex for larger systems, Markov models are mainly used for single components or a small group of system components.

Figure 2.5: More advanced state model.

The representation of components in the two-state Markov model results in a certain availability (A=PUP) and unavailability (U=PDOWN). A network of components then gives a reliability network in which the components are described by their availability/unavailability, like in figure 2.6. With simple mathematics, the reliability of the complete network can be calculated. This method is especially suitable for radial networks, but can also effectively be used for other parts of the power system. Figure 2.7 gives an example of a radial distribution network.

Figure 2.6: Series/parallel connection.

Figure 2.7: Radial distribution network.

System outages can be the result of various causes. In some systems, these various causes can not be represented by simple reliability models. In these cases, an investigation of the different failure modes that lead to a system failure has to be performed. The result can then be presented as a fault tree, like done in figure 2.9 for the typical substation configuration shown in figure 2.8. This method can effectively be used for the study of substations, where the different failure modes of the breakers and switches have to be analysed.

Figure 2.8: Typical substation configuration.

Figure 2.9: Fault tree analysis.

2.3.2 Methods for larger systems

For larger systems, the combination of the states of all the different components would lead to an extremely complicated model with a huge amount of system states. Therefore, special reliability methods exist for larger systems. In these methods, a selection is made of the possible system states that is assumed to be representative for the overall reliability behaviour. The two main approaches for this are state enumeration and Monte Carlo simulation.

In state enumeration, the states are defined according to their order of failure. This process is illustrated in figure 2.10. The lower-order states, with less component failures and a higher probability, are selected first. Then the higher-order states, with more component failures and a lower probability, are selected. It is assumed that the lowest-order states, with the highest probability, are representative for the reliability of the system. State enumeration is mainly used in systems with low failure

probabilities and when no additional information like probability distributions or switching sequence information is required.

Figure 2.10: State enumeration.

In a Monte Carlo simulation, the reliability of the system is determined by performing a simulation of the system. In this simulation, different random system states are defined. It is assumed that these random states are a sufficient representation for the reliability of the system. In order to be representative, the simulation has to simulate many system years. Monte Carlo simulation is mainly used in systems where more complicated failure combinations contribute to the system reliability and when additional information like probability distributions or switching sequence information is required.

Figure 2.11: Monte-Carlo Simulation.

In both methods, the defined states can be studied by performing a load flow, executing remedial actions and determining the load curtailment. From this, different reliability indices and the reliability of the complete system can be calculated.

The methods that were described in this section can now be summarised by table 2.1.

Table 2.1: Methods for probabilistic reliability analysis

Components and small systems Large systems Stress-strength model − representation of the failure behaviour

of components Markov models − model of the different operating states − used for components and simple

combinations of components Series/parallel networks − simple mathematic reliability

calculation of series/parallel connections

− especially suitable for radial (distribution) networks

Fault tree analysis − chart of the different causes that can

lead to a contingency (partial or full failure in the system)

− especially useful for substations, where different breaker and switch malfunctioning can lead to system failures

Stress-strength model − useful for the reliability of the

generation/load system State enumeration − system states are defined in order

from lower failure order, with high probability, to higher order, with lower probability

− used for systems with low failure probabilities and when no additional information like probability distributions or switching sequence information is required

Monte Carlo simulation − system states are defined by a

computer simulation − used for systems where more

complicated failure combinations contribute to the system reliability

2.3.3 Reliability indices

With the described methods, a variety of reliability indices can be calculated. For different parts of the power system, other sets of reliability indices exist. Some reliability indices reflect the reliability of a combined system, while others represent the generation system, the distribution system or the reliability economics.

The most common indices that are used for the reliability of a combined system are:

− LOLP (Loss Of Load Probability) The probability of a system situation in which load curtailments exist.

− LOLE (Loss Of Load Expectation) Expectation of the amount of time with load curtailments in a period.

− LOEE (Loss Of Energy Expectation) Expectation of the amount of energy that cannot be supplied to the load.

− EENS (Expected Energy Not Supplied) Expectation of the amount of energy that cannot be supplied to the load.

Indices that are often used for the generation system are:

− LOCP (Loss Of Capacity Probability) Probability of loosing a certain generation capacity.

− ESWE (Expected Surplus Wind Energy) Expected amount of wind energy that cannot be absorbed by the system.

− ELCC (Effective Load Carrying Capability) The amount of load that can be served by a (variable) energy source, while keeping the reliability of the system at the same level.

The generation system is often represented by the COPT (Capacity Outage Probability Table) in which the different capacity levels of the generation system and the corresponding probabilities are given.

Some indices that are used for the distribution system are:

− SAIFI (System Average Interruption Frequency Index) The amount of customer interruptions per customers served.

− CAIDI (Customer Average Interruption Duration Index) The average duration of a customer interruption.

− ASAI (Average Service Availability Index) Ratio of the available service duration and the demanded service duration.

For reliability economics, it is important to define the total costs and benefits of an

investment. Typically, the investment costs, the operation costs and the risk costs/benefits are defined. For the risk benefits, the costs of interrupted energy must be defined. A distinction can be made between the interruption of a generating facility, an industrial and a domestic customer. Indirect costs and benefits are often not considered.

Another reliability index that describes the reliability of a combined system is the

risk indicator. The risk is defined as the product of probability and severity. As the significance of an event depends on both its probability of occurrence and its consequences, this is a useful index.

2.4 Applications of probabilistic power system reliability analysis

Probabilistic reliability analysis can be used in different fields of activity in planning and operation of the power system, as shown in figure 2.12. The main fields are asset management, operational planning and reliability economics. These applications are described in more detail in this section.

Figure 2.12: Applications of probabilistic power system reliability analysis.

2.4.1 Asset management

In asset management, three issues are important, namely expansion, maintenance and adaptation of the power system. Based on the current situation and the expected growth and development of the power system, asset managers have to make decisions about which parts of the power system need to be expanded, maintained or adapted.

Not all system expansions do necessarily lead to an improvement in the system reliability. It is possible that due to economic reasons, the power flows in the system are redistributed after the system expansion and the effective system reliability remains the same as before the system expansion [4].

Different development objectives can lead to different solutions. For example, deterministic, security and probabilistic targets can all lead to different system expansions [5]. For the deterministic target, the system must be able to supply the maximum load level and for the security target, the system must be n-α secure. The probabilistic target states that the system must have a certain reliability level measured in the reliability index LOLE (Loss Of Load Expectation).

As deterministic rules are strongly integrated into the system operation practice and are still frequently used for network planning, it is often suggested that probabilistic approaches for the planning and operation are an extension of these deterministic rules [3]. Probabilistic system reliability is then regarded as a next selection criterion after the deterministic n-1 rule. Economical optimization then, is the following criterion.

Another issue in asset management is maintenance [6]. Maintenance of the components of the power system can influence the reliability of the power system in two different ways. First, the maintenance of components will increase the reliability of the components and eventually also the reliability of the complete system. Second, during maintenance, different components are taken out of the system and the reliability of the system will decrease. This latter is more a topic of operational planning. In most studies on maintenance, the effects of different maintenance policies on the reliability are compared and the maintenance costs are taken into account. Often, components are ranked on their importance for the system reliability or the expected economic benefits.

2.4.2 Operational planning

Operational planning deals with the scheduling of the generation and the transport of electricity in the available power system. After the liberalisation of the electricity market, the generation and the transmission of electrical energy are uncoupled. This has led to a situation in which the TSO (Transmission System Operator) is responsible for the overall reliability of the power system while the producers are only responsible for the generation of electricity [7]. To optimise their profit, generating companies plan the generation as economically as possible.

As a deterministic rule to secure the reliability of the power system during operation, a certain redundancy (operational reserve margin) in the generation is required. The level of the reserve margin is often discussed and is mostly chosen as the loss of the largest unit. The increasing amount of renewable, fluctuating sources will have an impact on the required reserve margin [8].

An important task of operational planning is maintenance scheduling. During maintenance, different components are taken out of the system. Consequently, the reliability of the system will decrease and the system is more operated to its limits. It

is essential to plan the maintenance in such a way that the reliability remains at a high level. Often, maintenance is scheduled by studying deterministic worst-case scenarios. Probabilistic reliability analysis can give more insight into the effects of the maintenance on the system reliability.

Probabilistic reliability analysis can play an important role in the operational planning. The challenge is to develop a method to analyse the reliability of the system in the actual and near-future situation. The results of the real-time analysis can help the system operator to localise critical situations and to find possible solutions for these situations.

2.4.3 Reliability economics

The costs are often the restricting factor for increasing the system reliability. In many situations, an optimum between the costs and the effects on reliability has to be found. The costs of reliability can be considered in individual situations as well as in general.

Knowledge about reliability economics can be useful when making decisions about possible system developments. When reliability analysis and economics are combined in a probabilistic cost/benefit analysis, decisions can be made about what the most optimal measures are.

It is important to study both the direct and indirect costs and benefits in a cost/benefit study. Direct costs and benefits may be for example the project costs and the reduction of not supplied energy. Indirect costs and benefits often consist of social wealth, welfare or loss of comfort. These costs and benefits are more difficult to define. A thorough study on these costs will lead to more optimal decisions.

More in general, one can also think of the real cost/worth of reliability. For what compensation are customers willing to accept some risk and what are customers willing to invest in a more reliable system [9]? In the liberalised market, an important question is who should be responsible for the system reliability [7].

References

[1] R. Billinion and R. N. Allan, Reliability Evaluation of Power Systems, 2nd ed. New York: Plenum Press, 1996.

[2] R. Billinton and R. N. Allan, "Power-system reliability in perspective," Electronics and Power, vol. 30, pp. 231-236, 1984.

[3] W. Li, Risk Assessment of Power Systems - Models, Methods, and Applications. Canada: Wiliey Interscience - IEEE Press, 2005.

[4] D. S. Kirschen, "Do Investments Prevent Blackouts?," in Power Engineering Society General Meeting, 2007. IEEE, 2007, pp. 1-5.

[5] C. Jaeseok, T. Mount, and R. Thomas, "Transmission System Expansion Plans in View Point of Deterministic, Probabilistic and Security Reliability Criteria," in Second International Conference on Innovative Computing, Information and Control, 2007. ICICIC '07, 2007, pp. 380-380.

[6] L. Bertling, "Reliability Centred Maintenance for Electric Power Distribution Systems," Ph.D. thesis, Dept. Elect. Power Engineering, Royal Institute of Technology, Stockholm, Sweden, 2002.

[7] C. Singh, M. Schwan, and W. Wellssow, "Reliability in Liberalized Power Markets - From Analysis to Risk Management - Survey Paper," in 14th Power System Computation Conference PSCC, Sevilla, 2002.

[8] A. M. L. L. da Silva, W. S. Sales, L. A. da Fonseca Manso, and R. Billinton, "Long-Term Probabilistic Evaluation of Operating Reserve Requirements With Renewable Sources," IEEE Transactions on Power Systems, vol. 25, pp. 106-116, 2010.

[9] R. Billinton and R. N. Allan, Reliability Evaluation of Power Systems, 2nd ed. New York: Plenum Press, 1996.