Distributed Processing of Reliability Index Assessment and Reliability-Based Network Reconﬁguration in Power Distribution Systems Fangxing Li, Member, IEEE

230 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 Distributed Processing of Reliability Index Assessment and Reliability-Based Network Reconﬁguration in Power Distribution Systems Fangxing Li, Member, IEEE

Abstract—Parallel and distributed processing has been broadly generation and transmission systems by Monte Carlo simu- applied to scientific and engineering computing, including various lation [12]. These previous works can be classified into two aspects of power system analysis. This paper first presents a dis- categories: applications of parallel processing [4]–[6], [12] tributed processing approach of reliability index assessment (RIA) for distribution systems. Then, this paper proposes a balanced and applications of distributed processing [7]–[12]. It appears task partition approach to achieve better efficiency. Next, the dis- that more recent works focus on distributed processing. This tributed processing of RIA is applied to reliability-based network is probably due to the latest development in network hardware reconfiguration (NR), which employs an algorithm combining and software, which makes distributed processing faster, more local search and simulated annealing to optimize system reliability. broadly available, and easier-to-implement than before. Testing results are presented to demonstrate the speeded execution of RIA and NR with distributed processing. This paper presents distributed processing schemes of reliability index assessment (RIA) and reliability-based network re- Index Terms—Network reconfiguration, parallel and distributed configuration (NR) for distribution systems. Here, the radial dis- processing, power distribution systems, reliability index assessment, scalability, simulated annealing. tribution system is addressed, since the majority of U.S. distribution systems are radial. In addition, system-level reliability indices are addressed because they are the primary reliability I. INTRODUCTION concerns of utilities. The discussion shows that RIA can be ARALLEL and distributed processing [1], [2] has signif- easily de-coupled and executed in parallel among different pro- P icant contributions to scientific and engineering computa- cessors to achieve speedup in wall-clock running time. The dis- tions, especially for time-critical or time-consuming tasks. Par- cussion also shows NR is mainly composed of iterative runs of allel processing is usually carried out in dedicated multiproces- RIA. Therefore, NR can be executed in parallel based on the sors with a global clock and shared memory, while distributed distributed processing of RIA. The proposed implementation processing is usually carried out at multiple workstations or of distributed processing considers the unbalanced computing computers connected to a network without a central clock and ability of different processors, a typical feature of heterogeneous shared memory. In distributed processing, message passing is a computers connected to a LAN or a similar network. common technique to share information since there is no shared This paper is organized as follows. Section II presents a con- memory. Although the performance of networked computers is troller-worker model based on message passing to share the not as competitive as a dedicated parallel computer, networked data and coordinate the activity of different processors. Sec- computers are less expensive and more broadly available such as tion III first discusses the principle of an analytical approach to in local area networks (LANs). As such, distributed processing assess distribution reliability and why it is highly parallelizable; is sometimes referred to as the low-end parallel processing. then presents a coarse-grained distributed processing scheme for It should be noted that the term “parallel” is used occasionally RIA. Section IV discusses the balanced task partition among dif- in this paper to indicate the concurrent execution of a computing ferent processors in order to achieve better performance and ef- task. The discussion in this paper is essentially based on dis- ficiency. Section V presents and discusses the testing results for tributed processing. Especially, like many other distributed pro- distributed processing of RIA. Section VI applies the distributed cessing approaches, the proposed approach employs the mes- processing of RIA to reliability-based NR, which employs an sage-passing scheme for data sharing among collaborating pro- annealed local search and RIA. Test results are also provided. cessors. Section VII concludes the paper. Parallel and distributed processing has been applied to power system computing in various areas [3]–[12], such as load II. CONTROLLER-WORKER MODEL flows [4]–[7], optimal power flows [8], [9], state estimation FOR DISTRIBUTED PROCESSING [10], contingency analysis [11], and reliability evaluation for Distributed processing has two fundamental units: computation and communication. Computation is referred to as the CPU activity to carry out the actual computing task, while commu- Manuscript received May 16, 2004. Paper no. TPWRS-00643-2003. nication is referred to as the overhead activity to transfer data The author is with ABB Inc., Raleigh, NC 27606 USA (e-mail: [email protected]; [email protected]). or share information among different collaborating processors. Digital Object Identifier 10.1109/TPWRS.2004.841231 Although communication is overhead indeed (and is desired to

0885-8950/$20.00 © 2005 IEEE LI: DISTRIBUTED PROCESSING OF RIA AND RELIABILITY-BASED NR 231 be minimum), it is normally a necessary part for distributed processing due to the lack of shared memory. This paper employs the controller-worker model [7] to coordinate computation and communication in distributed processing. As described in [7], the model is comprised of two types of processes, controllers and workers, which play different roles in the message-passing scheme. A controller is a process with the following responsibilities: • to accept user input; • to assign initial data to workers; • to invoke workers to execute tasks; Fig. 1. Input and output of RIA. • to send/receive intermediate data to/from workers; • to do system-wide, nonintensive calculations; (the failure on a component). Taking SAIFI as an example, this • to terminate the algorithm at the appropriate time and no- can be expressed as tify workers to stop. A worker is a process with the following responsibilities: • to receive initial data from a controller; (1) • to do intensive calculations upon a controller’s request; • to send/receive intermediate data to/from a controller; • to stop when notiﬁed. where In the approach here, there is only one controller, while there contribution to SAIFI from component ; are multiple workers. The controller is mainly responsible for total number of components. coordinating all workers and maybe some nonintensive compu- Since SAIFI is deﬁned as the number of customer inter- tations, while workers are responsible for intensive computa- ruptions that an average customer experiences during a year, tions [7]. This model implies a coarse-grained distributed pro- is the number of customer interruptions at an average cessing, in which parallelism occurs at the level of subroutine customer caused by failures on component . Hence, or group of subroutines. can be written as [18]

(2) III. DISTRIBUTED PROCESSING OF RIA where This work assumes N-1 contingencies. Further, only perma- failure rate per year of component ; nent component faults are considered in this section for sim- number of customers experiencing sustained interrup- plicity and illustration. tion due to a failure of component ; A survey of 205 U.S. utilities [13] showed that five relia- total number of customers. bility indices, SAIFI, SAIDI, MAIFI , CAIDI, and ASAI (or Combining (1) and (2), we have ), have been popularly employed in the utilities. The survey also mentioned other four less popular indices, CAIFI, CTAIDI, ASIFI, and ASIDI. This section takes SAIFI as an example to illustrate why the calculation of a reliability index can be divided into many parallelizable steps. The Ap- (3) pendix briefly illustrates that the other system reliability indices Since is a constant and is related to the component itself, can be assessed in a similar way. the above equation shows that the key to calculate SAIFI is to The purpose of RIA for distribution systems is to model each calculate . Since stands for the number of customers expe- system contingency and compute the reliability impact of each riencing sustained interruptions due to a failure of component contingency. It may be carried out through various approaches, , it can be evaluated by identifying which components will be such as analytical approach based on component contributions de-energized based on the system topology, protection scheme, [14], [15], failure mode and effect analysis (FMEA) [16], Monte and restoration. Although the evaluation of may be compli- Carlo simulation [17], [18], Markov modeling [19], and other cated, it is certain that the evaluation of is independent of practical approaches [20]. This work employs the analytical ap- . In other words, two processors can evaluate and proach in the previous work [14], [15] to evaluate reliability in- , respectively, in parallel, if both processors know the input of dices such as SAIFI, SAIDI, MAIFI , etc. Fig. 1 briefly illus- the system data. Hence, evaluation of SAIFI contributions from trates the input and output of RIA analysis. Appendix C also component and ,or and , can be carried out provides the fundamental principles for the analytical simula- at different processors independently. tion of RIA. More detailed algorithms can be found in [15]. As shown in the Appendix, similar conclusions can be easily With the assumption of N-1 contingencies, a reliability index drawn that the value of other indices is the sum of contributions can be viewed as the sum of contribution from each contingency from all individual components, or can be directly obtained as a 232 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 function of other indices. Hence, the evaluation of the component contributions can be carried out independently and concurrently. With this simple, but very important feature of reliability indices, a distributed processing approach for RIA is proposed. This approach employs the controller-worker model presented in the previous section. Assume there is a unique controller and workers. Also assume the system has components. The ap- Fig. 2. Partition of a task with s independent steps. proach is described as follows. 1) The controller reads in all input data including system This is typically true for parallel processing carried out in a ded- topology, component reliability rates, etc. icated supercomputer. However, sometimes the ideal environ- 2) The controller sends the input data to all workers. ment may not be available, especially in a “loosely coupled” dis- 3) The controller requests each worker to calculate the reli- tributed processing environment like heterogeneous networked ability index contributions from randomly selected computers. In this case, the computing power of different com- components. For instance, with the assumption that the puters may vary. An equal partition may lead some inefficiency component ID is randomly distributed, the first worker and task balancing needs to be considered for higher efficiency. computes reliability index contributions for components Assume a task consisting of 3000 steps, computationally 1to ; the second worker computes reliability index identical and logically independent, will be carried out at two contributions for components to ; and so different processors, P1 and P2. If running separately, P1 can on. complete the task in 100 s while P2 in 50 s. Ignoring overhead, 4) Each worker concurrently evaluates the reliability index an equal task partition (1500:1500) leads P1 to complete its contributions for its assigned components. assignment in 50 s and P2 in 25 s. This causes P2 to be idle for 5) Each worker sends the reliability index contributions of 25 s. The overall wall-clock running time is 50 s. its assigned components to the controller. To fully utilize the computing power, P2 should receive more 6) The controller adds up all reliability index contributions tasks than P1. If 1000 steps are assigned to P1 and 2000 steps to obtain the system reliability index. to P2, then each processor will finish its assignment in 33.3 s, The proposed coarse-grained distributed algorithm is highly or 16.7 s less than the previous equal task partition. It can be parallelized because of the following two reasons. easily proved that this 1000:2000 partition is the most efficient • The calculations of reliability index contributions among partition. different workers are highly independent. A worker does Based on the above analysis, the balanced task partition with not need any further communication with others once the computing ability adjusted for two processors is illustrated in actual calculation starts. Fig. 2. The weighting factors of task partition for each processor, • The communication is kept minimum. The major commu- and , are given by nication occurs only at the very beginning to distribute the input data (system topology, component reliability data, (4) etc) to workers. The collection of reliability index contributions calculated by workers needs much less communi- (5) cation time. It is noteworthy to mention that although this work assumes where N-1 contingencies, higher order contingencies can be applied time to complete the task with a single processor P1; as well based on the component (contingency) contribution ap- time to complete the task with a single processor P2. proach. The reason is that the basic unit of this approach is to Hence, each processor finishes its own assignment in evaluate the impact or contribution to reliability indices from s for 1 p.u. task. each contingency event. Because each contingency event can Equations (4) and (5) can be extended to multiple processors. be assessed independently, RIA can be carried out in parallel. Equation (6) shows the weighting factor for the th processor. It Due to the same reason, similar parallelization approach can be can be easily verified that each processor completes its assign- applied to meshed network RIA, although more implementation ment simultaneously in efforts are needed than for radial system RIA. (6) IV. BALANCED TASK PARTITION WITH THE CONSIDERATION OF COMPUTING ABILITY Parallel or distributed processing usually partitions a task into where many small units processed at each individual participating pro- weighting factor for processor ; cessors. It is a common practice that the task is equally parti- time to complete the task with a single processor . tioned among processors, with the ideal assumption of a bal- The above discussion is based on two assumptions. anced computing environment. That is, all processors can finish 1) Each step within a task is independent. the same amount of computing work in the same time duration. 2) Each step is computationally equal. LI: DISTRIBUTED PROCESSING OF RIA AND RELIABILITY-BASED NR 233

The first assumption is true for the RIA task because each TABLE I RIA step, i.e., the evaluations of the contribution to a reliability SIZES OF FIVE TEST SYSTEMS index from each individual component, is independent. The second assumption may not be precisely true for the RIA algorithm because the evaluation of the reliability index contribution from a component depends on connectivity, protection, restoration, etc. For example, it takes less CPU time to evaluate SAIFI contribution from a component in a feeder with 50 components than a component in a feeder with 100 components. However, the second assumption shall be roughly true for a large TABLE II distribution system in the scale of thousands of components, be- TIME (SECONDS) TO COMPLETE A SEQUENTIAL RIA AT EACH cause each processor is assigned a large number of randomly INDIVIDUAL MACHINE selected components. Hence, an average step carried out at one processor shall be computationally equal to an average step at another processor. The above discussion makes it apparent that Step 3 in the distributed processing of RIA should be modified slightly as follows: The th processor should be responsible for randomly selected components instead of components. of scalability are presented. Here the scalability is measured as It should be noted that there are alternative ways for task bal- speedup and efficiency, defined by the following equations [1], ancing such as dynamic assignment, which balances the compu- [2]: tation among processors at run-time based on assignment-completion loops [11]. However, in general, it requires more syn- (7) chronization and communication overhead. It is more suitable (8) for a dynamic task with computation time very unpredictable where or in a wide range. Since the computation-dominant RIA appli- wall-clock time to complete the best sequential algo- cation completes in a stable manner (in steps of component rithm on a single processor; contribution evaluation), the proposed approach tends to be ef- wall-clock time to complete the parallel or distributed fective for RIA application. The reason is that there is no mul- algorithm on multiple processors; tiple, iterative run-time communication involved, because each number of processors. worker receives its total assignment at the start of the algorithm. The above speedup definition does not indicate which pro- Implementation is also simplified. Meanwhile, efficiency can be cessor is associated with. This implies each processor is iden- ensured with the task partition adjusted by weighting factors. tical for typical parallel processing environment. Since this work is carried out in processors with different computing capability, V. T EST RESULTS OF RIA DISTRIBUTED PROCESSING is taken as the average of the sequential RIA running time at A. Test Systems, Environment, and Procedure all participating processors. This section presents testing results of the distributed pro- This section also presents an illustrative test to verify the per- cessing of RIA. Here the five popular system-level reliability formance improvement with balanced task partition. The test is indices, SAIFI, SAIDI, MAIFI , CAIDI, and ASAI [13], are performed in the mid-sized system SYS-3. considered. The tests are explored in several actual distribution B. Test Results systems with sizes from 3835 components to 11 026 components. This is shown in Table I. Table II shows the running time in seconds to complete a se- The test environment is a Local Area Network with up to 6 quential RIA algorithm with a single processor. It shows that computers, each with Pentium III processor. The speeds of the these networked computers have different computing power. 6 processors vary from 450 MHz to 1 GHz. The network speed This is used for balanced task partition in the distributed pro- is 10 M bps. The memory sizes vary from 128 to 256 MB. The cessing. distributed RIA processing is implemented in Visual C++ with Tests for distributed processing are carried out for five sce- TCP/IP protocol for message passing. narios with 2–6 processors working in parallel. The processor The testing procedure is described as follows. First, the test assignment is as follows. is examined for the RIA algorithm in a sequential processing Two processors: P1, P2. mode. That is, the RIA algorithm is executed in a single pro- Three processors: P1, P2, P3. cessor for each of the six processors. The running time of each Four processors: P1, P2, P3, P4. individual processor will be used to compute the weighting Five processors: P1, P2, P3, P4, P5. factor for balanced task partition based on (6). Then, the RIA is Six processors: P1, P2, P3, P4, P5, P6. executed in the distributed processing mode with balanced task In the distributed processing scenario, each processor runs partition in 2, 3, 4, 5, and 6 processors, respectively. The results a worker process. In addition, one of the processors runs the 234 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005

TABLE III TABLE IV SPEEDUP OF DISTRIBUTED PROCESSING OF RIA SPEEDUP OF DISTRIBUTED PROCESSING WITH TWO PARTITIONS FOR THE SYS-3 SYSTEM

reliability-based optimization such as NR. NR is accomplished by closing normally open switches and opening normally closed switches. Since the NR is usually very time-consuming (from tens of minutes to hours), the application of distributed processing may be very beneficial to improve the wall-clock running time. Reliability-based NR is to identify the optimal network configuration without any financial investment to achieve the highest possible reliability. Although various formulations may be applied to NR, this work takes a weighted sum of multiple reliability indices as the objective function [15], [21]

(9) Fig. 3. Efficiency of distributed processing of RIA. The constraints could be component loadings and voltage controller process as well. The purpose is to fully utilize the drop limits. Penalties can be added into the objective function if computing ability of that processor, because a controller is not constraints are violated [21]. computationally intensive while a worker is. In fact, when the Since NR is a nonlinear, noncontinuous optimization worker is calculating the reliability index contribution, the con- problem, many combinatory techniques like local search, tabu troller is essentially idle except waiting for the feedbacks from search, simulated annealing and genetic algorithm have been all workers. Typically, the most powerful processor is selected applied to NR. This paper employs an approach presented to host the controller and a worker. in [15] and [21], which combines local search and simulated Table III and Fig. 3 present the speedup and efficiency, respec- annealing [22], to demonstrate that the previous distributed tively, of the distributed processing of RIA. It is shown that the processing of RIA can be easily applied to NR. The traditional, distributed processing can speed the wall-clock execution time sequential algorithm is described in details in previous works of RIA. It is also shown that, in general, the speedup and the [15], [21]. Here it is sketched as follows. efficiency increase when the system size increases. When the 1) Set initial parameters for simulated annealing such as the number of involved processors increases up to 6, the speedup starting temperature (T) and the annealing rate (R). keeps increasing at a considerable rate, although the efficiency 2) Identify the objective function of the initial config- drops. uration. 3) Set the above obj as the temporary best objective, OBJ. C. Tests for Equal and Balanced Task Partitions 4) For each tie switch: A test run of distributed RIA processing with the equal task a) identify a new configuration by performing a tie switch partition is carried out for comparison against the balanced task shift; partition. The test is executed in the mid-sized system SYS-3 b) perform an RIA to calculate the new objective function, for illustrative purpose. obj’, for the new configuration; c) generate a random number ; In Table IV, the second row shows the speedup with equal d) check whether the new configuration is acceptable or task partition, while the third row shows the speedup with bal- not: if or , then ’; anced task partition. The last row shows the improved speedup otherwise, shift the tie switch to its original position; in percentage. It is clearly demonstrated that the balanced task e) Go to 4a) to perform switch shift for remaining adjacent partition improves the efficiency of the distributed processing switches. for RIA, if the processors have various computing capabilities. 5) Repeat 4) for all tie switches. 6) Change annealing factor by setting . VI. DISTRIBUTED ROCESSING OF ELIABILITY ASED P R -B 7) If there is no change of OBJ since last change of T, stop. NRETWORK RECONFIGURATION Otherwise, go to step 4). The distributed processing for RIA proposed in Section III, As the above description shows, the NR algorithm requires together with the balanced task partition, can be easily applied to multiple runs of Step 4), especially Step 4b) to complete a RIA LI: DISTRIBUTED PROCESSING OF RIA AND RELIABILITY-BASED NR 235 run for each new configuration. Normally, there may be hun- TABLE V dreds, even thousands, of new configurations to examine in large SCALABILITY OF DISTRIBUTED PROCESSING FOR NR ON THE SYSTEM SYS-3 systems. Even though network simplification may be employed to reduce the running time to some extent, the NR algorithm is still very time-consuming. Therefore, it is very beneficial if distributed processing can be applied to NR to reduce the wall- clock running time. The description of the distributed processing implies that ex- Practical tests show that the repeated RIA executions of Step ecutions of NR may yield different results and require different 4b) typically consume more than 98% CPU time of the overall number of RIA runs because of the random number generation NR algorithm. Hence, the key to parallelize NR is to parallelize in Step 4c). To examine the efficiency of distributed processing, the RIA executions. With the annealing algorithm, it is very dif- the completion time of a NR test is normalized to each RIA run. ficult to carry out the multiple instances of RIA in parallel. This It should be noted that if the random numbers are identically is because a new configuration is determined only after the RIA generated among different NR tests, the result of each NR test of the previous configuration is completed. The acceptance or should be the same regardless of distributed processing or not. rejection of the previous configuration, together with the pertur- The number of RIA runs within each NR test should be the same, bation mechanism, may affect the new configuration. However, too. there is a simple yet effective approach to parallelize the sequen- Table V shows the speedup of the distributed processing of tial version of NR. The approach is to parallelize each individual NR. It appears that the scalability here is higher than that of the RIA using the distributed RIA processing presented previously, distributed processing of RIA. The reason is that less relative rather than to parallelize different RIA instances. As such, the overhead is involved in distributed processing of NR. Detailed following three principles are proposed for the distributed pro- explanations are given as follows. cessing of NR. The NR algorithm contains many iterative RIA runs. The initial bulk data transfer (Step 2 in the distributed processing of • The controller knows the global parameters like T, R, and RIA in Section III) is necessary only in the first RIA run. In the OBJ. All steps except 4b) will be sequentially carried out follow-up runs, only a small amount of information needs to be at the controller without the involvement of the workers. transferred from the controller to the workers. The information • The responsibility of the workers is to collaboratively is the change of network topology such as which switches are complete Step 4b), the RIA, in parallel for each new con- opened or closed. There is no need to re-send the whole system figuration. The workers know the system data in order to information. Therefore, the initial data transfer, which is a con- complete RIA and do not have any knowledge about the siderable overhead for RIA algorithm, is negligible in the NR global parameters for NR. Here, balanced task partition is algorithm. Hence, it is apparent that the higher speedup should considered in the distributed processing of RIA. be observed in the distributed processing of NR algorithm. • The controller needs to transfer the bulk system data (components, topology, protection schemes, etc) to the workers in the first RIA run. In the following RIA run, the con- VII. CONCLUSION troller only needs to notify the workers about the topo- The RIA for distribution systems can be carried out in par- logical changes after each new configuration is identified. allel, because the contributions to a reliability index from dif- Hence, the workers will be able to complete another RIA ferent components are independent. A balanced task partition run for the new configuration without unnecessary bulk scheme may be applied to improve efficiency, especially in a data transfer. distributed processing environment where the computing abili- The above distributed processing of NR can be easily imple- ties of participating processors are usually different. mented by reusing the previous approach of RIA distributed pro- The distributed processing of RIA can be applied to a NR al- cessing. Good efficiency is expected even though Step 4b) is the gorithm, which is to optimize the system reliability based on an- only step carried out in parallel because it dominates over the nealed local search. Since the kernel of the NR algorithm mainly other steps in CPU consumption. consists of multiple RIA runs, it can be easily implemented on It should be noted that the annealed local search algorithm is top of the distributed processing of RIA. selected for illustrative purpose. Other optimization algorithms The proposed distributed processing of RIA and NR should like a genetic algorithm may work for NR as well, but it is be practically beneficial if applied to utility distribution systems, likely that reliability-based optimization depends on the RIA al- which may have tens, or even hundreds, of thousands of compo- gorithm to evaluate the system reliability for each new config- nents. It should be especially attractive for NR application since uration. Therefore, the proposed distributed processing for NR NR adds a dimension of computational complexity on top of may be applied to other similar reliability-based optimization RIA. algorithms. The distributed processing of reliability-based NR is imple- APPENDIX mented as an extension of the sequential version of the pre- A. System Reliability Indices vious works [15], [21]. Then, It is tested at the mid-sized system SYS-3 for demonstrative purpose. To achieve acceptable so- The analytical approach considers a reliability index as the lution quality without unnecessary computation time, an an- sum of contributions from failures of individual components nealing rate of 0.9 and a starting temperature of 0.3 are selected. during the course of a year. This Appendix gives the following 236 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 equations as a formal method to calculate reliability indices. TABLE AI These equations are expected to make it easy to understand why OCCURRENCE OF INTERRUPTION AT LOAD POINTS DUE TO N-1 CONTINGENCIES the analytical approach is parallelizable. SAIFI, SAIDI, and MAIFI can be calculated with (A1)–(A3). Equation (3) is repeated here as (A1) for com- pleteness. These three equations can also be found in [18]

(A1)

(A9) (A2)

(A3) (A10) where failure rate of component ; where number of customers experiencing sustained interrup- total number of customers experiencing at least 1 in- tion due to a failure of component ; terruption per year; sustained interruption durations for all customers due total connected kVA served; to a failure of component ; interrupted kVA due to a failure of component ; sustained interruption duration for customer due to a interrupted kVA weighted by interruption hours due to failure of component ; a failure of component ; number of customers experiencing temporary interrup- interrupted kVAfor customer due to a failure of com- tion (event) due to a failure of component ; ponent . (Note: ). total number of customers; Other important reliability indices, though not shown in [13], total number of components. include Expected Energy Not Served (EENS) and Average En- Three other popular reliability indices, CAIDI, ASAI, and ergy Not Served (AENS). They can be obtained with ASUI, can be directly derived from SAIFI and SAIDI. They are given as follows: (A11) (A4) (A12) (A5) where is the power factor of customer . (A6) B. Load Point Reliability Indices Some less popular reliability indices, such as CAIFI, CTAIDI, ASIFI, and ASIDI [13], can still be obtained using Although this work does not address load point reliability in- the contribution approach. They are given by dices [16], [17], they can be calculated with slight refinement of the analytical approach based on component contributions. The refinement is as follows: when a component failure is ana- lyzed, a temporary record is kept for each load point to identify: 1) whether it will be interrupted and 2) the duration of the inter- (A7) ruption. Table AI shows a simple example of the impact to load points from component failures under N-1 contingencies. If a load point experiences an interruption after a component fails, the corresponding entry is set to “1”. Otherwise, it is set to “0” if (A8) no interruption at the load point. Hence, the annual interruption rate at each load point, , is the sum of its column, weighted LI: DISTRIBUTED PROCESSING OF RIA AND RELIABILITY-BASED NR 237 by component failure rate. If the values are replaced with in- C. Description of the Analytical Approach of RIA Simulation terruption durations, then we can obtain the annual load point The algorithm of the analytical approach of RIA is outlined interruption duration , which is the weighted column sum. as follows. The average interruption time can be obtained as . This is summarized as follows: 1) For a permanent fault at component C (repair time ): a) Fault isolation: An upstream search is performed to find (A13) the nearest protection or reclosing device, P, which op- erates to isolate the fault. b) Upstream restoration: If there is a switching device S (A14) between P and C, S is opened in minutes to restore components between P and S. All restored (A15) components experience a (sustained) interruption of minutes. If this is less than the threshold of momentary interruption in the case that where S is automated, the interruption is classified as a mo- if a failure of component causes an interruption at mentary interruption event. load point ; otherwise 0; c) Downstream restoration: If there is an alternate power interruption duration at load point if component source through a normally open switch (NOS) and fails. there is another normally closed switch (NCS) be- It is noteworthy to mention that the sum of the row in tween NOS and C, all components between the two Table AI, weighted by the number of customers at each load switches experience a (sustained) interruption of point shall be , which is the contribution factor used to minutes calculate SAIFI. If the values in the table are outage durations, due to downstream restoration. If the time is less than then the sum of the row weighted by the number of customers the momentary interruption threshold because of au- is , which is used to calculate SAIDI. tomated switching, the interruption is classified as a Since such tables are intermediate results and consumes lots momentary interruption event. of memory, it is not necessary to implement the tables. In the d) All isolated and unrestored components experience a actual implementation, it is only necessary to keep records of sustained interruption of MTTR minutes. the accumulated value of the weighted column (or row) sum for Note: load point reliability indices (or for system reliability indices as done in this work). The interruption rate at load point can be employed to calculate , the total number of customers experiencing at least one interruption per year, which is used in (A7)–(A8). This is 2) For a temporary fault: given by a) Fuse-saving: If a temporary fault can be cleared by an upstream reclosing device, R, with fuse-saving scheme, all components downstream of R experience (A16) a momentary interruption event. b) Fuse-clearing: If the there is no fuse-saving protection, the fuse blows and isolates its downstream compo- where nents. The interrupted downstream components may total number of load points; be restored through back-feed as described in 1c). number of customers at load point . The above approach shall be applied to each component to In the above equation, is the probability that load identify the interruption type and duration at each load/customer point experiences at least one interruption per year. This is be- point if the specific component fails. Hence, the contribution cause a component with a constant interruption rate follows factors to reliability from a component failure, such as a Poisson process [15], [18]. The probability density function of and so on, can be easily obtained. its being interrupted times per year is given as follows: The above approach can be extended to address more com- plicated cases like operation failures, transfer switches, and distributed generations. Details can be found in [15]. (A17)

Hence, the probability of being interrupted at least once per ACKNOWLEDGMENT year is given by The author is very grateful to Dr. R. E. Brown for his thoughts and discussions in power distribution reliability issues. The au- (A18) thor would also like to thank the reviewers and the editor for 238 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005

their comments and suggestions, which helped the author to re- [13] IEEE/PES Working Group on System Design, “A survey of distribution vise and improve this paper. reliability measurement practices in the U.S.,” IEEE Trans. Power De- livery, vol. 14, no. 1, pp. 250–257, Jan. 1999. [14] G. Kjolle and K. Sand, “RELRAD—An analytical approach for distribution system reliability assessment,” IEEE Trans. Power Delivery, vol. 7, no. 2, pp. 809–814, Apr. 1992. REFERENCES [15] R. E. Brown, Electric Power Distribution Reliability. New York: Marcel Dekker, 2001. [1] V. Kumar, A. Grama, A. Gupta, and G. Karypis, Introduction to Parallel [16] R. Billinton and R. N. Allan, Reliability Evaluation of Power Systems, Computing: Design and Analysis of Algorithms. Reading, MA: Ben- 2nd ed. New York: Plenum, 1996. jamin-Cummings (Addison-Wesley), 1994. [17] R. Billinton and P. Wang, “Teaching distribution system reliability eval- [2] U. Manber, Introduction to Algorithms—A Creative Ap- uation using Monte Carlo simulation,” IEEE Trans. Power Syst., vol. 14, proach. Reading, MA: Addison-Wesley, 1989. no. 2, pp. 397–403, May 1999. [3] D. J. Tylavsky et al., “Parallel processing in power systems computa- [18] F. Li, R. E. Brown, and L. A. A. Freeman, “A linear contribution factor tion,” IEEE Trans. Power Syst., vol. 7, no. 2, pp. 629–638, May 1992. model and its applications in Monte Carlo simulation and sensitivity [4] A. Abur, “A parallel scheme for the forward/backward substitutions in analysis,” IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1213–1215, Aug. solving sparse linear equations,” IEEE Trans. Power Syst., vol. 3, no. 4, 2003. pp. 1471–1478, Nov. 1988. [19] R. E. Brown, S. Gupta, S. S. Venkata, R. D. Christie, and R. Fletcher, [5] J. Q. Wu and A. Bose, “Parallel solution of large sparse matrix equations “Distribution system reliability assessment using hierarchical markov and parallel power flow,” IEEE Trans. Power Syst., vol. 10, no. 3, pp. modeling,” IEEE Trans. Power Delivery, vol. 11, no. 4, pp. 1929–1934, 1343–1349, Aug. 1995. Oct. 1996. [6] S. D. Chen and J. F. Chen, “Fast load flow using multiprocessors,” Int. [20] S. R. Gilligan, “A method for estimating the reliability of distribution J. Elect. Power & Energy Syst., vol. 22, no. 4, pp. 231–236, 2000. circuits,” IEEE Trans. Power Delivery, vol. 7, no. 2, pp. 694–698, Apr. [7] F. Li and R. P. Broadwater, “Distributed algorithms with theoretic scal- 1992. ability analysis of radial and looped load flows for power distribution [21] R. E. Brown, “Distribution reliability assessment and reconfiguration systems,” Elect. Power Syst. Res., vol. 65, no. 2, pp. 169–177, 2003. optimization,” in Proc. IEEE PES Transmission and Distribution Conf., [8] R. Baldick, B. H. Kim, C. Craig, and Y. Luo, “A fast distributed imple- vol. 2, 2001, pp. 994–999. mentation of optimal power flow,” IEEE Trans. Power Syst., vol. 14, no. [22] H. D. Chiang, J. C. Wang, O. Cockings, and H. D. Shin, “Optimal ca- 3, pp. 858–864, Aug. 1999. pacitor placements in distribution systems: Part 1: A new formulation [9] B. H. Kim and R. Baldick, “A comparison of distributed optimal power and the overall problem,” IEEE Trans. Power Delivery, vol. 5, no. 2, pp. flow algorithm,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 599–604, 634–642, Apr. 1990. May 2000. [10] R. Ebrahimian and R. Baldick, “State estimation distributed processing,” IEEE Trans. Power Syst., vol. 15, no. 4, pp. 1240–1246, Nov. 2000. [11] J. R. Santos, A. G. Exposito, and J. L. M. Ramos, “Distributed contin- Fangxing Li (M’01) received the B.S.E.E. and M.S.E.E. degrees from Southeast gency analysis: Practical issues,” IEEE Trans. Power Syst., vol. 14, no. University, Nanjing, China, in 1994 and 1997, respectively, and the Ph.D. degree 4, pp. 1349–1354, Nov. 1999. from Virginia Tech, Blacksburg, Virginia, in 2001. [12] C. L. T. Borges, D. M. Falcao, J. C. O. Mello, and A. C. G. Melo, “Com- He is presently a Senior Consulting R&D Engineer at ABB Inc., Raleigh, posite reliability evaluation by sequential Monte Carlo simulation on NC, where he specializes in computational methods and applications in power parallel and distributed processing environments,” IEEE Trans. Power systems, especially in power distribution analysis and energy market simulation. Syst., vol. 16, no. 2, pp. 203–209, May 2001. Dr. Li is a member of Sigma Xi.