Abstract
Most existing distribution networks are difficult to withstand the impact of meteorological disasters. With the development of active distribution networks (ADNs), more and more upgrading and updating resources are applied to enhance the resilience of ADNs. A two-stage stochastic mixed-integer programming (SMIP) model is proposed in this paper to minimize the upgrading and operation cost of ADNs by considering random scenarios referring to different operation scenarios of ADNs caused by disastrous weather events. In the first stage, the planning decision is formulated according to the measures of hardening existing distribution lines, upgrading automatic switches, and deploying energy storage resources. The second stage is to evaluate the operation cost of ADNs by considering the cost of load shedding due to disastrous weather and optimal deployment of energy storage systems (ESSs) under normal weather condition. A novel modeling method is proposed to address the uncertainty of the operation state of distribution lines according to the canonical representation of logical constraints. The progressive hedging algorithm (PHA) is adopted to solve the SMIP model. The IEEE 33-node test system is employed to verify the feasibility and effectiveness of the proposed method. The results show that the proposed model can enhance the resilience of the ADN while ensuring economy.
2023.
IN recent years, the frequent occurrence of extreme weather events has battered the power system, causing customers to experience different degrees of power outages after damaging the power grid infrastructure, and resulting in enormous economic losses [
As the ADN is still vulnerable to natural disasters, ensuring the rapidity and effectiveness of fault restoration has become a research priority in the power industry [
In the PLS, taking precautionary measures such as hardening lines [
In contrast, the two-stage programming divides the planning cycle into different stages and solves the optimal solution of the model according to the growth of different factors during the planning cycle. For the two-stage programming problem of enhancing the resilience of ADNs, more uncertainties in distribution lines and loads caused by disastrous weather events need to be considered. Considering the shorter time scale of the emergency support process compared with the restoration process, uncertainties in the operation state of lines, the load level and the initial charge state of ESSs can all have impacts on the outcome of dispatch implementation. Robust optimization and stochastic programming models are widely used. In terms of robust optimization, [
At present, the planning problem with a single stage or a certain scenario is generally a small-scale stochastic mixed-integer programming (SMIP) model, which can be directly solved by optimization solvers such as Yalmip or Gurobi. The multi-stage planning problem is generally a large-scale SMIP model, which is difficult to directly call the solver to calculate. Robust optimization is generally solved by Benders decomposition algorithm or column-and-constraint generation (C&CG) algorithm. The primal problem is decomposed into the main problem and the sub-problem to obtain the optimal solution of the primal problem alternately [
Therefore, in order to enhance the resilience of ADNs, this paper establishes a two-stage SMIP model to minimize the upgrading and operation cost of ADNs by considering random scenarios referring to different operation states of ADNs. In the first stage, the system planner makes investment decisions: hardening existing distribution lines, upgrading automatic switches, and deploying energy storage resources. The second stage is to evaluate the operation cost of ADNs to ensure the supply of loads after network reconfiguration during meteorological disasters and the optimal utilization of ESSs in different scenarios. Besides, a new modeling method is proposed to address the uncertainty of operation state of distribution lines. Based on historical statistics, multiple representative scenarios are generated by Latin hypercube sampling, and the similar scenarios are reduced by adopting a K-means clustering method. The PHA is used to solve the proposed model. The contributions of this paper are outlined below.
1) The best access point of the ESS in different operation scenarios are comprehensively considered to exert its value as much as possible by dividing the operation scenarios of ADNs into normal, severe, and extreme weather conditions. In normal scenarios, the ESS is used to improve the operation economics of ADNs. During the fault recovery period, the ESS is considered as the emergency power supply to the critical loads through network reconfiguration; and in the PLS, the safety support of ESSs is played combining with the results of the risk assessment of ADNs. Compared with existing planning models that only consider extreme scenarios, the proposed model not only enhances the resilience of ADNs during meteorological disasters, but also balances safety and economy.
2) In the proposed model, a modelling method for describing the uncertainty of the operation state of distribution lines is constructed to handle the logical constraints among variables such as the initial state of distribution lines, the existence of automatic switches on the line, the state of switches, and the fault state of lines in a certain scenario. The proposed method reduces the multivariate coupled problem of operation state of distribution lines to a linear model in a canonical representation, which makes the proposed model simpler and easier to solve.
The remainder of this paper is organized as follows. Section II classifies the operation scenarios of ADNs. Section III describes the two-stage SMIP model. Section IV generates the scenarios of ADNs and introduces the solution algorithm of the proposed model. Section V utilizes the modified IEEE 33-node test system to verify the proposed model. Finally, Section VI concludes this paper.
For the operation analysis of ADNs, the short-term risk level in the future is often predicted by obtaining the external operation environment information and combining with the operation parameters of lines, which is called predictive evaluation. For the planning analysis, the estimated value can be obtained after the data from long-term operation records are analyzed, which is called statistical analysis. In this paper, the ESS is applied to the planning model to improve the flexibility of the ADN, and historical statistical data are applied to the typical operation scenarios of the ADN to simplify the failure rate model [
The main causes for the failure of ADNs are insulation damage, external force damage, and natural disasters. Among natural disasters, mountain fires, thunderstorms, and typhoon have greater impacts on ADNs. Therefore, this paper divides the operation scenarios of ADNs into normal, severe, and extreme weather conditions. ADN is in a normal operation state under normal weather condition, and in a fault operation state in response to meteorological disasters under severe and extreme weather conditions. Different operation states of ADNs throughout the year for comprehensive planning are considered in this paper.
The energy storage technologies currently used in power grids can be divided into four categories: electrical energy storage, mechanical energy storage, chemical energy storage, and thermal energy storage [
ESS can effectively realize the conversion, storage, and utilization of electrical energy, and it is a kind of critical means to improve the flexibility, economy, and safety of the power grid [
To simplify the model, this paper mainly considers the benefits of the ESS from peak shaving and valley filling under normal weather condition:
(1) |
Under meteorological disasters, the resistance and recovery process of the ADN [

Fig. 1 Resistance and recovery process of ADN.
There are two special problems of ESS planning when considering the meteorological disasters. First, compared with distributed power generation, ESS planning has strong time coupling constraints, so its modeling is more complex. Second, the limited investment budget will limit the extensive allocation of ESSs. An ESS can provide a short-term emergency response to pick up the load and reduce economic losses during the fourth stage. After an ADN enters infrastructure recovery, the grid maintenance personnel will take recovery measures such as transferring supply, and ESSs may not be able to continue discharging due to its own capacity limitations, so this paper only considers the participation of ESSs in the emergency response state.
Aiming at responsiveness and catastrophic severity in different states during the resilience recovery process, the evaluation index, i.e., load loss rate (LLR) [
(2) |
The objective of this paper is to improve the benefits of ESSs in terms of peak shaving and valley filling and to reduce the cost following two aspects: ① investment cost of hardening distribution lines, upgrading automatic switches, and deploying ESSs; and ② the cost of load shedding in response to extreme disasters during the emergency period.
Correspondingly, the objective function of the planning model (in unit of year) is as follows:
(3) |
(4) |
(5) |
The cost of ESSs includes equipment deployment cost and annual operation maintenance cost , as shown in (6)-(8). The equipment deployment cost of ESSs is related to the configured capacity and the maximum discharge power.
(6) |
(7) |
(8) |
Since the operation conditions of ADNs are divided into different scenarios, the annual operation cost of ADNs is the lowest sum of the operation cost in different scenarios, as shown in (9). The operation cost of ADNs in scenario s is the cost of load shedding minus the benefits of ESSs in terms of peak shaving and valley filling in (10).
(9) |
(10) |
Based on the failure rate of historical statistics, the fault state of distribution lines in a scenario is obtained by sampling. It is worth noting that hardening existing distribution lines can only reduce the failure rate, but cannot fully ensure normal operation. To manage the risk in a more realistic way, when a line is hardened, the failure rate is assumed to be 1/10 of that before the hardening. The uncertainty of line fault state is decoupled by two independent parameters, which can be generated in advance to represent the line fault state weather it is hardened.
(13) |
At present, most ADNs are not fully automated, that is, not all lines have automatic switches. In addition, the tie line of ADNs does not participate in the normal operation. The initial state of the distribution line, whether there is an automatic switch on the line, and the fault state of the line in a certain scenario will affect the final operation state of the line, which makes it difficult to model the actual state of the line. In order to solve this problem, this paper establishes the logical relationship between variables, as shown in the
1 | 1 | 1 | 0 | 0 |
0 | 0 | 0 | ||
0 | 1 | 1 | ||
0 | 1 | 0 | ||
0 | 1 | |||
0 | 1 | 1 | 0 | 0 |
0 | 0 | 0 | ||
0 | 1 | 1 | ||
0 | 1 | 0 | ||
0 | 0 |
(14) |
(15) |
(16) |
(17) |
(18) |
(19) |
If a fault occurs on a normal line, the line will be out of service. If no fault occurs, the actual running state of the line is determined by the state of the line switch, i.e., if the switch is closed, the line is in operation, and if the switch is open, the line is out of service. If there is no switch on a normal line, the line is out of service when a fault occurs. The above results correspond to all cases () in
(22) |
Since (22) is a nonlinear constraint, the linearization method [
(23) |
(24) |
(25) |
(26) |
(30) |
(31) |
(32) |
(33) |
(34) |
(35) |
(36) |
Constraints (30)-(36) always ensure the radiality of the ADN, and a specific explanation can be found in [
Constraint (38) limits the inability of ESSs to be charged and discharged simultaneously. Constraints (39) and (40) limit the charging and discharging rates of ESSs, respectively. Constraints (41) and (42) are the charging state constraints of ESSs. Constraints (43) and (44) are the reactive power constraints of ESSs. Constraint (45) limits the initial charging state of ESSs.
(38) |
(39) |
(40) |
(41) |
(42) |
(43) |
(44) |
(45) |
is 0.5 under normal weather condition. It is assumed that the SOC of ESSs can be adjusted before a disaster occurs.
Due to the uncertainty of faulty lines and load state, the SMIP model proposed in this paper is an optimization problem that contains all random scenarios. To ensure a balance between computational accuracy and efficiency, this paper first obtains fault probabilities of lines based on historical statistics, then adopts Latin hypercube sampling to generate representative operation scenarios, and finally uses K-means clustering to reduce the scenarios.
In this paper, a large number of scenarios of ADNs are generated by sampling. The scenario generation considers two uncertain factors.
Based on historical statistical load data, the load data of typical days are selected as the base load profiles. Assume that all nodes adopt the same typical load curve, which can be expressed as follows:
(52) |
(53) |
The failure rates of ADNs are determined based on time and weather-dependent fault statistics [
(54) |
The calculation method of typical daily failure rate under different weather conditions is as follows:
(55) |
After calculating , the fault state of the line is obtained according to the sampling result. That is, if a random number between 0 and 1, i.e., , , otherwise, ; if , , otherwise, .
Among the scenarios generated by sampling, the number of some scenarios is small while the number of other scenarios is large. First, the failure rate of lines is low under normal weather condition, so most of normal scenarios are exactly the same. Second, the fault locations in severe and extreme scenarios may be the same, and even if the fault locations are different, the unsupplied loads obtained by the second-stage optimization are close to each other. Therefore, this paper adopts K-means clustering to reduce the scenarios [
The unsupplied load at node i can be obtained by solving the two-stage problem in scenario s, as shown in (56). Then, the unsupplied load vector can be expressed as (57).
(56) |
(57) |
K-means clustering aims to partition the NS-dimensional vector into k (k<NS) sets so as to minimize the within-cluster sum of distance :
(58) |
is the function of calculating variance. After the clustering, if a set contains multiple scenarios, one scenario among can be randomly selected. The probability of this representative scenario is the sum of all scenarios within this cluster:
(59) |
(60) |
(61) |
The value of k is 20-40 with a step of 5. When k satisfies (61), k0 be regarded as a proper value because it indicates that σ(k) begins to saturate. Eventually, the reduced k scenarios are fed back into the SMIP problem.
The two-stage SMIP model established in this paper can be expressed as follows:
(62) |
s.t.
(63) |
x implicitly implements the non-anticipative constraints that avoid allowing decisions to depend on the scenario. By introducing copies of x, the block-angular structure leads to the so-called scenario formulation of the SMIP model:
(64) |
s.t.
(65) |
represents the non-anticipative constraint, which guarantees the first-stage decision vector x independent of scenarios. Finally, this scenario formulation ((64) and (65)) decomposes the large-scale SMIP problem into scenario subproblems with the non-anticipative constraints. The PHA is used to solve the large-scale SMIP problem due to the uncertainty of multi-scenario. The steps are as follows.
Step 1: .
Step 2: for all , .
Step 3: .
Step 4: for all , .
Step 5: .
Step 6: for all ,
Step 7: .
Step 8: for all , .
Step 9: .
Step 10: if , where is the termination threshold, go to Step 5; otherwise, terminate.
The size of ρ in the PHA directly affects the convergence and solution speed of the model [
The test is conducted based on the modified IEEE 33-node test system to verify the validity of the proposed model, using MATLAB R2018b with YALMIP toolbox on a computer with an Intel Core i5-8400 processor and 16 GB of memory. The SMIP model is solved by Gurobi 9.5.1. The mixed-integer programming (MIP) gap is set to be 1%. The single time span is set to be 15 min.
The modified IEEE 33-node test system is shown in

Fig. 2 Modified IEEE 33-node test system.
Measure | Candidate position | Value |
---|---|---|
Hardening lines | All line sections | |
Upgrading automatic switches | Pre-selected lines | |
Deploying ESSs | All nodes | , , |
It is assumed that there are 300 typical days of normal weather condition, 10 occurrences of severe weather condition, and 5 occurrences of extreme weather condition throughout the year. Each occurrence of the severe and extreme weather conditions may last for 1 to 3 days. Since this paper discusses the emergency response period under disastrous weather conditions, a typical day is considered to represent the severe and extreme weather conditions, respectively. Besides, the emergency period is assumed as 2 hours during a fault.
The load curves of each typical day under different weather conditions are shown in

Fig. 3 Typical load curves of each typical day under different weather conditions.
Session | Time | Tariff (¥/kWh) |
---|---|---|
1 | 00:00-08:00 | 0.3377 |
2 | 08:00-14:00 | 0.6648 |
17:00-19:00 | ||
22:00-24:00 | ||
3 | 14:00-17:00 | 1.0900 |
19:00-22:00 |
Since the number of scenarios for solving the proposed SMIP model should be at least 50 [
Normal weather | Severe weather | Extreme weather | |||
---|---|---|---|---|---|
Number of scenarios | Number of faulty lines | Number of scenarios | Number of faulty lines | Number of scenarios | Number of faulty lines |
32 | 0 | 8 | 2 | 7 | 7 |
18 | 1 | 10 | 3 | 9 | 8 |
14 | 4 | 13 | 9 | ||
13 | 5 | 11 | 10 | ||
4 | 6 | 9 | 11 | ||
1 | 7 | 3 | 12 |
As failure rates of distribution lines are low under normal weather condition, there is no faulty line in most scenarios. Even if a fault occurs on the line, the ADN could operate normally by operating the automatic switch to disconnect the fault and resupply the load, so 50 scenarios under normal weather condition are simplified into one scenario, i.e., the normal operation of the ADN in a typical day (24 hours). However, the number of faulty lines is large under severe and extreme weather conditions, and the ADN cannot solve the load shedding problem by itself, so ESSs can be considered for emergency support. But considering the capacity limitation of ESSs, this paper only considers the emergency response period before grid maintenance.
After 50 failure scenarios for severe and extreme weather conditions are clustered, respectively, the sensitivity of within-cluster sum of distance for different distances k is shown in

Fig. 4 Sensitivity of within-cluster sum of distance.
The penalty factor of PHA is set to be 10000, being slightly smaller than . After 87 iterations, the planning results of the proposed method are successfully obtained, as shown in
Cost (¥) | Cost of load shedding (¥) | Benefit of ESSs (¥) | ||
---|---|---|---|---|
Hardening lines | Upgrading automatic switches | Deploying ESSs | ||
840000 | 42400 | 519840 | 205977 | 1092711 |

Fig. 5 Optimal results of proposed method.
The results show that the first measure involves an investment cost of ¥840000 to harden distribution lines L1-2, L2-3, L4-5, L7-8, L16-17, L2-19, L19-20, L3-23, L23-24, and L24-25; the second measure involves an investment cost of ¥42400 to deploy automatic switches on L9-15, L12-22, L18-33, and L25-29; and the third measure involves an investment cost of ¥519840 to deploy ESSs at nodes 4, 7, 13, 18, 31, and 32. Over the course of a year, the third measure benefits ¥1092711 through peak shaving and valley filling under the normal weather condition. Due to disastrous weather conditions, the number of faulty lines is higher, so even though the investment cost is higher, the emergency supply of load needs to be picked up as much as possible after the lines are hardened and automatic switches are upgraded. After the three measures are planned, the cost of load shedding for the year is ¥205977.

Fig. 6 Planning results of structures involving reconfiguration during failure period for scenario-9 and scenario-16. (a) Scenario-9 under severe weather condition. (b) Scenario-16 under extreme weather condition.
In scenario-9, L2-3 and L19-20 are no longer faulty after line hardening, only L13-14, L27-28, and L29-30 are faulty, and the automatic switches on lines L25-29, L9-15, and L18-33 have been upgraded and are able to close quickly to participate in the operation of the distribution network, so that all loads are recovered. In scenario-16, L2-19, L3-23, and L24-25 are no longer faulty after line hardening, but there are still 6 lines, i.e., L9-10, L14-15, L20-21, L26-27, L28-29, and L32-33, that are faulty and the automatic switches on L12-22 and L18-33 are closed and eventually most of the loads are recovered. It can be observed from

Fig. 7 SOC of ESS3 in scenario-16.
Case 1: only hardening distribution lines and upgrading automatic switches without ESSs.
Case 2: hardening distribution lines and upgrading automatic switches responding to extreme weather condition, and deploying ESSs only at critical load nodes.
Case 3: using the proposed method, i.e., considering hardening distribution lines, upgrading automatic switches, and deploying ESSs.
The optimal planning results of the three cases are shown in
Case | Objective | Cost (¥) | Cost of load shedding (¥) | Benefit of ESSs (¥) | ||
---|---|---|---|---|---|---|
Hardening lines | Upgrading automatic switches | Deploying ESSs | ||||
1 | 2463256 | 2184000 | 53000 | 186256 | ||
2 | 794794 | 924000 | 53000 | 433200 | 295186 | 910592 |
3 | 515506 | 840000 | 42400 | 519840 | 205977 | 1092711 |

Fig. 8 Optimal planning results of Case 1.
It can be observed that if only hardening distribution lines and upgrading automatic switches are considered, although the cost of load shedding can be effectively reduced and the resilience of the ADN can be enhanced, the overall investment cost is larger without ESSs. Therefore, the planning results are less economical. The optimal planning results of Case 2 are shown in

Fig. 9 Optimal planning results of Case 2.
This paper proposes a novel two-stage SMIP model to enhance the resilience of ADNs in view of the fact that the existing distribution network is not able to withstand the damage caused by disastrous weather events. The first stage is to make decisions according to the measures of hardening existing distribution lines, upgrading automatic switches, and deploying ESSs. The second stage is to evaluate the operation cost of ADNs considering the cost of load shedding due to disastrous weather and the benefits of ESSs under different weather conditions. Logical constraints are formulated among variables such as the initial state of distribution lines, the existence of automatic switches on the line, the state of switches, and the fault state of lines in a certain scenario to portray the actual operation state of the line. This paper divides the year-round operation environment of ADNs into normal, severe, and extreme weather conditions. As an important resource to participate in enhancing the resilience of ADNs, the ESS can benefit from peak shaving and valley filling under the normal weather condition. Under severe and extreme weather conditions, the ESS can also ensure the continuous power supply for loads as much as possible. Overall, the ESS can improve the economy and security of ADNs under normal, severe, and extreme weather conditions. This paper exploits the potential of the ESS in terms of safety support, and balances its safety and economy to provide emergency power supply for loads. The modified IEEE 33-node test system is employed to verify the feasibility of the proposed method.
This paper proposes a new idea for ESS planning that can be applied to ADNs where disastrous weather occurs frequently to enhance the resilience. However, due to the limited scale of ESS, other flexible resources of ADNs, such as microgrids and flexible loads, can be considered in the future research.
Nomenclature
Symbol | —— | Definition |
---|---|---|
A. | —— | Indices |
—— | Line index | |
—— | Node indexes | |
—— | Resilience level of active distribution networks (ADNs) | |
—— | Resilience level of ADNs in resilient state | |
—— | Resilience level of ADNs in post-event degraded state | |
—— | Scenario index | |
—— | Time index | |
B. | —— | Sets and Matrices |
—— | Mean of points in | |
—— | Set of unsupplied load vectors | |
—— | Scenario of | |
—— | Cost coefficient vector | |
—— | Associated cost coefficient vector | |
—— | Set of lines | |
—— | Set of nodes | |
—— | Set of problem constraints | |
—— | Set of scenarios | |
SW | —— | Set of lines with switches |
—— | Set of time spans | |
x | —— | Vector of the first-stage decisions must be made before the scenario is known |
ys | —— | Vector of decisions made after the first stage, or as a result of the scenario realization |
C. | —— | Parameters |
, | —— | Charging and discharging efficiencies of energy storage systems (ESSs) |
—— | Recovery factor of ESSs | |
—— | Electricity price during time t | |
—— | Power factor of ESSs | |
λij | —— | Failure rate of line |
—— | Failure rate under weather condition a in year x of line | |
—— | Load level uncertainty for a stochastic scenario, which is assumed to follow a normal distribution | |
—— | Total annual operation cost of ADNs | |
—— | Number of typical days throughout one year when scenario s is under normal, severe and extreme weather conditions | |
—— | Penalty cost for shedding load at node i | |
—— | Annual cost for hardening distribution line | |
—— | Annual cost for adding an automatic switch on line | |
, | —— | Power and capacity costs of ESSs |
—— | Annual operation-maintenance cost of per unit power capacity of ESS | |
—— | Total annual capital cost of hardening distribution lines | |
—— | Total annual capital cost of upgrading automatic switches | |
—— | Total annual cost of deploying ESSs | |
—— | Total annual equipment deployment cost of ESSs | |
—— | Total annual equipment operation-maintenance cost of ESSs | |
—— | Energy capacity of ESS at the end of time t | |
—— | A large constant | |
—— | Limited investment of ESSs | |
nij | —— | Number of statistical years of line |
Nij | —— | Number of failures of line |
—— | Number of failures of line caused by weather condition a in year x | |
—— | Probability of occurrence of scenario s | |
, | —— | Rated power and capacity of ESSs |
—— | Resistance and reactance of line | |
—— | Rated apparent power from node i to node j during time t in scenario s | |
—— | Initial state of charge of ESS | |
, | —— | The minimum and maximum allowable states of charge |
—— | Duration of weather condition a in year x of the same historical period | |
—— | Square of voltage magnitude at node i | |
, | —— | Squares of lower and upper bound of allowable voltage magnitude at node i |
D. | —— | Variables |
—— | A binary variable, which is equal to 1 if the state of line is closed during time t in scenario s and 0 otherwise | |
—— | A binary variable, which is equal to 1 if the state of line is damaged during time t in scenario s and 0 otherwise | |
—— | A binary variable, which is equal to 1 if line is operating at time t0 in scenario s and 0 otherwise | |
—— | A binary variable, which is equal to 1 if the line switch on line is closed during time t in scenario s and 0 otherwise | |
—— | A variable that represents the fault state of line during time t in scenario s if it is hardened | |
—— | A variable that represents the fault state of line during time t in scenario s if it is not hardened | |
—— | Benefits of ESSs from peak shaving and valley filling | |
—— | Current of line during time t in scenario s | |
—— | Active and reactive load curve values during time t | |
Pi, Qi | —— | Reference values of active and reactive loads at node i |
, | —— | Active charging and discharging power of ESS at node i during time t in scenario s |
, | —— | Active and reactive power flows from node i to node j during time t in scenario s |
, | —— | Active and reactive power generated by distributed generator (DG) connected to node i during time t in scenario s |
, | —— | Upper bounds of active and reactive power of DG connected to node i |
, | —— | Actual active and reactive loads at node i during time t in scenario s |
, | —— | Active and reactive load sheddings at node i during time t in scenario s |
, | —— | Reactive charging and discharging power of ESS at node i during time t in scenario s |
—— | Binary variables, which are equal to 1 if ESS is charging or discharging during time t in scenario s and 0 otherwise | |
—— | A binary variable, which is equal to 1 if line is hardened and 0 otherwise | |
—— | A binary variable, which is equal to 1 if line has an existing switch and 0 otherwise | |
—— | A binary variable, which is equal to 1 if new line switch is added on line and 0 otherwise | |
—— | A binary variable, which is equal to 1 if line has switch and 0 otherwise | |
—— | A binary variable, which is equal to 1 if ESS is connected to node i and 0 otherwise |
Appendix
Line | Failure rate (occurrence/day) | ||||
---|---|---|---|---|---|
Number | From | To | Normal | Severe | Extreme |
1 | 1 | 2 | 0.0024 | 0.0327 | 0.2383 |
2 | 2 | 3 | 0.0131 | 0.1748 | 0.6395 |
3 | 2 | 19 | 0.0097 | 0.1297 | 0.5490 |
4 | 3 | 4 | 0.0101 | 0.1351 | 0.5717 |
5 | 3 | 23 | 0.0217 | 0.1903 | 0.3285 |
6 | 4 | 5 | 0.0050 | 0.0664 | 0.2808 |
7 | 5 | 6 | 0.0189 | 0.2522 | 0.2671 |
8 | 6 | 7 | 0.0273 | 0.2651 | 0.5450 |
9 | 6 | 26 | 0.0277 | 0.1701 | 0.5660 |
10 | 7 | 8 | 0.0052 | 0.0697 | 0.2949 |
11 | 8 | 9 | 0.0099 | 0.1327 | 0.5616 |
12 | 9 | 10 | 0.0389 | 0.5204 | 0.3020 |
13 | 10 | 11 | 0.0144 | 0.1920 | 0.3124 |
14 | 11 | 12 | 0.0157 | 0.2095 | 0.3865 |
15 | 12 | 13 | 0.0198 | 0.2646 | 0.3195 |
16 | 13 | 14 | 0.0342 | 0.2570 | 0.4335 |
17 | 14 | 15 | 0.0194 | 0.2595 | 0.2980 |
18 | 15 | 16 | 0.0043 | 0.0581 | 0.2460 |
19 | 16 | 17 | 0.0399 | 0.2332 | 0.2563 |
20 | 17 | 18 | 0.0109 | 0.1452 | 0.6143 |
21 | 19 | 20 | 0.0188 | 0.2513 | 0.4634 |
22 | 20 | 21 | 0.0120 | 0.1600 | 0.6768 |
23 | 21 | 22 | 0.0238 | 0.3183 | 0.3470 |
24 | 23 | 24 | 0.0237 | 0.3176 | 0.3440 |
25 | 24 | 25 | 0.0054 | 0.0720 | 0.3045 |
26 | 26 | 27 | 0.0075 | 0.1007 | 0.4263 |
27 | 27 | 28 | 0.0281 | 0.2754 | 0.5885 |
28 | 28 | 29 | 0.0213 | 0.2851 | 0.2063 |
29 | 29 | 30 | 0.0134 | 0.1799 | 0.3613 |
30 | 30 | 31 | 0.0258 | 0.2454 | 0.4616 |
31 | 31 | 32 | 0.0082 | 0.1101 | 0.4658 |
32 | 32 | 33 | 0.0090 | 0.1209 | 0.5115 |
33 | 21 | 8 | 0.0330 | 0.2090 | 0.4400 |
34 | 9 | 15 | 0.0330 | 0.2090 | 0.4400 |
35 | 12 | 22 | 0.0330 | 0.2090 | 0.4400 |
36 | 18 | 33 | 0.0133 | 0.1773 | 0.3500 |
37 | 25 | 29 | 0.0133 | 0.1773 | 0.3500 |
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