Electric Power Grid Reliability Evaluation
Description
More details
Other editions
Additional editions
Persons
DR. CHANAN SINGH is a Regents Professor and Irma Runyon Chair Professor of Electrical & Computer Engineering at Texas A&M University.
DR. PANIDA JIRUTITIJAROEN is the head of Corporate Strategy and Business Development for Press Quality Co., Ltd., in Bangkok, Thailand. (Formerly a faculty at the National University of Singapore).
DR. JOYDEEP MITRA is a Professor of Electrical Engineering at Michigan State University.
Content
Preface xiii
Acknowledgments xv
Figures xvii
Tables xxi
Part I Concepts and Methods in System Reliability 1
1 Introduction to Reliability 3
1.1 Introduction 3
1.2 Quantitative Reliability 4
1.3 Basic Approaches for Considering Reliability in Decision-Making 6
1.4 Objective and Scope of This Book 8
1.5 Organization of This Book 9
2 Review of Probability Theory 11
2.1 Introduction 11
2.2 State Space and Event 11
2.3 Probability Measure and Related Rules 16
2.4 Random Variables 25
2.5 Jointly Distributed Random Variables 31
2.6 Expectation, Variance, Covariance and Correlation 32
2.7 Moment Generating Function 36
2.8 Functions of Random Variables 39
Exercises 51
3 Review of Stochastic Process 53
3.1 Introduction 53
3.2 Discrete-Time Markov Process 57
3.3 Continuous-Time Markov Process 72
Exercises 80
4 Frequency-Based Approach to Stochastic Process 81
4.1 Introduction 81
4.2 Concept of Transition Rate 82
4.3 Concept of Frequency 83
4.4 Concept of Frequency Balance 91
4.5 Equivalent Transition Rate 100
4.6 Coherence 102
4.7 Conditional Frequency 104
4.8 Time-Specific Frequency 109
4.9 Probability to Frequency Conversion Rules 110
Exercises 115
5 Analytical Methods in Reliability Analysis 117
5.1 Introduction 117
5.2 State Space Approach 117
5.3 Network Reduction Method 139
5.4 Conditional Probability Method 147
5.5 Cut-Set and Tie-Set Methods 152
Exercises 164
6 Monte Carlo Simulation 165
6.1 Introduction 165
6.2 Random Number Generation 166
6.3 Classification of Monte Carlo Simulation Methods 167
6.4 Estimation and Convergence in Sampling 174
6.5 Variance Reduction Techniques 178
Exercises 182
Part II Methods of Power System Reliability Modeling and Analysis 185
7 Introduction to Power System Reliability 187
7.1 Introduction 187
7.2 Scope of Power System Reliability Studies 187
7.3 Power System Reliability Indices 188
7.4 Considerations in Power System Reliability Evaluation 190
8 Generation Adequacy Evaluation Using Discrete Convolution 193
8.1 Introduction 193
8.2 Generation Model 193
8.3 Load Model 205
8.4 Generation Reserve Model 208
8.5 Determination of Reliability Indices 210
8.6 Conclusion 212
Exercises 213
9 Reliability Analysis of Multinode Power Systems 215
9.1 Introduction 215
9.2 Scope and Modeling of Multinode Systems 215
9.3 System Modeling 216
9.4 Power Flow Models and Operating Policies 222
10 Reliability Evaluation of Multi-Area Power Systems 227
10.1 Introduction 227
10.2 Overview of Methods for Multi-Area System Studies 227
10.3 The Method of State Space Decomposition 229
10.4 Conclusion 245
Exercises 245
11 Reliability Evaluation of Composite Power Systems 247
11.1 Introduction 247
11.2 Analytical Methods 247
11.3 Monte Carlo Simulation 250
11.4 Sequential Simulation 250
11.5 Nonsequential Simulation 254
11.6 Testing of States 262
11.7 Acceleration of Convergence 263
11.8 State Space Pruning: Concept and Method 263
11.9 Intelligent Search Techniques 268
11.10 Conclusion 272
12 Power System Reliability Considerations in Energy Planning 273
12.1 Introduction 273
12.2 Problem Formulation 275
12.3 Sample Average Approximation (SAA) 279
12.4 Computational Results 282
12.5 Conclusion and Discussion 288
13 Modeling of Variable Energy Resources 291
13.1 Introduction 291
13.2 Characteristics of Variable Energy Resources 292
13.3 Variable Resource Modeling Approaches 293
13.4 Integrating Renewables at the Composite System Level 301
14 Concluding Reflections 305
Bibliography 309
Index 321
1
Introduction to Reliability
1.1 Introduction
The term reliability is generally used to relate to the ability of a system to perform its intended function. The term is also used in a more definite sense as one of the measures of reliability and indicates the probability of not failing by the end of a certain period of time, called the mission time. In this book, this term will be used in the former sense unless otherwise indicated. In a qualitative sense, planners and designers are always concerned with reliability, but the qualitative sense does not help us understand and make decisions while dealing with complex situations. However, when defined quantitatively it becomes a parameter that can be traded off with other parameters, such as cost and emissions.
There can be many reasons for quantifying reliability. In some situations, we want to know what the reliability level is in quantitative measures. For example, in military or space applications, we want to know what the reliability actually is, as we are risking lives. In commercial applications, reliability has a definite trade-off with cost. So we want to have a decision tool for which reliability needs to be quantified. The following example will illustrate this situation.
Example 1.1 A system has a total load of 500 MW. The following options are available for satisfying this load, which is assumed constant for simplicity:
- 5 generators, each with 100 MW;
- 6 generators, each with 100 MW;
- 12 generators, each with 50 MW.
The question we need to answer in terms of design and operation aspect is: Which of these alternatives has the best reliability?
A little thinking will show that there is no way to answer this question without some additional data on the stochastic behavior of these units, which are failure and repair characteristics. After we obtain this data, models can be built to quantify the reliability for these three cases, and then the question can be answered.
1.2 Quantitative Reliability
Most of the applications of reliability modeling are in the steady state domain or in the sense of an average behavior over a long period of time. If we describe the system behavior at any instance of time by its state, the collection of possible states that the system may assume is called the state space, denoted by S.
In reliability analysis, one can classify the system state into two main categories, success or failure states. In success states the system is able to do its intended function, whereas in the failed states it cannot. We are mostly concerned with how the system behaves in failure states. The basic indexes used to characterize this domain are as follows.
Probability of failure
Probability of failure, denoted by pf, is the steady state probability of the system being in the failed state or unacceptable states. It is also defined as the long run fraction of the time that system spends in the failed state. The probability of system failure is easily found by summing up the probability of failure states as shown in (1.1):
(1.1)where
- pf system unavailability or probability of system failure;
- Y set of failure states, Y?S;
- S system state space.
Frequency of failure
Frequency of failure, denoted by ff, is the expected number of failures per unit time, e.g., per year. This index is found from the expected number of times that the system transits from success states to failure states. As will be seen clearly in Chapter 4, this index can be easily obtained by finding the expected number of transitions across the boundary of subset Y of failure states.
Mean cycle time
Mean cycle time, denoted by Tf, is the average time that the system spends between successive failures and is given by (1.2). This index is simply the reciprocal of the frequency index:
(1.2)Mean down time
Mean down time, denoted by TD, is the average time spent in the failed states during each system failure event. In other words, this is the expected time of stay in Y in one cycle of system up and down periods. This index can be found from (1.3):
(1.3)Mean up time
Mean up time, denoted by TU, is the mean time that the system stays in the up states before system failure and is given by (1.4):
(1.4)There are several other indices that can be obtained as a function of the above indices, and these will be discussed in Chapter 5.
There are also applications in the time domain, say [0, T]. For example, at time 0, we may be interested in knowing the probability of not having sufficient generation at time T in helping decide the start of additional generation. The following indices could be used in such situations:
-
Probability of failure at time T
This indicates the probability of being in the failed state at time T. This does not mean that the system did not fail before time T. The system may have failed before T and repaired, so this only indicates the probability of the system being in a failed state at time T. -
Reliability for time T
This is the probability that the system has not failed by time T. -
Interval frequency over [0, T]
This is the expected number of failures in the interval [0, T]. -
Fractional duration
This is the average probability of being in the failed state in interval [0, T].
The most commonly computed reliability measures can be categorized as three indices as follows.
- Expected value indexes: These indices involve
Expected Power Not Supplied (EPNS) or Expected Unserved Energy (EUE).
- Probability indices such as
Loss of Load Probability (LOLP) or Loss of Load Expectation (LOLE).
- Frequency and duration indices such as
Loss of Load Frequency (LOLF) or Loss of Load Duration (LOLD).
1.3 Basic Approaches for Considering Reliability in Decision-Making
Having quantified the attributes of reliability, the next step is to see how it can be included in the decision process. There are perhaps many ways of doing it, but the most commonly used are described in this section. It is important to remember that the purpose of reliability modeling and analysis is not always to achieve higher reliability but to attain the required or optimal reliability.
Reliability as a constraint
Reliability can be considered a constraint within which other parameters can be changed or optimized. Until now this is perhaps the most common manner in which reliability considerations are implemented. For example, in generation reliability there is a widely accepted criterion of loss of load of one day in 10 years.
Reliability as a component of overall cost optimization
The conceptual relationship between cost and reliability can be appreciated from Figure 1.1. The overall cost is a combination of the investment cost and the cost of failures to the customers. The investment cost would tend to increase if we are interested in higher levels of reliability. The cost of failures to the customers, on the other hand, tends to decrease with increased level of reliability. If we combine these costs, the total cost is shown by the solid curve, which has a minimum value. The reliability at this minimum cost may be considered an optimal level; points to the left of this would be dominated by customer dissatisfaction, while points to the right may be dominated by investment cost considerations.
Figure 1.1 Trade-off between reliability and cost.
It can be appreciated that in this type of analysis we need to calculate the worth of reliability. In other words, how much do the customers think that interruptions of power cost them? One way of doing this is through customer damage function, like the one shown in Figure 1.2.
Figure 1.2 Customer damage function (compiled from data in [1]).
The customer damage function provides the relationship between the duration of outage and the interruption cost in $/kW. The damage function is different depending on the type of customer. The damage function is clearly nonlinear with respect to the duration, increasing at much higher rates for longer outages. The frequency and duration indices defined earlier can be combined to yield the cost of interruptions using (1.5):
(1.5)where
n number of load points in the system; Li load requirement at load point i in kW; fi failure frequency at load point i in number of occurrence per year; ci(di) customer damage function at load point i in $ per kW in terms of outage duration di; di outage duration at load point i in hours.