**6. Simulation, analysis, and discussion of the results**

In a previous work [19], the EDN of the metropolitan Rome, area (Italy), was deeply investigated by extensive calculations enabling to estimate its resilience score, according to the definition reported in Section 3. RecSIM has been used to study the behavior of the whole EDN of the metropoli-Rome EDN that is a large EDN grid composed of 139 PS and 14938 SS distributed along 1607 MV lines.

The power grid has 6348 telecontrolled SS (i.e., 42% of the total SS) and 1012 automated SS (i.e., 6% of the total SS). Considering the MV lines, the power grid has 1447 MV lines that contain at least 1 telecontrolled SS (90%) and 510 MV lines containing at least 1 automatized SS (31%). The considered power grid is set in the so-called normal configuration that is:


When referred to "normal configuration," we will refer to equal (1)–(5) conditions. The reported simulation has the character of a "stress test." Two different stress schemes have been adopted: the *unbiased* perturbation scheme and the *heuristic* scheme.

In the *unbiased* perturbation scheme, each electrical substation (SSi i = 1, *N*), one at a time, has been set in the damage state and the resulting impact of electrical crisis estimated in terms of the *<sup>i</sup>* defined in Eq. (3). **Figure 5** reports the distribution function *D***<sup>1</sup>** () for all resulting *<sup>i</sup>* . This simulation will be referred to as the "(*N***−** 1) analysis," as it involved the set in the damage state of a single SS (at a time).

**Figure 5.**

*D(1) (Γ) distribution of the resulting impacts as functions of the impacts, for a N − 1 analysis, i.e., in the event of one node damaged.*

**43**

*Modeling Resilience in Electrical Distribution Networks DOI: http://dx.doi.org/10.5772/intechopen.85917*

The impact distribution *D*(*i*)

*R* = *a*(**1**)

distribution function of the values such as *D(2)( )*, *D(3)( )*, etc.

resilience score, which has been associated to its normalized integral Eq. (6): the larger the integral of the distribution, the lower the resilience. In fact, for an infinitely resilient network, each damage should correspond to the lowest possible (or vanishing) impact in the terms expressed by Eq. (3). The overall system resilience could be estimated as a series of terms each one representing the contribution

+ *a*(**2**)

While the terms *R(i)* will be achieved by applying Eq. (6) to the different *D(i)(***Γ***)*, the terms *a(i)* can be related to the probability of the event; this would produce a series of progressively smaller terms which will reduce the impact of the high order contributions to the total value. The first terms related to (*N* **−** 1) and (*N* **−** 2) events would thus dominate the series in Eq. (7) which will provide an *unbiased* estimate of the global resilience of the network when perturbations are imposed following an exhaustive scheme rather than a heuristic method. A different perturbation scheme (the *heuristic* one) has been also applied to provide a further possible perturbation scheme aiming to realize a resilience assessment which, in such case, will be measured by estimating Eq. (6): this will be done through the use of a distribution function *D(h)()* resulting from the application of the *heuristic* perturbation scheme. The *heuristic* scheme has been thus designed and

Instead of producing systematic damages (as in the *unbiased* scheme), we have produced "educated" damage scenarios where SS have been set in the damaged state as a function of their effective rate of faults (as declared by the electrical operator). The heuristic perturbation scheme is thus carried out in the following way. Let us assume to know the *rate of faults per day* **ρI** of each SS of the network, expressed in terms of the average number of times that the SS have been recorded to be out of order. Statistics have been collected along several years and the number of observed faults normalized over the number of days of observation. The ρI value could be thus assimilated to the daily probability that the specific substation goes in a damaged state. The cause of SS fault could be different: the SS could be hit by some external event (i.e., natural hazard and/or its consequences) or by some internal event (i.e., the disruption of some component). The statistical fault rate per component does not distinguish between the origins of fault; we will thus consider this

The *heuristic* perturbation scheme has been thus applied to the network "normal configuration" by simulating *M* working days: in each day of operations, the damaged state of each SS has been sampled (as in a Monte Carlo scheme) by extracting a random number *ri (ri = [0, 1])* and by comparing it with the **ρ**i value: if *ri <* ρi, the i-th SS is put in the damaged state where it remains unperturbed elsewhere. The SS set in the damage state have been put simultaneously in the damaged state, in order to simulate the worst-case scenario. This procedure is repeated *n* times to scan each

This procedure generates very few damages, as the rate of faults of the substations is usually particularly small. However, it generates cases where one (or even more than one) substation will result in a damaged state. This procedure thus allows to sample

*R*(**2**)

+ *a*(**3**)

*R*(**3**)

+ …. (7)

with that obtained through the

toward resilience for different (and progressively large) perturbations:

*R*(**1**)

applied to compare the resulting resilience score *R*(*h*)

fault rate as an "intrinsic" property of the EDN element.

SS and then repeated *M* times to simulate different working days.

use of the *unbiased* perturbation scheme.

The same stress test could be repeated by setting in the damage state two, three, or more SS simultaneously, in a way intended to generate crisis situation of higher impact (although with a lower probability of occurrence. Each case will produce a

() functions do provide the generating function for the

*Modeling Resilience in Electrical Distribution Networks DOI: http://dx.doi.org/10.5772/intechopen.85917*

*Infrastructure Management and Construction*

so-called normal configuration that is:

ent networks)

ery operations

restoration

*heuristic* scheme.

function *D***<sup>1</sup>**

crisis estimated in terms of the *<sup>i</sup>*

() for all resulting *<sup>i</sup>*

The power grid has 6348 telecontrolled SS (i.e., 42% of the total SS) and 1012 automated SS (i.e., 6% of the total SS). Considering the MV lines, the power grid has 1447 MV lines that contain at least 1 telecontrolled SS (90%) and 510 MV lines containing at least 1 automatized SS (31%). The considered power grid is set in the

given fraction of telecontrolled, automatic, and frontier SS)

1.A specific topology of the network (consisting of a given number of SS, with a

2.With the switches along the medium-voltage lines located in specific points,

3.The telecontrolling BTS providing services to a certain extent (in our simulation we consider a default fraction of unavailable BTS leading telecontrolling functionality unavailable—apparently a "physiological" conditions of depend-

4.A given number of technical crews available in the field for the manual recov-

5.Standard times for the solution of the different actions to be performed for SS

When referred to "normal configuration," we will refer to equal (1)–(5) conditions. The reported simulation has the character of a "stress test." Two different stress schemes have been adopted: the *unbiased* perturbation scheme and the

In the *unbiased* perturbation scheme, each electrical substation (SSi i = 1, *N*), one at a time, has been set in the damage state and the resulting impact of electrical

*(Γ) distribution of the resulting impacts as functions of the impacts, for a N − 1 analysis, i.e., in the event of* 

analysis," as it involved the set in the damage state of a single SS (at a time).

defined in Eq. (3). **Figure 5** reports the distribution

. This simulation will be referred to as the "(*N***−** 1)

**42**

**Figure 5.** *D(1)*

*one node damaged.*

The same stress test could be repeated by setting in the damage state two, three, or more SS simultaneously, in a way intended to generate crisis situation of higher impact (although with a lower probability of occurrence. Each case will produce a distribution function of the values such as *D(2)( )*, *D(3)( )*, etc.

The impact distribution *D*(*i*) () functions do provide the generating function for the resilience score, which has been associated to its normalized integral Eq. (6): the larger the integral of the distribution, the lower the resilience. In fact, for an infinitely resilient network, each damage should correspond to the lowest possible (or vanishing) impact in the terms expressed by Eq. (3). The overall system resilience could be estimated as a series of terms each one representing the contribution toward resilience for different (and progressively large) perturbations:

$$\mathbf{R} = \mathbf{a}^{(1)}\mathbf{R}^{(1)} + \mathbf{a}^{(2)}\mathbf{R}^{(2)} + \mathbf{a}^{(3)}\mathbf{R}^{(3)} + \dots \tag{7}$$

While the terms *R(i)* will be achieved by applying Eq. (6) to the different *D(i)(***Γ***)*, the terms *a(i)* can be related to the probability of the event; this would produce a series of progressively smaller terms which will reduce the impact of the high order contributions to the total value. The first terms related to (*N* **−** 1) and (*N* **−** 2) events would thus dominate the series in Eq. (7) which will provide an *unbiased* estimate of the global resilience of the network when perturbations are imposed following an exhaustive scheme rather than a heuristic method.

A different perturbation scheme (the *heuristic* one) has been also applied to provide a further possible perturbation scheme aiming to realize a resilience assessment which, in such case, will be measured by estimating Eq. (6): this will be done through the use of a distribution function *D(h)()* resulting from the application of the *heuristic* perturbation scheme. The *heuristic* scheme has been thus designed and applied to compare the resulting resilience score *R*(*h*) with that obtained through the use of the *unbiased* perturbation scheme.

Instead of producing systematic damages (as in the *unbiased* scheme), we have produced "educated" damage scenarios where SS have been set in the damaged state as a function of their effective rate of faults (as declared by the electrical operator). The heuristic perturbation scheme is thus carried out in the following way. Let us assume to know the *rate of faults per day* **ρI** of each SS of the network, expressed in terms of the average number of times that the SS have been recorded to be out of order. Statistics have been collected along several years and the number of observed faults normalized over the number of days of observation. The ρI value could be thus assimilated to the daily probability that the specific substation goes in a damaged state. The cause of SS fault could be different: the SS could be hit by some external event (i.e., natural hazard and/or its consequences) or by some internal event (i.e., the disruption of some component). The statistical fault rate per component does not distinguish between the origins of fault; we will thus consider this fault rate as an "intrinsic" property of the EDN element.

The *heuristic* perturbation scheme has been thus applied to the network "normal configuration" by simulating *M* working days: in each day of operations, the damaged state of each SS has been sampled (as in a Monte Carlo scheme) by extracting a random number *ri (ri = [0, 1])* and by comparing it with the **ρ**i value: if *ri <* ρi, the i-th SS is put in the damaged state where it remains unperturbed elsewhere. The SS set in the damage state have been put simultaneously in the damaged state, in order to simulate the worst-case scenario. This procedure is repeated *n* times to scan each SS and then repeated *M* times to simulate different working days.

This procedure generates very few damages, as the rate of faults of the substations is usually particularly small. However, it generates cases where one (or even more than one) substation will result in a damaged state. This procedure thus allows to sample

#### **Figure 6.**

*Comparison of the D() distribution values for the N − 1 and N − 2 unbiased scheme simulation resulting from the simulation via the heuristic scheme (red = unbiased (N − 2), purple = unbiased (N − 1), and black = heuristic scheme).*

(among the manifold of possible damaged network states) those states where one or more SS are simultaneously damaged, in agreement with the rate of faults of the different stations. Over *nh* = 1515 damaged configurations were obtained with the Monte Carlo sampling, of which 1163 were constituted by a single damaged SS; 296 with 2 damaged SS; 49 with 3 damaged SS; 5 with 4 damaged SS; and 2 with 5 damaged SS.

**Figure 6** summarizes all the results obtained thanks to the simulations by using the (*N* **−** 1) and the (*N* **−** 2) *unbiased* scheme and the *heuristic* perturbation scheme. In all simulations (both for the *unbiased* and for the *heuristic* schemes), the same number of technical crews *C* available for the service restoration has been assumed (*C* = 2). The three curves, however, derive from simulations scheme which have produced different amounts of crisis scenarios whose impacts have been measured through Eq. (3). In fact, for the *unbiased* (*N* **−** 1) simulation, a number of crisis scenarios *n* equal to the number of nodes *N* have been produced (*n(N* **<sup>−</sup>** *1)* = *N* = 13,618). In the case of the unbiased (*N* **−** 2) simulation, a number of crisis scenarios *n(N* **<sup>−</sup>** *2)* = 271,581 have been produced (this number corresponds to the total number of double faults occurring along the same medium tension line).

For the heuristic perturbation scenario, the number of cases was, in turn, *nh* = 1515 as previously stated. The most relevant feature of the three distributions must be observed in the impact dimension. The perturbations produced by using an unbiased (*N* − 2) scheme produce very large effects, as they tend to involve a large number of SS, which impose a sequence of interventions (with the provided number of technical crews *k* available, not all SS could be simultaneously repaired).

The estimate of the corresponding *R(1)*, *R(2)*, and *R(h)* [through the use of Eq. (6)] appropriately renormalized all the distributions. Application of Eq. (6) to the three different distribution functions provides the following values: *R(1)* = 2.17 × 10<sup>−</sup><sup>2</sup> , *R(2)* = 7.60 × 10<sup>−</sup><sup>3</sup> , and *R(h)* = 1.78 × 10<sup>−</sup><sup>2</sup> .

It is interesting, in turn, to notice that crises produced by the *heuristic* scheme (i.e., involving SS which have shown a large propensity to fault), although in some cases involving more than a SS produces impacts which, even in the largest cases, are of the same dimension of those produced by worst cases in the (*N* − 1) unbiased simulation. This is probably due to the fact that more vulnerable SS are located

**45**

has revealed two main results.

*Modeling Resilience in Electrical Distribution Networks DOI: http://dx.doi.org/10.5772/intechopen.85917*

large number of connected customers).

**7. Conclusions**

along lines, which do not produce relevant outages in case of fault (either for the presence of a few not remotely controlled SS and/or for the presence of a not too

Different scores are the results of the different adopted simulation schemes. Rather than the absolute resilience score, what should be estimated which might have a technological meaning are "resilience score variations": when the same network (and/or its management properties) is modified, the same simulation scheme can be adopted and the resilience score measured again. The difference of the resilience score (before and after the modifications) will provide an indication on if modifications have (or have not) produced benefits to the overall network resilience.

The work presented in this chapter that built a great amount of work done on the same topic [18, 19] presents the RecSIM system and its relevant capabilities to represent and simulate real urban system and in particular problems related to the reconfiguration of electric distribution systems following faults. In particular two major achievements are highlighted, one related to the possibility to account for a number of issues, which have not been appropriately considered in the resilience assessment process in the existing literature, and the second concerning the viability of implementing RecSIM (and its scalability) to large, real EDN. In particular reference has been made in the paper to the case study of Rome city that has a quite large distribution network containing more than 13,500 electrical substations.

As for the general achievements in the area of the models for estimating resilience

of EDN, a novel, computable scheme has been identified, on which the RecSIM engine, described in the paper, is based on. The RecSIM model considers different factors encompassing all the phases of risk management, including the technological properties of the network, the fault management procedures, and the network interdependency with the telecontrol network. In many cases of previous works on the same topic (recalled in Section 2), the resilience estimates have been done by using models which considered just the electrical response of the network, thus disregarding the topological and technological features of the network, as well as the management skills and procedures, and the external and environmental constraints. The EDN management model behind the RecSIM tool, in turn, is able to reconstruct the impact of a crisis by considering all the abovementioned factors (recalled in Section 4) which play a critical role in determining the overall systemic resilience of the EDN. Moreover, the possibility of relating the resilience to the distribution of impacts generated by a range of possible perturbations, described in this chapter, provides a further improvement to the prosed approach. Many different perturbation schemes could be therefore investigated, and a resilience score, more suitable for to the user's requirements, can be therefore assessed. Last but not the least, this scheme could also be prone to be modified by varying the outage impact metrics. Whereas in this work the outage impact Γ was assessed in terms of the KPI adopted by the Italian regulatory agency [Eq. (3)], it can be expressed by considering further metrics, able to account, for instance, the

economic losses or the level of wealth reduction caused to the citizens [19].

As from the analysis of the data resulting from the case study analyzed, i.e., the Rome city EDN, the profile of the impact distribution functions resulting from the different simulations made on the basis of the *unbiased* and the *heuristic* schemes

Firstly, the *unbiased* (*N* − 2) scheme provides the worst-case scenario. The simultaneous damage of two SS residing along the same medium-voltage line, produces (as expected) impacts of a significant severity since several other SS are involved.

along lines, which do not produce relevant outages in case of fault (either for the presence of a few not remotely controlled SS and/or for the presence of a not too large number of connected customers).

Different scores are the results of the different adopted simulation schemes. Rather than the absolute resilience score, what should be estimated which might have a technological meaning are "resilience score variations": when the same network (and/or its management properties) is modified, the same simulation scheme can be adopted and the resilience score measured again. The difference of the resilience score (before and after the modifications) will provide an indication on if modifications have (or have not) produced benefits to the overall network resilience.
