# An Efficient Methodology to Identify Relevant Multiple Contingencies and Their Probability for Long-Term Resilience Studies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed Methodology

#### 2.1. The Context: The RELIEF Risk-Based Resilience Assessment Framework

#### 2.2. Requirements for Application to Large Systems

- Ensure scalability and computational efficiency for high values of the number of lines to be treated (for example, a whole department of the Italian transmission system contains up to 900 lines);
- Consider the correlation between failures, in fact that the same weather event (e.g., wet snowstorm or strong wind) can affect several lines in the same time frame.

#### 2.3. Overview of the Proposed Methodology

**Pillar 1: Clustering of grid components**

- Correlation Matrix Calculation (stage 1): This stage accounts for the possibility of multiple line failures due to a single event (like wet snow events). A correlation matrix is built, based on historical weather events (see Section 3.1), to quantify the likelihood of lines failing together during a specific time frame (e.g., an hour).
- Highly Correlated Line Clustering (stage 2): Based on the correlation matrix, this stage identifies groups of lines such that multiple line failures within each group are more likely than between groups (see Section 3.2). This clustering helps focus the contingency identification process on the most probable combinations of line failures.

**Pillar 2: Identification of relevant contingencies and their probability**

- Contingency Identification within Clusters (stage 3): Within each identified cluster, this stage pinpoints relevant contingencies, representing specific combinations of line failures that have a significant probability of occurring together (see Section 4.3). To ensure efficiency, negligible probability scenarios are excluded.
- Multiple Contingency Probability Estimation (stage 4): This stage calculates the probability of each identified contingency within the clusters, exploiting the correlation between line failures and the individual failure probabilities of each line (see Section 4.4).

## 3. Clustering of Correlated Lines

#### 3.1. Calculation of the Correlation Matrix

_{events}× N] event matrix M evaluated as in (1), where N

_{events}is the number of relevant weather events and N is the total number of lines considered in the analysis. An event is considered relevant if a specific intensity threshold has been overcome at least for one line.

- n
_{11}is the number of severe events for which both lines L1 and L2 are affected by a weather variable exceeding a threshold Th (e.g., in m/s for wind and kg/m for wet snow); - n
_{10}is the number of severe events for which line L1 is affected while line L2 is not affected by a weather variable exceeding a threshold Th; - n
_{01}is the number of severe events for which line L2 is affected while line L1 is not affected by a weather variable exceeding a threshold Th; - n
_{00}is the number of severe events for which neither line is affected by a weather variable exceeding a threshold Th.

_{ij}[20].

#### 3.2. Clustering

- Create clusters with a user-defined threshold of minimum internal correlation between the lines in each cluster;
- Create clusters with a maximum cardinality (N
_{MAX, LI}); - Subdividing clusters that are too large (with cardinality higher than N
_{MAX, LI}) into smaller clusters of maximum cardinality N_{MAX, LI}based on topological information.

- Step 1: identification of the clusters based on the correlation matrix R;
- Step 2: identification of subclusters using topological indications;
- Step 3: aggregation of individual clusters through a mix of topological indications and correlation factors.

#### 3.2.1. Step 1: Clustering of the Lines Based on the Correlation Matrix R

- The parameters of minimum value of the intra-cluster average correlation (${\rho}_{min,s1}^{intra}$) and the distance limit between two distinct clusters (${D}_{max,s1}^{inter}$) are set;
- The distance matrix between the lines is calculated, defined as D =
**1**—R’, where R’ is the correlation matrix with all zeros set along the main diagonal and**1**is an N × N matrix entirely filled with ones; - N groups are defined, each containing one line of the set;
- Lines i and j are identified s.t. D(i, j) = min(min(D));
- Lines i and j are grouped into a new group “N + 1”;
- The cophenetic distance is calculated between the group made up of i and j and the remaining N − 2 groups (excluding the groups related to rows i and j). Ψ is defined as the set of N − 2 groups. This distance is defined as max(max(D([i j], h)));
- The matrix D is updated, in particular D(N + 1, h) and D(h, N + 1) with h ∈ Ψ;
- The rows and columns associated with the original groups i and j are deleted;
- Steps 3 to 7 are repeated until one of the following conditions occurs:
- The minimum value of the intra-cluster mean correlation (calculated as the average value of the linear correlation coefficient between any pair of lines belonging to the same cluster) becomes less than a threshold ${\rho}_{min,s1}^{intra}$;
- The minimum distance between two distinct clusters becomes greater than a threshold ${D}_{max,s1}^{inter}$.

- Steps 2–9 are repeated on several pairs of parameters ${\rho}_{min,s1}^{intra}$, ${D}_{max,s1}^{inter}$, defined as the Cartesian product of two sets containing reasonable values for both of the aforementioned parameters, in order to select the pair that provides the best performance indicator according to the indications at step 11;
- The pair of parameters that ensures the highest performance index is selected, indicated as the weighted sum of the 5% quantiles of the silhouette coefficient and the internal correlation value of the non-single clusters. This guarantees the best separation between the groups and at the same time a good cohesion within the groups. Values between 0.50 and 0.70 for the silhouette indicate reliable groups, while values between 0.7 and 1 very reliable groups (as they are cohesive and well separated from the others).

#### 3.2.2. Step 2: Splitting of Wide Clusters According to Topological Information (Cutsets)

_{MAX, LI}) into smaller clusters, making sure that the lines, whose simultaneous tripping causes the disconnection of primary substations, are kept in the same cluster (because such events highly affect resilience indicators such as expected energy not served). This sub-division leverages topological information about the grid to ensure the resulting clusters maintain meaningful relationships between lines.

- The cutsets of lines that lead to the disconnection of the substations (primary substations, PSs) of the network portion are identified. Then, the connectivity matrix A of dimensions N × N
_{PS}is defined where N is the number of lines and N_{PS}is the number of PSs of the considered network, s.t. A(i, j) = 1 if PS j is terminal 1 of line i and A(i, j) = −1 if PS j is terminal 2 of line i, A(i, j) = 0 otherwise; - For each PS j, a vector X
_{j}of dimensions N_{PS}× 1 is set such that X_{j}(j) = −1, it is 1 otherwise; - Y
_{j}= A × X_{j}gives a vector NL × 1 where the non-zero terms represent the minimum subset of lines which cuts PS j; - For each cluster larger than N
_{MAX, LI}:- The sub-matrix S of matrix A corresponding to the lines belonging to the cluster to be disaggregated is obtained;
- The columns of S are sorted according to the decreasing number of non-zero terms. We obtain the S
_{ord}reordered matrix; - The sub-cluster is identified as the set L
_{sc}of lines associated with the first column of the S_{ord}matrix. The lines of the sub-cluster L_{sc}are discarded from the lines of the original cluster L_{co}, redefining the matrix S on the basis of the set L_{co’}of the remaining lines to be clustered where card(L_{co’}) = card(L_{co}) − card(L_{sc}), “card” being the “cardinality” operator which indicates the number of elements of the set. - tems ii and iii are repeated on the new matrix S, updating the set of L
_{co’}lines to be clustered until the set L_{co’}has a cardinality lower than or equal to N_{MAX, LI}; - The subclusters are reaggregated on the basis of the following criteria:
- The subclusters in pairs form a cutset of the network;
- The sum of the dimensions of the reaggregated subclusters are at most equal to N
_{MAX, LI}.

_{R}among the NG clusters: this matrix results from the application of steps 1 and 2 to the original correlation matrix R.

#### 3.2.3. Step 3: Cutset-Oriented Re-Aggregation of Clusters

- There are still some groups composed of a single line that have high correlations with already clustered lines;
- There may be relatively small clusters that can be increased with a small decrease in the intra-cluster correlation.

- To increase cluster size with little detriment on cluster internal correlation;
- To aggregate unpaired lines with already clustered lines.

- For each cluster i1 = 1 … NG − 1, the clusters i2 = i1 + 1 …. NG are analyzed and the following are calculated:
- The mean value of the correlation between each pair of clusters i1 and i2 (defined as the arithmetic mean of the absolute values of the linear correlation coefficients, each calculated on a different pair of lines, one belonging to cluster i1 and the other to cluster i2) reported in position (i, j) of matrix M
_{R}; - The maximum value of the correlation between two lines of clusters, i1 and i2;
- The best candidates for aggregation between clusters are identified according to the following criteria:
- Clusters that fully define a cutset (topological clusters) and have an average inter-cluster correlation at least equal to a fixed value ${\overline{\rho}}_{s3}^{inter,topo}$;
- Clusters that have an inter-cluster correlation no less than a fixed value ${\overline{\rho}}_{s3}^{inter}$;
- Topological clusters that have a maximum correlation value at least equal to a fixed value equal to ${\rho}_{max,s3}^{inter,topo}$.

- Two matrices, M1 and M2, of dimension NG × NG are defined s.t. in position (i, j) if they contain 1 or 0, respectively, depending on whether the following criteria are verified or not:
- criterion c.i or c.ii for matrix M1;
- criterion c.iii for matrix M2.

- The conditionality matrix M
_{C}between clusters i1 and i2 is also defined such that M_{C}(i1, i2) = 0 if the sum of the dimensions of clusters i1 and i2 is greater than a defined value (parameter N_{MAX, LI}); it is 0 otherwise; - For the candidates selected in point 1.c, the Matrix_total of the performance indicators is calculated as in (3):$$Matrix\_total=\left[\left(1-\alpha \right)\cdot {M}_{R}+\mathrm{max}(M1,M2)\cdot \left(\alpha \right)\right]\times {M}_{C}$$
- The previous steps 1 and 2 are repeated until:
- the residual correlation between the “single” groups and the multi-line groups does not fall below an established threshold (${\rho}_{residual,s3}^{inter}$), or;
- the maximum number of iterations ${N}_{max,s3}^{it}$ is exceeded.

#### 3.3. Management of Greenfield Lines and Partially Buried Lines

_{pi}partially buried lines and N

_{green}greenfield lines are shown below:

- Consider the clusters C
_{h}h = 1 … NC identified in the pre-intervention analysis; - Modify matrix M in the columns of the partially buried lines q = 1 … N
_{pi}; - The partially buried lines that are part of singleton clusters in the pre-intervention analysis remain in the singleton cluster;
- Increase the number of matrix M columns by adding the columns of greenfield lines i = 1 … N
_{green}(on the basis of the hypothetical layout of the greenfield lines, the events of exceeding the threshold in the greenfield lines are counted considering the same weather events of the pre-intervention analysis); - Calculate the correlation coefficients R
_{qj}between each partially buried line q = 1 … N_{pi}not belonging to singleton clusters and the other lines j; - Calculate the correlation coefficients R
_{ij}between the greenfield line i = 1 … N_{green}and the other lines j; - Greenfield line i is attributed to cluster h*, which has the highest median value calculated on the absolute values of the coefficients R
_{is}with s ∈ C_{h}, i.e., $\underset{h=1\dots Nc}{\mathrm{max}}\left[\underset{s\in {C}_{h}}{median}\left(\left|{R}_{is}\right|\right)\right]$; - Steps 6 and 7 are repeated for all greenfield lines.

## 4. Selection of Contingencies and Probability Computation

- Computation of the probability of the “AND” event of multiple line trippings;
- Iterative filtering based on total probability theorem;
- Calculation of the probability of occurrence of retained contingencies.

#### 4.1. Probabilistic Modeling of Multiple Contingencies

_{j}

_{,1hr}, is given by (4).

_{1hr}) for each line, estimated using weather data and vulnerability models, is then used to derive the average hourly probabilities of various contingencies, as described in the following.

#### 4.2. Copula-Based Computation of Contingency Probability

_{j}= {line L

_{j}tripping due to initiating event};

_{j}= {overcoming of a critical weather variable threshold at line L

_{j}}.

_{j}) is the probability of a failure in the j-th line. In the sequel, for the sake of brevity the following notation is used: P(F

_{j}) = p

_{j}.

_{i}values, considering a suitable copula family. Different family copulas have been tested (e.g., empirical, Clayton and Frank copulas), and we have found that the Gaussian copula is the best tradeoff between accuracy and computational speed for a reliable evaluation of the multiple contingency probabilities.

#### 4.3. Filtering of Failure Combinations Based on Total Probability Theorem

_{h}trippings and n_nt

_{h}not trippings, its probability is always lower than the probability of the AND of the n_t

_{h}trippings thanks to the total probability theorem. Thus, the contingencies that include any of the discarded AND combinations are also discarded by the algorithm.

#### 4.4. Calculation of the Probability of Occurrence of Multiple Contingencies

_{t}of trippings and a set S

_{nt}of not-trippings, i.e., P(S

_{t}and S

_{nt}), can be written as an algebraic sum of the cumulative distribution of probability of the copula C (copula CDF) evaluated at suitable points according to the general formula indicated in (8).

_{1}, …, s

_{q}, where s

_{j}can be x

_{j}or x

_{j}− 1.

_{ctg h, 1yr}is a function of the hourly probability of the same h-th contingency P

_{ctg h, 1hr}by means of (10).

## 5. Case Study

#### 5.1. Test System and Simulations

#### 5.2. Simulation CLU: Line Clustering for Wet Snow Events

_{MAX, LI}lines each. Importantly, these clusters still exhibit a strong internal connection between lines, with a minimum correlation value of around 0.5.

#### 5.3. Simulation PRO: Tests on the Algorithm for Contingency Probability Computation

_{CTG,i}is the hourly probability of i-th contingency involving line j and ${N}_{CTG,i}^{(j)}$ is the number of contingencies involving line j. The metrics used to assess the fulfillment of such a condition is the percentual error in (12) between the line failure probability derived from the failure return periods RP

_{j}, i.e., $P\left({F}_{j}\right)=1/\left(R{P}_{j}*8760\right)$ and the reconstructed failure probability ${P}_{R,j}={\displaystyle \sum _{i=1}^{{N}_{CTG,i}^{(j)}}{P}_{CTG,i}}$.

- Different numbers of lines in the cluster, with the same correlation matrix R;
- Different values for the return periods, with the same cluster cardinality and matrix R;
- Different correlation matrices R among the lines, with the same line cardinality.

#### 5.3.1. Subcase (a): Comparison of the Two Algorithms with Different Line Cardinalities

#### 5.3.2. Subcase (b): Comparison of the Two Algorithms with Different RP Values

#### 5.3.3. Subcase (c): Comparison of the Two Algorithms with Different Correlation Levels

#### 5.3.4. Some Remarks

- Enhanced speed and accuracy: compared to algorithm A, Botev’s algorithm boasts significantly faster execution times and demonstrably higher accuracy, at least for contingency probability computations in power system resilience assessment. This improved accuracy is particularly crucial, as even small deviations in contingency probabilities can have substantial consequences.
- Robustness across scenarios: algorithm B exhibits remarkable stability to various input characteristics. Its calculation time remains independent of the specific line failure probabilities within a cluster, unlike algorithm A, which can be sensitive to these values. This robustness ensures reliable performance across diverse situations.
- Accuracy maintained for large clusters: even when dealing with very large clusters (tested up to a size of 15, often considered the upper limit) and significant variations in line reliability parameters (RPs), algorithm B delivers exceptional accuracy. This characteristic makes it ideal for real-world power system analysis, where large clusters and diverse RP values are common.
- Adaptability to correlation matrices: Botev’s algorithm maintains its high accuracy regardless of the correlation matrix configuration. It performs equally well with highly correlated, moderately correlated, or weakly correlated line failures, providing a versatile solution for various power system scenarios.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Bowtie conceptual scheme of the RELIEF risk-based resilience assessment framework [19].

**Figure 2.**Workflow of the proposed methodology for the efficient enumeration of multiple contingencies for resilience analyses.

**Figure 3.**Difference between (

**a**) two failures occurring at different time instants over the year and (

**b**) a N-2 contingency.

**Figure 4.**Main results after stage 1 (

**a**) and after stage 3 (

**b**): size of the clusters (top left) and level of intra-c luster correlation (bottom left).

**Figure 5.**Percentual errors on the line failure probabilities for a cluster with cardinality equal to 12 using algorithm A (

**a**) and algorithm B (

**b**).

L2 | Not L2 | Totals | |
---|---|---|---|

L1 | n_{11} | n_{10} | n_{1*} |

not L1 | n_{01} | n_{00} | n_{0*} |

Totals | n_{*1} | n_{*0} |

Sim ID | Description | Goal |
---|---|---|

CLU | Three-step clustering algorithm application to the set of 647 lines, assuming a maximum cardinality N_{MAX, LI} = 12. | To verify the representativeness of the identified clusters with respect to historical weather events. |

PRO | Copula-based algorithm application via two alternative algorithms to one specific cluster of the line set analyzed in sim CLU. | To verify the accuracy and the robustness in the copula CDF computation method adopted in the methodology. |

Probability | 0.05 | 0.25 | 0.5 | 0.75 | 0.95 |

Quantile | 0.2176 | 0.6806 | 0.9671 | 1.0000 | 1.0000 |

**Table 4.**Quantiles of the distribution of mean values of the internal correlation of the topological cutsets.

Probability | 0.05 | 0.25 | 0.5 | 0.75 | 0.95 |

Quantile | 0.3162 | 0.4709 | 0.6005 | 0.7857 | 0.9901 |

Probability | 0.05 | 0.25 | 0.5 | 0.75 | 0.95 |

Quantile | 0.5272 | 0.6583 | 0.7826 | 0.9901 | 0.9901 |

Subcase | Set of RPs (Year) | Cluster Cardinality | Minimum Correlation in the Cluster |
---|---|---|---|

(a) | 22, 2743, 32, 33, 38, 46, 959, 1374, 150, 150, 10, 72, 400, 90, 120 | 5, 7, 10, 12, 14, 15 | >0.9 |

(b) | 22, 100(*), 32, 33, 38, 46,959, 1374, 50(*), 50(*), 10, 72, 400, 90, 120 | 10, 12 | >0.9 |

(c) | Same as subcase (a) | 12 | >0.9, 0.8, 0.5, 0.3 |

Maximum Percentage Error (%) | Computational Time (s) | |||
---|---|---|---|---|

Cluster Cardinality | Algorithm A | Algorithm B | Algorithm A | Algorithm B |

3 | 4.43 × 10^{−13} | 4.51 × 10^{−13} | 0.9 | 1.5 |

5 | 3.2 | 5.72 × 10^{−13} | 3.6 | 2.7 |

7 | 10.5 | 1.35 × 10^{−11} | 16 | 5.8 |

10 | 413.5 | 8.93 × 10^{−10} | 217.9 | 38 |

12 | 1945 | 9.30 × 10^{−9} | 1067 | 175 |

Maximum Percentage Error (%) | ||
---|---|---|

Cluster Cardinality | Algorithm A | Algorithm B |

10 | 228 | 2.41 × 10^{−10} |

12 | 1430 | 2.75 × 10^{−8} |

Computational Time (s) | Speed up Ratio | ||
---|---|---|---|

Cluster Cardinality | Algorithm A | Algorithm B | |

10 | 164 | 38 | 4.3 |

12 | 816 | 175 | 4.7 |

**Table 10.**Maximum percentual errors with different correlation levels—algorithms A and B, subcase (c).

Correlation Level | Algorithm A, % | Algorithm B, % | ||
---|---|---|---|---|

Maximum Percentual Error, % | Computational Time, s | Maximum Percentual Error, % | Computational Time, s | |

Very high (min corr > 0.9) | 41.7 | 40.0 | 6.91 × 10^{−11} | 9.9 |

High (min 0.8) | 59.5 | 35.0 | 0.21 | 10.0 |

Medium (min 0.5) | 22.2 | 12.3 | 0.03 | 10.1 |

Low (min 0.3) | 4.0 | 8.2 | 2.86 × 10^{−3} | 10.1 |

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**MDPI and ACS Style**

Ciapessoni, E.; Cirio, D.; Pitto, A.
An Efficient Methodology to Identify Relevant Multiple Contingencies and Their Probability for Long-Term Resilience Studies. *Energies* **2024**, *17*, 2028.
https://doi.org/10.3390/en17092028

**AMA Style**

Ciapessoni E, Cirio D, Pitto A.
An Efficient Methodology to Identify Relevant Multiple Contingencies and Their Probability for Long-Term Resilience Studies. *Energies*. 2024; 17(9):2028.
https://doi.org/10.3390/en17092028

**Chicago/Turabian Style**

Ciapessoni, Emanuele, Diego Cirio, and Andrea Pitto.
2024. "An Efficient Methodology to Identify Relevant Multiple Contingencies and Their Probability for Long-Term Resilience Studies" *Energies* 17, no. 9: 2028.
https://doi.org/10.3390/en17092028