Patent Application
20230021610
The present invention relates generally sequencing fault prevention in complex systems, for example, electrical power grid systems, supply networks, healthcare system, among others.
Faults (or failures) are unavoidable in complex systems such as supply networks, smart grids, and healthcare systems. Faults have major cost implications and can cause catastrophic events. For instance, the annual cost of power interruptions in the U.S. is estimated between $22 and $135 billion (LaCommare and Eto, 2004) and is increasing; a healthcare insurance company covering 10 million members is estimated to pay $400 million a year in overpayments (Anand and Khots, 2008) due to unintentional (mistakes made by patients, providers, and insurance companies) and intentional errors (fraudulent claims). Many faults in a system are caused by a few faulty sources which require repair; other faults are repaired once the faulty sources are repaired. For instance, products flow from supplier A to customer B in a supply network. A fault occurs if B does not receive certain products by a predetermined time. This fault, however, is often caused by faults at A or other manufacturing and distribution entities along the path from A to B. The fault at B is repaired if other faulty entities are repaired. Similarly, a fault occurs in a smart grid if a bus cannot provide electricity to a customer. If the bus itself is damaged, it must be repaired. It is also possible that the fault at the bus is caused by faults at generators, transmission lines, distribution lines, and/or transformers; the fault is repaired if other faulty components are repaired. The goal of this research is to design efficient fault prevention and repair (FPR) sequencers to prevent faults from occurring and minimize their damage.
An FPR sequencer determines which faults are to be repaired first and which faults are to be repaired next. The cost of faults includes repair cost and damage caused by faults. Because all faulty sources must be repaired, the repair cost of faulty sources may be assumed to be the same regardless of the sequence of repairs. The damage caused by a fault is often proportional to the time for which the fault exists, which depends on the FPR sequencer. A fault causes less damage if it is repaired early. A complex system has multiple faults and there may be a crippling or cascading effect when a few sources become faulty.
Research on efficient fault repair and infrastructure recovery focused mostly on smart grids and highway systems. For example, power-flow models were developed (Ang, 2006; Salmeron et al., 2004) to identify optimal or near-optimal repair sequences for electrical power grids. Fault repair is part of fault management, in which the state-of-the-art development is automated fault detection and diagnostics (Chen and Nof, 2012, 2014, 2015; Nof and Chen, 2015, 2017). An example of fault management is the emergence of the smart grid, which is a form of electricity network that utilizes digital technology and has the self-detection and self-diagnostics ability. Organizations such as the Electric Power Research Institute (EPRI) have invested heavily in grid operations and planning to help improve real-time situation awareness, wide area protection and control performance, and the capability to handle extreme events and restore the system (EPRI, 2012). The issue of how to prevent and repair faults with an optimal sequence is important but largely left open (Ang, 2006; Jin et al., 2018). The FPR problem is prevalent in many systems. Hospitals are entangled in insurance claim denials due to various faults in the claim process. Insurance companies are concerned about faults in claims and take a conservative approach in processing claims. For instance, FICO (FICO, 2012) developed the Insurance Fraud Manager to detect fraud, abuse, and error in healthcare claims before payment. The result is an unreasonably long delay for many justified payments. One reason for this costly but common practice is that faults are not efficiently corrected after they are detected.
Many systems may be described with mathematical models, e.g., the “scale-free network” depicts electrical power grids (Barabasi and Albert, 1999) and the “random network” depicts transportation networks (Barabasi, 2002; Chen, 2009; Jeong, 2003). Since faulty components are part of a system, mathematical models of systems may be adapted to depict fault networks. Most methods developed earlier exclusively deal with the repair of a single fault (e.g., Dimitrov et al., 2004; Sim and Endrenyi, 1993). Limited research (Ang, 2006; Salmeron et al., 2004) focused on optimal repair sequences. Models of fault networks and fault prevention through repair have not been studied. Previous research (Alizadeh and Sriramula, 2017; Chen, 2009; Chen and Nof, 2012; Sanislav et al., 2018) suggested tools to effectively detect, diagnose, and predict multiple faults (conflicts and errors). Given a network of faults which have been detected, diagnosed, or predicted, this research aims at modeling the fault network and designing the FPR sequencers to prevent faults and minimize the total damage.
The methodology applied in this research is also part of the effort to control network operations through structural search (Dawande et al., 2011). Structural search is a process to search for useful subsets of nodes in a network. For instance, to fight terrorism and prevent epidemic spread through populations with limited resources, critical nodes in a terrorist network or a population must be identified for removal or isolation to disrupt information or disease diffusion. For another example, to promote healthy behaviors in social networks (Parsa and Chen, 2013), a subset of a population, i.e., an influential set of opinion leaders and innovators, needs to be identified to maximize the speed and scale of promotion. The primary goal in structural search is to identify useful structures in networks. In addition, the sequence of operations is of great importance. For example, which node in a terrorist network is removed first and which one is removed next have significant impact on terrorist activities. With regard to an FPR sequencer, the useful structure, i.e., a fault network, is known and comprises all faulty sources and other affected faulty nodes. The sequence of repair, however, needs to be determined. Efficient FPR sequencers designed in this research help advance our understanding of optimizing operations sequences for a useful structure.
There is a need, therefore for efficient FPR sequencing.
At least some embodiments described herein of address the foregoing need by providing an FPR sequencer that uses defined fault networks to determine prevention and/or repair sequences that reduce damage from failures and help prevent the costliest failures.
One embodiment is a non-transitory computer-readable medium encoded with a computer-readable program which, when executed by a processor, will cause a computer to execute a method of managing an electrical smart grid. The method includes acquiring the failures data, the repair resources information, the plurality of failure prevention goals, and the plurality of failure repair goals from the electrical smart grid. The method also includes supervising the electrical smart grid based on such information.
The supervising includes identifying failure networks and structures of the electrical smart grid, and determining failure types associated with the failure networks of the electrical smart grid. The supervising further includes generating a plurality of failure prevention and repair sequences, a plurality of failure prevention goals, and a plurality of failure repair goals. Each failure prevention and repair sequence of the plurality of failure prevention and repair sequences is associated with the failure networks and the failure types. The supervising further includes emulating the plurality of failure prevention and repair sequences.
Such emulation includes using failures data and repair resources information acquired from a memory device system, and analyzing the plurality of failure prevention and repair sequences to determine and select a set of failure prevention and repair sequences and associated repair resources to achieve the plurality of failure prevention goals and the plurality of failure repair goals. The emulation further includes validating the set of failure prevention and repair sequences and the associated repair resources, thereby producing a validated set of failure prevention and repair sequences and validated associated repair resources. The method also includes applying the validated set of failure prevention and repair sequences and the validated associated repair resources to the electrical smart grid, thereby managing the electrical smart grid. The method also includes updating the failure networks and structures, the plurality of failure prevention and repair sequences, and the associated repair resources of the electrical smart grid.
In some embodiments a system includes a failure network modeler, and a sequencer. The failure network modeler generates a failure network model identifying a plurality of nodes associated with at least one failure node, the failure network further includes a plurality of failure nodes. The sequencer is configured to generate an FPR sequence based on the failure network and one or more parameters. The system further causes at least one generated FPR sequence to be displayed or transmitted to a device having a display.
The above described features and advantages, as well as others, will become more readily apparent to those of ordinary skill in the art by reference to the following detailed description and accompanying drawings.
The FPR sequence system 30 includes failure network model generator 32, an FPR sequencer 34, a display 36 and a communication circuit 38. Electronic power conditioning and control of the production and distribution of electricity are important aspects of the smart grid. The failure network modeler 32 generates a failure network model identifying a plurality of nodes associated with at least one failure node. As will be discussed below in detail, the failure network model includes a plurality of failure nodes. The sequencer 34 is configured to generate multiple FPR sequences, each of which is a schedule of the allocation of repair resources to the failure nodes, based on the failure network model and one or more parameters. The system further causes at least one generated FPR sequence to be displayed on display 36 or transmitted via communication circuit 38 to a device having a display, for example, the remote computer device 40, or even back to the control center 22 of the smart grid.
The FPR sequence system 30 may include a general purpose computer or other computing device that executes program instructions to perform the operations of the failure network modeler 32 and the FPR sequencer 34, as well as other operations described further below. To this end the FPR sequence system 30 includes a processing circuit 30a, and memory 30b, among other things. The memory 30b is or includes, among other things, a non-transitory computer-readable medium encoded with a computer-readable program which, when executed by a processor of the processing circuit 30a, will carry out the operations of the failure network modeler 32, the FPR sequencer 34, as well as others as described below in connection with
The embodiment of
Failure networks can be defined using a set of values. The nomenclature of those values and the values used in defining FPR sequences is set forth below in Table 1.
)
)
)
Reference is made to
In general, nodes in a complex system represent machines, equipment, workstations, computers, generators, control units, and other components each of which is modeled as a separate entity. Links between nodes represent the flow of products, services, or information. In the smart grid example of
Referring again to the example of
The number of arcs connected to a faulty node is the degree of the node (Angeles Serrano and De Los Rios, 2007; Dorogovtsev et al., 2001). The IN degree, δIN, of a faulty node is the number of arcs that point at the node. The OUT degree, δOUT, of a faulty node is the number of arcs that originate from the node. Faulty nodes in a network belong to three groups: leaf, internal, and root nodes. A faulty node j is (a) a root node if its fault is not caused by fault(s) at any other faulty node, i.e., siδjIN=0; (b) a leaf node if it does not cause fault(s) at any other faulty node, i.e., δjOUT=0; and (c) an internal node when δjIN>0 and δjOUT>0. A faulty node j is both a root and a leaf node if δjIN=δjOUT=0; this node is an orphan node because it is not connected to any other nodes. A root node requires repair; an internal or a leaf node is repaired or prevented (from having a failure) if and only if all its causes are repaired or prevented.
In
A fault at an internal or leaf node may be partly caused by faults at other faulty nodes and partly due to faults that occur locally. Suppose the fault at node 305 in
Three networks may be defined. Let G (W, ) represent a complex system where W is a set of nodes (vertices) and
is a set of links in the system. |W| is the total number of nodes in W. |W| is an integer and |W|>0. Since faulty nodes are usually linked through arcs, let G (V, A) represent a directed network of faulty nodes in the system where V is a set of faulty nodes and A is a set of arcs. |V| is the total number of faulty nodes in V. |V| is an integer and |V|≥0. V∈W, A∈
, and |V|≤|W|. Let G(VF, AF) represent a directed fault network including pseudo nodes. |VF|≥|V|, VF∩n W=V, and AF ∩
=A. |VF| is an integer and |VF|≥0. There are three types of nodes vj's, vj∈VF: root nodes vr's, internal nodes vi's, and leaf nodes vl's. Let R, I, and L represent a set of root nodes vr's, internal nodes vi's, and leaf nodes vi's, respectively. |R|, |I|, and |L| are integers. |R|≥0; |I|≥0; and |L|≥0. Any FPR sequence must repair all root nodes vr's. vi's and vl's are repaired or prevented if and only if vr's are repaired. Depending on when vr's are repaired, vi's and vl's may be prevented. Time zero, i.e., t=0, is defined to help evaluate the FPR sequences. In practice, the time at which the first fault occurs is often defined as t=0. Let tc represent current time and tj represent the time Vj becomes faulty; tc, tj≥0. Suppose t10<tc<t9 in
COROLLARY 1: In a directed network G (VF, AF) of faulty nodes, tj≤tj′ if (vj, vj′)∈AF. All vi's and vi's, Vi ∈I and Vl∈L, are repaired or prevented if and only if all vr's, Vr ∈R, are repaired. R∪I∪L=VF. |R|≤|VF|, |I|≤|VF|, and |L|≤|VF|. |R|+|I|+|L|≥|VF|.
Many network structures are available and may be used to model complex systems and fault networks. A random network (Erdos and Renyi, 1959; Solomonoff and Rapoport, 1951) follows a degree distribution
wherein n is the total number of nodes, d is the degree of a node or the number of links (arcs or edges) connected to the node, and p is the probability that a pair of nodes are connected. The maximum number of links in a random network is ½n(n−1). The mean degree
Referring now specifically to
Referring again to
In step 206, the FPR sequencer 34 and/or the failure network modeler 32 determines the failure types in the networks. Failure types can, for example, be random failures, cascading failures, cascading failures with backup capacity, and a combination of these failures. As will be discussed below, the failure types affect selection of the FPR sequence.
The FPR sequencer 34 may suitably determine the failures types by analyzing relationships between and among failures in failure networks using the times and locations of failures, determining causation between and among failures, calculating the numbers of failures that occur and will occur due to causation, and determining the portion of nodes in the complex system having failures due to causation and the portion of nodes in the complex system that will have failures due to causation. Examples describing the characterization of failures is discussed further below.
In step 208, the FPR sequencer 34 generates at least one FPR sequence based on the information from steps 202, 204, 206 and further information received from the smart grid control centers 22 such as failure prevention goals and failure repair goals. In the embodiment described herein, the FPR sequencer 34 can generate FPR sequences based on various types of sequencing methods. The FPR sequencing method (also referred to as sequencer) may be distributed or centralized. The centralized FPR sequence method (or sequencer) is noted herein as FPR-C. The decentralized FPR sequencers include that with random selection (FPR-DR), one that minimizes total damage (FPR-DD), and one that maximums preventability (FPR-DP).
In at least some embodiments, the FPR sequencer 34 selects one or more of the sequencers based on the type of fault, the failure prevention goals and the failure repair goals. As discussed below in detail, different sequencers can be more advantageous in certain types of faults when fault repair (i.e. repair cost) is to be minimized. Other sequencers may be more advantageous when fault prevention is to be maximized, and/or another type of fault is present. The selection of sequencers by the FPR sequencer 34 based on these factors evolves over time as the system 10 acquires a knowledge base of sequencing methods, faults, goals, and actual results. In at least one embodiment, after step 208, the FPR sequencer 34 has generated a set of candidate FPR sequences, using one or more the FPR sequencers FPR-C, FPR-DR, FPR-DD, and FPR-DP.
Further detail on the various FPR sequencers that can be executed in step 208 are provided below.
The FPR sequencer 34 may use a centralized FPR sequencer (FPR-C), which repairs one root node at a time. For each root node, the FPR-C compares the required repair resources and available repair resources. If the required repair resources are less than or equal to available repair resources, the root node is repaired; otherwise the root node is not repaired. The FPR-C has the centralized control of repairs and does not employ parallelism (simultaneous repairs of multiple root nodes). The FPR-C is expected to have the worst performance with the lower bounds (maximum D and minimum P) for the performance of all FPR sequencers.
The FPR-C sequencer:
If it is determined to use the FPR-C sequencer in step 208, then the FPR sequencer 34 will perform the above-described operations multiple times to generate a plurality of candidate FPR sequences. In particular, because Step 1 involves selecting a random root node, the FPR-C sequencer can generate multiple different FPR sequences. In some implementations, such as the one discussed below in connection with
The FPR sequencer 34 using a decentralized sequence method generates repair sequences for multiple root nodes at the same time. The number of root nodes which can be repaired simultaneously is subject to available repair resources. There are three types of decentralized FPR sequence methods depending on how root nodes are chosen for repairs and the objective of a FPR sequencer.
The Decentralized FPR Sequencer with Random Selection: FPR-DR
The FPR sequencer 34 may use a decentralized FPR sequencer with random selection (FPR-DR) in step 208. The decentralized FPR-DR sequencer repairs multiple root nodes at the same time. The FPR-DR randomly selects root nodes for repair. When available repair resources are sufficient, the FPR-DR repairs all root nodes at the same time, which provides the best performance, i.e., the upper bound (minimum D and maximum P) for the performance of all FPR sequencers.
If it is determined to use the FPR-DR sequencer in step 208, then the FPR sequencer 34 will perform the above described operations multiple times to generate a plurality of candidate FPR sequences. Accordingly, in step 208 (when the FPR-DR sequencer is used), the FPR sequencer 34 generates instead a statistically significant sample number N of different candidate sequences using the FPR-DR operations described above. In other words, the FPR sequencer 34 generates N candidate sequences by executing the above operations to completion N times.
The FPR sequencer 34 in step 208 may also use the decentralized FPR sequence method (FPR-DD), which aims to minimize D for a fault network. The FPR-DD sequencer guarantees that D is minimized for a fault network comprised of disconnected components, each of which has one root node (LEMMA 1).
Let vr and vr′ represent two root nodes in a fault network G(VF, AF). vr is a direct or indirect cause of total nr leaf nodes vl's, nr>0; there exists at least one path from vr to any vl. All vl's are repaired or prevented if and only if vr is repaired, i.e., any vl is not caused directly or indirectly by any root node other than vr. The damage caused by vl over one time unit is dl. The total damage caused by vl's is
Suppose the difference between the times at which faults occur is much smaller than the repair time for a faulty node, i.e., mr>>|tl/i/r−tl′/i′/r′|. Since t0 represents the time at which the FPR sequence begins repairs and t0≥0, mr+t0>>|tl/i/r−tl′/i′/r′|. Because cl−tl≥mr+t0, cl−tl/i/r−tl′/i′/r′| for ∀vl. The total damage caused by vl's is Σldl(cl−tl). For vr′, the total damage caused by vl′'s is Σl′dl′(cl′−tl′); vr′ is a direct or indirect cause of total nr′ leaf nodes vl′'s, nr′>0. Because the difference between the times at which faults occur is small, tl/i/r≈tl′/i′/r′=t, the total damages caused by vl's and vl′'s are Σldl(cl−t) and Σl′dl′(cl′−t), respectively. If vr is repaired before vr′, i.e., cl−t=mr and cl′−t=mr+mr′, the total damage caused by vl's and vl′'s is nrmr
As with the other sequencers, if it is determined to use the FPR-DD sequencer in step 208, then the FPR sequencer 34 will perform the above described operations multiple times to generate a plurality of candidate FPR sequences. Accordingly, in step 208 (when the FPR-DD sequencer is used), the FPR sequencer 34 generates instead a statistically significant sample number N of different candidate sequences using the FPR-DD operations described above. In other words, the FPR sequencer 34 generates N candidate sequences by executing the above operations to completion N times.
The FPR sequence method (FPR-DP) executed by the FPR Sequencer 34 aims to maximize P for a fault network. The FPR-DP guarantees that P is maximized for a fault network comprised of disconnected components, each of which has one leaf node (LEMMA 2).
Let vl and vl′ represent two leaf nodes in a fault network G (VF, AF). Faults at vl and Vl′ have not occurred and may be prevented. vl is caused by total nl root nodes vr's, nl>0; there exists at least one path from any vr to vl. vl is repaired or prevented if and only if all nl Vr's are repaired. Any vr does not cause faults at other leaf nodes other than vl. Similarly, vl′ is caused by total nl′ root nodes vr′'s, nl′>0; there exists at least one path from any vr′ to vl′. Any vr′ does not cause faults at other leaf nodes other than Vl′.
Repairing root nodes may be able to prevent faults at leaf nodes from occurring. Assume that tl≈tl′>0, i.e., faults at leaf nodes occur at the same time, and mr≈mr′>0, i.e., repair time for any root node is the same. Without losing generality, assume that nl≤nl′. Therefore mrnl≤mr′nl′. mrnl is the minimum required time to repair or prevent faults at vl. mr′nl′ is the minimum required time to repair or prevent faults at vl′. t0 is the time at which the FPR sequence begins repairs; t0≥tr and t0≥tr′. The time at which faults at the leaf nodes occur, tl or tl′, falls into four intervals: tl/l′<mrnl+t0, mrnl+t0≤tl/l′<mr′nl′+t0, mr′nl′+t0≤tl/l′<mrnl+mr′nl′+t0, and tl/l′≥mrnl+mr′nl′+t0.
If tl/l′<mrnl+t0, tl/l′<mr′nl′+t0 because mr′nl′. Neither vl nor vl′ can be prevented. P=0. If mrnl+t0≤tl/l′<mr′nl′+t0, vl′ cannot be prevented. To maximize P, vr's are repaired before the repair of vr′'s.
if all nlvr's are repaired. P=0 if vr′'s are repaired first or a mix of vr's and vr′'s are repaired such that not all nl vr's are repaired by tl/l′. If mr′nl′+t0≤tl/l′<mrnl+mr′nl′+t0, either vl or vl′ can be prevented, but not both.
if all nl vr's are repaired first or all nl′vr′'s are repaired first. P=0 if a mix of vr's and vr′'s are repaired; not all nlvr's are repaired by time tl/l′ and neither are nl′vr′'s. If tl/l′≥mrnl+mr′nl′+t0, both vl and vl′ are prevented and
regardless of the FPR sequence. In summary, repairing all nl vr's first always maximizes P.
This completes the proof of LEMMA 2.
As with the other sequencers, if it is determined to use the FPR-DP sequencer in step 208, then the FPR sequencer 34 will perform the above described operations multiple times to generate a plurality of candidate FPR sequences. Accordingly, in step 208 (when the FPR-DP sequencer is used), the FPR sequencer 34 generates instead a statistically significant sample number N of different candidate sequences using the FPR-DP operations described above. In other words, the FPR sequencer 34 generates N candidate sequences by executing the above operations to completion N times.
Referring again to the operations of
In step 210, the FPR sequencer 34 uses the failures data and the repair resources information acquired from the memory device 30b to analyze the plurality of candidate failure prevention and repair sequences generated in step 208. The FPR sequencer 34 determines and selects a set of failure prevention and repair sequences and, in some cases, associated repair resources, to achieve the plurality of failure prevention goals and the plurality of failure repair goals. The analysis includes determining the damage values D and preventability values P for each FPR sequence generated in step 208.
To this end, the FPR sequencer 34 operates on the basis of an FPR model that defines the goals of the FPR sequence. Since repair cost is the same regardless of the FPR sequence, the two objectives of FPR are to (a) minimize total damage caused by faults; and (b) prevent the maximum number of faults from occurring. The first objective reflects economic consequences of faults and the second objective reflects social impacts of faults. Damage is a financial measure whereas preventing faults indicates service quality. Two metrics are defined to measure the performance of an FPR sequencer: total damage D (D≥0) and preventability P (0≤P≤1). Let cj represent the time vj is repaired or prevented. cj<tj indicates vj is prevented; cj>tj indicates vj is repaired. Let pj indicate whether vj is prevented. pj=0 if cj>tj; pj=1 if cj<tj. Since time is continuous, the probability that cj=tj is zero, i.e., cj≠tj. Preventability P is the percentage of faults which are prevented.
may be expressed in a closed form:
Let dl represent the damage caused by vl over one time unit; dl≥0. The damage caused by vl is dl (cl−tl)(1−pl) assuming that dl is approximately the same over a short period of time cl−tl. The total damage D=Σldl(cl−tl)(1−pl), or δlOUT=0. The objectives of FPR are to minimize D and maximize P. The FPR problem is described as a multi-objective optimization model (Eq. (1)):
cj=max(cj′) indicates that the time at which vj (vi or vl) is repaired or prevented depends on when all its direct causes vj′'s are repaired or prevented. For vr's, cr's are determined by the FPR sequencer 34. The decision variables in Eq. (1) are cr's for vr's, which are times at which root nodes are repaired. ci's for vi's and cl's for vl's are determined by cr's. dl's, |VF|, and tj's including tl's are parameters. A feasible solution to Eq. (1) is an FPR sequence that repairs all vr's. The goal is to identify efficient points, each of which achieves objective function values D and P that are together superior to what can be achieved by all other feasible solutions. Whether an FPR sequence is an efficient point depends on the parameters and topology of fault networks. Both objective functions in Eq. (1) are nonlinear and not differentiable, and constraints are nonlinear. The model in Eq. (2) rewrites Eq. (1) and admits only linear constraints, but the two objective functions remain nonlinear and not differentiable. Heuristic FPR sequencers and simulation experiments need to be developed to identify and validate efficient sequences.
The FPR sequencer analysis of step 210 is illustrated by way of another example of a fault network 400 shown in
In Table 2, multiple subscripts in mr represent the summation of repair times. For instance, m1,2,3,4=m1+m2+m3+m4. Some FPR sequences, e.g., the FPR sequences 1 and 2, have the same D and P. Let a pair of brackets, , represent that a group of FPR sequences have the same D and P. There are 10 such groups in Table 2:
1,2
,
3,4
,
5,6
,
7,8,9,10
,
11,12,13,14
,
15,16
,
17,18
,
19,20
,
21,22
, and
23,24
. Table 3 shows the comparison for one node 6 between two groups
1,2
and
3,4
. The group
3,4
causes more damage and has smaller preventability than
1,2
.
1,2
is better than
3,4
in terms of both D and P, which can be expressed as
1,2
≥
3,4
. Other comparisons among the 10 groups show that
1,2
≥
5,6
,
1,2
≥
11,12,13,14
,
15,16
≥
3,4
,
5,6
≥
11,12,13,14
,
21,22
≥
7,8,9,10
,
23,24
≥
7,8,9,10
,
17,18
≥
19,20
,
23,24
≥
19,20
, and
23,24
≥
21,22
. Total eight out of 24 FPR sequences, or four out of 10 groups of FPR sequences,
1,2
,
15,16
,
17,18
, and
23,24
, have better performance in D and P than other FPR sequences. Depending on the values of dl's, mr's, t0, and tl's, one or more of the eight FPR sequences minimize D and maximize P. This example clearly indicates that the optimal FPR sequence is determined by the structure of a fault network and parameters in FPR (Eqs. (1) and (2)). Four FPR sequencers are developed to produce various FPR sequences and they are illustrated in the next two sections.
1,2
and
3,4
.
3,4
-
1,2
The result from step 210 is that a subset of the candidate FPR sequences developed in step 208 appear to best achieve certain goals, such as minimizing D, maximizing P, or striking a desired balance between the two, based on the stored failure prevention goals and failure damage goals.
Referring again to
A plurality of simulations or “experiments” are run on each candidate FPR sequence. Each of the experiments varies information that which would not be known already to the FPR sequencer 34 and FPR modeler 32. Such information can include which nodes have failures, when failures occur, total repair resources, repair resources required for each failure, and the time it takes to repair a failure. By way of example, during actual execution of the FPR sequence on the smart grid 20 of
The results of each simulation can be generated as a graph, as is typical in Monte Carlo simulations.
The FPR sequencer 34 then provides the generated FPR sequence(s) to the display 36 for display, and/or to the communication circuit 38. The communication circuit 38 may then communicate the information to any external computing device 40 (or control center 22). Thereafter, in step 214, the control center 22 and/or computing device 40 can apply the validated set of FPR sequences and the repair resources on the smart grid 20. For example, the control center can display the generated sequence(s) and repair units can be dispatched to failure nodes of the smart grid 20 based on one of the generated sequence(s).
The FPR sequence system 30 and/or the control center 22 thereafter, in step 216, updates the failure networks and structures, the plurality of failure prevention and repair sequences, and the associated repair resources of the smart grid. To this end, the results of applying the FPR sequences in step 214 are reviewed and recorded as part of the updates.
To compare and validate the FPR sequencers for efficacy in different failure modes, Monte-Carlo simulation experiments (Nasiruzzaman et al., 2014) are designed and conducted using AutoMod (Applied Materials, 1988-2009). Many real-world complex systems may not satisfy conditions in LEMMA 1 or LEMMA 2. The objectives of the experiments are to examine whether (a) the FPR-C results in the highest total damage D and lowest preventability P; (b) the FPR-DD minimizes D; (c) the FPR-DP maximizes P; and (d) the FPR-DR performs better than the FPR-C but worse than the FPR-DD and FPR-PP.
Whether the FPR-DD can minimize D and the FPR-DP can maximize P depend on conditions in LEMMA 1 and LEMMA 2. In LEMMA 1, it is assumed that (a) each leaf node, except the orphan node, has only one root node; (b) nodes become faulty at almost the same time; (c) damage caused by failures at each leaf node is approximately the same; and (d) repair resources, e.g., repair personnel, required for each root node is approximately the same. In LEMMA 2, it is assumed that (a) each root node, except the orphan node, has only one leaf node; (b) all leaf nodes become faulty at almost the same time; and (c) repair resources, e.g., repair personnel, required for each root node is approximately the same.
The structure of a fault network is derived from the structure of a complex system, and rarely satisfies condition (a) in either LEMMA 1 or LEMMA 2. In general, a root node may have multiple leaf nodes and a leaf node may have multiple root nodes. Condition (b) in LEMMA 1 and LEMMA 2 specifies the type of failures in a complex system. Three types of failures, random, cascading, and cascading with backup capacity, are studied in the experiments. Most random failures are independent of each other and occur over a long period of time. Random failures do not satisfy condition (b) in either LEMMA 1 or LEMMA 2. A cascading failure in a complex system (Nedic et al., 2006) occurs in a relatively short period of time and includes multiple faults most of which are caused by a few faulty sources (root nodes in a fault network; Hoffmann and Payton, 2014). A cascading failure satisfies condition (b) because nodes become faulty almost at the same time. A cascading failure with backup capacity does not satisfy condition (b) since leaf nodes become faulty at different times depending on the amount of backup capacity each leaf node has. Conditions (c) and (d) in LEMMA 1 and condition (c) in LEMMA 2 are valid for many complex systems. A fault network's properties along the four conditions in LEMMA 1 and LEMMA 2 determine the structure of the fault network.
In each experiment, a fault network is first generated; an FPR sequencer is used to generated an FPR sequence, which is emulated to prevent and repair faults. D and P are calculated for each experiment. The experiments use the electrical power grid of the Western United States (Watts and Strogatz, 1998), which has 4,941 nodes including generators, transformers, and substations. In a simulation experiment, each node has 0.1 probability of becoming faulty. Resources required to repair failures at a root node are assumed to be randomly and uniformly distributed between 3 and 10 units. For example, to repair faults at a node may require a crew of 6 people, i.e., 6 units of resources. The damage per second caused by faults at a node is randomly and uniformly distributed between $5 and $15. Each simulation experiment emulates an FPR sequence for 24 hours.
Total repair resources affect the performance of FPR sequencers. The FPR-C repairs one root node at a time. Since the maximum amount of resources needed to repair a root node is 10 units, total repair resources for the FPR-C are 10 units, which are sufficient for the repair of any root node. The decentralized FPR sequencers, FPR-DD, FPR-DP, and FPR-DR repair multiple root nodes at the same time. The electrical power grid of the Western United States has 4,941 nodes and each nodes has 0.1 probability of having faults in the experiments. There are on average 494 nodes that become faulty in an experiment. Since only root nodes, including orphan nodes that are both root and leaf nodes, require repair, different levels of total repair resources are applied in the experiments according to the number of root nodes.
Many complex systems have random failures most of which occur independent of each other. In the simulation experiments, the time at which random failures occur is uniformly distributed between 0 and 86,400 seconds (24 hours=86,400 seconds). One-hundred experiments are conducted for each combination of an FPR sequencer and a certain amount of total repair resources, which are 10 units for the FPR-C. It is necessary to determine the maximum required total repair resources (MRT), which is the amount of resources sufficient to repair all root nodes simultaneously. The simulation experiments show that the maximum number of root nodes with random failures is 523 with a mean of 464 and a standard deviation of 19. Since a root node requires at most 10 units to repair, the MRT for random failures is about 5,000 units. For scalability evaluation, 14 levels of total repair resources are used in the experiments for each of the FPR-DD, FPR-DP, and FPR-DR: 10, 50, 100, 200, 300, 400, 500, 1,000, 1,500, 2,000, 2,500, 5,000, 7,500, and 10,000 units. A decentralized FPR sequencer is expected to have the best performance when total repair resources are at or greater than the MRT. The two levels, 7,500 and 10,000 units, are included in the experiments to validate the best performance of an FPR sequencer. Total 4,300 experiments (100 experiments for FPR-C+3 decentralized FPR sequencers×14 levels of total repair resources×100 experiments) are conducted to compare and validate the performance of FPR sequencers for random failures.
Table 4 summarizes experiment results, which provide several important findings for managing random failures:
A cascading failure may occur within a few minutes to a few hours (Andersson et al., 2005). For example, major failures in the U.S.-Canadian blackout of Aug. 14, 2003 occurred in less than an hour. In the simulation experiments, the time at which faults as part of a cascading failure occur is uniformly and randomly distributed between 42,300 and 44,100 seconds (43,200±900), i.e., faults occur within 30 minutes. The simulation experiments show that the maximum number of root nodes is 94 with a mean of 66 and a standard deviation of 9. Since a root node requires at most 10 units to repair, the MRT for a cascading failure is about 1,000 units. For scalability evaluation, 10 levels of total repair resources are used in the experiments for each of the FPR-DD, FPR-DP, and FPR-DR: 10, 50, 100, 200, 300, 400, 500, 1,000, 1,500, and 2,000 units. Total 3,100 experiments (100 experiments for FPR-C+3 decentralized FPR sequencers×10 levels of total repair resources×100 experiments) are conducted to compare and validate the performance of FPR sequencers for cascading failures.
Table 5 summarizes experiment results, which provide several important findings for managing cascading failures:
Cascading Failures with Backup Capacity
Many critical nodes in a complex system have backup capacity in case of failures. For example, consumers in a smart grid can have backup power that provides uninterrupted power supply when there is a random failure or a cascading failure. Backup power may be fueled by gasoline, diesel, propane, natural gas, battery, and other energy sources. Some provide protection against failures and others require a short period of time, for example, 30 seconds, to resume power supply. Backup power may last for a few minutes to a few days depending on its capacity and power usage. In theory, some generators provide an endless electricity supply using natural gas from the utility company. In practice, however, these generators require periodical maintenance, for example, replacing engine oil, or cooling. There is always a limit on how long backup power can continuously supply electricity.
In the simulation experiments, the time at which root and internal nodes become faulty is uniformly and randomly distributed between 42,300 and 44,100 seconds, which is the same for root and internal nodes in a cascading failure without backup capacity (See above). The time at which leaf nodes become faulty is uniformly and randomly distributed between 46,800 and 86,400 seconds, i.e., leaf nodes with backup power become faulty approximately between 1 hour and 12 hours after their corresponding root nodes become faulty. The simulation experiments show that the maximum number of root nodes is 102 with a mean of 66 and a standard deviation of 9. Since a root node requires at most 10 units to repair, the MRT for a cascading failure with backup capacity is about 1,000 units. For scalability evaluation, 10 levels of total repair resources are used in the experiments for each of the FPR-DD, FPR-DP, and FPR-DR: 10, 50, 100, 200, 300, 400, 500, 1,000, 1,500, and 2,000 units. Total 3,100 experiments (100 experiments for FPR-C+3 decentralized FPR sequencers×10 levels of total repair resources×100 experiments) are conducted to compare and validate the performance of FPR sequencers for cascading failures with backup capacity.
Table 6 summarizes experiment results, which provide several important findings for managing cascading failures with backup capacity:
Four fault prevention and repair sequencers, including a centralized sequencer, FPR-C, and three decentralized sequencers, FPR-DD, FPR-DP, and FPR-DR, are developed to sequence the prevention and repair of three different types of faults in a complex system, including random failures, cascading failures, and cascading failures with backup capacity. The FPR-DD aims to minimize total damage caused by faults. The FPR-DP aims to maximize preventability, the percentage of faults prevented from occurring. The FPR-DR randomly selects faults for simultaneous repairs. All four FPR sequencers are implemented in a software program to generate FPR sequences, and compare and validate their performance. The electrical power grid of the Western United States is studied in total 10,500 experiments to examine the performance of the four FPR sequencers. Results show that either the FPR-DD or FPR-DP should be used to prevent and repair faults in complex systems; both sequencers minimize total damage and maximize preventability.
Total repair resources affect the performance of three decentralized FPR sequencers. Total repair resources have different thresholds for different types of failures and performance metrics. Below a threshold, increasing total repair resources improves the performance of an FPR sequencer. Above the threshold, increasing total repair resources does not further improve the performance of the FPR sequencer. The threshold of repair resources is measured as a percentage of repair resources sufficient to simultaneously repair all root nodes, which are root causes and must be repaired directly using repair resources. Below is a summary of thresholds for three types of failures:
The experiments results show that applying the FPR-DD and FPR-DP results in almost the same total damage and preventability, although the FPR-DD aims to minimize total damage and the FPR-DP aims to maximize preventability. The FPR-DD is developed based on LEMMA 1, which assumes that any fault has at most one root cause. The FPR-DP is developed based on LEMMA 2, which assumes that a root cause only causes at most one faulty leaf node. In real-world complex systems, these two assumptions are hardly true. A root cause may cause multiple faulty leaf nodes whereas a faulty leaf node may be caused by multiple root causes. This is a reason that the FPR-DD and FPR-DP perform almost the same. On the other hand, both perform better than the FPR-C and FPR-DR; the latter is also a decentralized FPR sequencer that randomly selects root nodes for repairs. This finding suggests that parallelism (simultaneous repairs of multiple root nodes) and FPR sequencers that take advantage of the structure of a fault network help improve the performance of FPR sequencers. Future research may develop other FPR sequencers and experiment with additional complex systems to further identify how different FPR sequencers perform in different systems.
Total repair resources significantly affect the performance of FPR sequencers up to a point. Increasing repair resources improves the performance of FPR sequencers until the amount reaches a threshold. To maximize preventability requires less resource than minimizing total damage. This observation provides an important insight for managing faults in complex systems. For instance, in a transportation system with ongoing traffic problems in certain areas, a primary objective is to prevent congestions in other areas. To resolve each traffic problem as they occur may reduce damage but may not be necessary to prevent congestions elsewhere. As long as repair resources are sufficient to resolve a certain percentage of all traffic problems (69.93% if it is a cascading failure and 3.32% if traffic problems are random failures and mostly independent), any of the three decentralized FPR sequencers can prevent the maximum number of congestions from occurring.
Another important insight regarding random failures is that only a small fraction of maximum required total repair resources, 6.63% based on the simulation experiments, is needed to minimize total damage and maximize preventability. A cascading failure may be catastrophic but it may happen relatively rarely. Most complex systems routinely experience random failures that occur sporadically over a long period of time. The simulation experiments indicate that their long term damage is higher than that of cascading failures, although the latter attract much more attention to the public for their broad impacts. It is not true that more repair resources always reduce damage caused by faults and prevent more faults from occurring. Random failures may be effectively and efficiently managed using the FPR-DD or FPR-DP with a relatively small amount of repair resources.
The simulation experiments use the electrical power grid of the Western United States to obtain the thresholds for repair resources. Additional experiments may be conducted in the future to fine tune the thresholds with more inputs from the system. Other complex systems may have different threshold values and may also be studied in the future.
This application claims the benefit of U.S. provisional patent application Ser. No. 62/942,259, filed Dec. 2, 2019, which is incorporated in its entirety herein by reference.
| Filing Document | Filing Date | Country | Kind |
|---|---|---|---|
| PCT/US2020/062887 | 12/2/2020 | WO |
| Number | Date | Country | |
|---|---|---|---|
| 62942259 | Dec 2019 | US |
