A power grid topology can be represented by a graph GpN , Eq, where i denotes a node within the node set N and pi, jq the link between node i and j in the line set E [73]- [66]- [163]- [74]. Denote with NLthe number of loads, with Nl the number of lines and
with Ng the number of generators in G.
Optimal-Power-Flow (OPF) methods can used to solve the network power dispatch problem [89]- [71]. In the adopted formulation, loads can be curtailed if necessary. This has indeed very high cost for the grid and will occur only if the cost minimization problem can not be solved otherwise, for instance, if load demand exceeds power capacity or to avoid line overloads. Mathematically, the problem is defined as follows:
min
Pg,Lcut
f pPg, Lcutq (6.1)
where the cost function depends on the power generated and the load curtailed (Lcut).
6.2.1 Power Grid Resilience Index
The Expectation of Energy-not-Supplied (ErENSs), has been commonly used as a reli- ability index in a number of studies as well is previous chapter of this thesis. Although initially conceived as a reliability indicator, it has been claimed it is also suitable to the resilience concept [44]. Indeed, it is worth recalling that ErENSs is not full capable of capture relevant resilience features, such as how fast the performance of different hazards in potential future scenarios and lack an economic dimension. The design of compre- hensive resilience measures capable of capturing all relevant features is a challenging research topic itself and is further discussed in the concluding session of this thesis. For the aim of this framework, the ErENSs is a sophisticated enough indicator of resilience and can be obtained by averaging contributions of Ns independent simulations:
ErEN Ss “ Ns ř i“1 EN Si Ns (6.2)
where the Energy-not-Supplied, ENS, for a given simulation period Tsim, here consid-
ered to be 1 year, is obtained as follows: EN S “ Tsim ÿ t“1 ÿ iPN Lcut,i,t¨ t (6.3)
where Lcut,i,t is the load curtailed at each time t and each node i, obtained solving the
minimisation problem in Eq. (6.1). The ENS due to a failure event f can be obtained as Třf
t“1
ř
iPN
Lcut,i,t¨ t, where Tf is the duration of the failure event.
6.2.2 Weather-Dependent Failures
In this work, weather events can trigger failures in the network. The network state is therefore identified by the combination of a ‘normal weather’ failure mode and the ‘severe weather’ failure mode. A normal weather model represents weather-independent effects such as ageing, malicious attacks or manufacturing errors in general. A severe weather model describes failures which are triggered by extreme weather conditions, for instance, lightning-induced dielectric breakdowns or wind-driven structural failures. High winds storms and lightning storms are here considered and, for simplicity but without loss of generality, the entire network is assumed facing the same weather con- ditions. The random occurrence of failures in ‘normal’ weather conditions is modelled as a Homogeneous Poisson process (HPP) [4]:
P pNfptq “ kq “
rλn¨ tsk
k! e
´λn¨t k “ 0, 1, .., N (6.4)
where λn rh¨kmocc s represents the line failure rate in normal weather conditions,
P pNfptq “ kq is the probability that k failures occur in the network in the period p0, ts
and Nfptq is the number of failures per km of grid line occurring in the period p0, ts
and measured in rocc kms.
The severe weather events (e.g. high wind and lightning storms) are affected by uncertainty due to climate changes and inherent variability. In general, events are more likely to occur during specific periods of the year. The occurrence of severe weather events is modelled by a Non-Homogeneous Poisson Process (NHPP) [4]:
P pNeptq “ kq “
rVeptqsk
k! e
´Veptq k “ 0, 1, .., N (6.5)
where Veptq represents the time dependent occurrence rate of the severe weather event
e, P pNeptq “ kqis the probability that k events occur in the period p0, ts and Neptq is
obtained as:
Veptq “
żt 0
vept1qdt1 (6.6)
where vept1q is the time-varying occurrence rate of the event e. In this model, lightning
storms and high wind speeds are considered as threatening environmental conditions e P tlg, wu. In Ref. [4], the ratios vwptq and vlgptq of occurrence of these conditions
are evaluated on a monthly basis and assumed to be stepwise constants, as depicted in Fig. 6.1.
Gen Feb Mar Apr May Jun Jul Aug Sep Opt Nov Dec
0 0.005 0.01 0.015 0.02 0.025 Occur ance R ate
Figure 6.1: The variable occurrence rate of wind storm events, solid line, and lightning events, dashed line (data taken from [4]).
Once a severe weather scenario occurs (i.e high wind and/or lightning storm), its intensity and duration are described by characteristic historically fitted, probability distribution functions. Specifically, the wind storm intensity is obtained as follows [4]:
Wwptq “ Wcrt` ∆wptq (6.7)
where Wwptq is the wind speed intensity at time t for the wind event w, Wcrt is the
‘critical’ wind speed assumed to be 8 rm
ss and ∆w is a random surplus of the critical
wind speed threshold. The intensity of a lightning storm is quantified by its lightning ground strike density Ngptq, measured as the number of ground-flashes (or ground-
strikes) per unit of time and area r occ
h¨km2s. The variability of the ground flash density
is assumed log-normally distributed, with parameters fitted based on historical records. The probabilistic model for wind storm duration (Dw), lightning storm duration (Dlg) and the respective intensities are summarised in Table 6.1.
High wind speeds can directly or indirectly damage the line structure, e.g. by friction-induced fatiguing of structure’s joints or by moving/breaking trees branches in
Distribution Scale(a) Shape(b)
Dw Weibull 9.89 1.17
Dlg Weibull 0.96 0.85 ∆wptq Weibull 1.23 1.05
Mean(µNg) Standard Deviation(σNg)
Ngptq log-Normal -5.34 1.07
Table 6.1: Probability distributions for intensity and duration of severe weather events [4].
the proximity of the line. Lightning strikes can damage lines, for instance triggering insulator dielectric breakdown. By considering an overhead line as made of subcom- ponents in series (the insulation and the mechanical structure) and assuming that the individual subcomponent failure depends on different physical phenomena, the total failure rate can be obtained as follows [4]:
λptq “ λn` λwpWwptqq ` λlgpNgptqq (6.8)
were λw is the contribution to the total line failure rate per km due to high wind speed
at time t and λlg the lightning storms contribution. Note that it is possible, although
very unlikely, to face the simultaneous occurrence of a lightning storm and a high wind event.
The contribution to the line failure rate due to high wind is expressed as follows [4]: λwpWwptqq “ λn ˆ Wwptq2 W2 crt ´ 1 ˙ αw (6.9)
were αw is a regression parameter obtained from failure data. The failure rate due
to a wind event has a strong relation with the wind intensity, following a quadratic law. It can be observed that for wind speed less or equal to the critical wind speed (Wwptq ď Wcrt) the wind contribution to the failure rate is null.
The contribution to the line failure rate due to lightning is obtained as follows [4]: λlgpNgptqq “ λnβlgNgptq (6.10)
βlg is a regression coefficient fitted on historical data and the line failure rate is linearly
related to the lightning event intensity. According to [23], the failure rate of the generic line λiptq expressed in [occh ] can be obtained from the total failure rate by multiplying
it by the line length (li).
6.2.3 Weather-Dependent Repairs
Delays in the repair of power grid components can be caused by, for instance, ineffective communication between all of the parties involved (e.g. non-cooperative landowner) or harsh weather which slows down the identification of the trouble location (e.g. heli-
copters can not be sent to find the fault location and report back). Recently a weather- dependent repair model has been proposed, reflecting the realistic sense that the effi- ciency of repairing crews is strongly affected by the external weather conditions [23]. The model assumes that: (1) a crew of repairmen is dispatched with no delay as soon as a failure in one line occurs, (2) the network becomes fully functional as soon as failed lines are replaced, (3) the time of the failure transient is negligible with respect to the time to repair. It is clear that the time needed to fully replace an overhead line increases if the crew operates in harsh weather conditions. In particular if wind and/or lightning events occur, the normal average repair speed (νnorm) is assumed to decrease
accordingly to the intensity of the severe weather condition. The repair speed can be defined as follows [23]: νrepair “ $ ’ ’ ’ & ’ ’ ’ % νnorm 1`η¨pWwptq´Wcrtq, if Wwptq ě Wcrt & Ng “ 0 νnorm 1`ψ¨Ng, if Wwptq ă Wcrt & Ng ą 0 νnorm r1`η¨pWwptq´Wcrtqs`r1`ψ¨Ngs, if Wwptq ě Wcrt & Ng ą 0 (6.11)
where ψ and η are positive parameters. The normal speed νnorm is set equal to 20 [%h]
(i.e. 5 [h] are needed to replace a line in normal weather conditions), whilst ψ and η are set to 40 and 0.4 following engineering judgement [23]. Also in this work, a line under repair is not subject to further failures (i.e. its time-to-replacement monotonically decreases) whilst repaired/working lines are subject to failures.
6.2.4 Probabilistic Load Demand
Load demands generally display time and space correlations, i.e. the variability of the different power demands depends on the time of the day and their relative position within the network. Stochastic models for load demand have been designed to account for time correlation, e.g. [87]. The model employed here considers daily variability of the average load demand and neglects seasonal and holiday effects. The aggregated load connected to a node i at a time t (Liptq) can be described by a Normal distribution
with parameters fitted on historical data [122]: f pLiptqq “ 1 a p2πqσLiptq e´ Liptq´µLiptq 2σLiptq2 (6.12)
where Liptqis the load demand at node i at hour of the day t, µLiptqis the load mean