Fig. 6.2 presents the flow chart of the proposed framework, where the Artificial Neural Network is employed to speed up the calculations. The algorithm starts by inputting parameters of the stochastic model (e.g. shape and scale parameters, critical wind speed, failure rates, etc.) and two main blocks can be seen. The first block is, in essence, an event sampler whilst the second uses the emulator to calculate the load curtailed for each sampled failure event f. Then, the ENS is obtained by Eq. 6.3.
First, ‘normal’ time to failures and occurrence time of severe weather events are sampled using HPP and NHPP, respectively. Then, a vector of Times-to-Events (TTEs) is obtained by sorting the occurrence times and recording the type of event (i.e. normal failure, lighting and/or strong wind). Then, a sequential Monte Carlo (S-MC) starts, see as example Refs. [23]- [143]. This iterative procedure terminates if the maximum simulation time is reached (t ą 8760 [h]) or all the sampled events (normal failure and severe weather) have been analysed. The S-MC procedure is summarised as follows:
t=MU e=cU f=M
+HPP> Sample Normal failure Events in [M U Tsim]
+NHPP> Sample Extreme Weather Events in [M U Tsim]
and sort the Time to Events +TTE>
Set t equal to TTE+e> Is the event i
a normal Failure?
yes
no Sample Extreme Weather
Intensity and Duration +Td>
Compute failure rates and Sample TTF
Is tATTF+f>> tATd ?
no yes
Sample line failures Xf
and load profile Lffor t
E
ve
n
ts
gH
is
to
ry
S
am
p
le
r
ANN
Sample line failures Xf
and load profile Lffor t
set t=tATdU f=fAc
Update Repair Speed and Time to Repair +TTR> Is t> Tsim?
yes no
Compute Load Curtailed
using Surrogate Model Compute ENS+f> usingL
cut+f> and TTR+f> and
sum each contribution
Output: EnergyRNotRSupplied in [MUT
sim]
Update Repair Speed and Time to Repair +TTR>
Input:
f=fAc
Is e the last event ? or
e=eAc
Figure 6.2: A simplified flow-chart for the resilience analysis by sequential Monte Carlo simulation. The probabilistic model is used to sample failures and repairs whilst the Artificial Neural Network is used to compute load curtailments.
1 Set t equal to the occurrence time of first event, TTE(e “ 1) and failure index f “ 0. If e is a ‘normal’ failure, go to point 2 otherwise go to point 3.
2 Set f “ f ` 1 and sample a line i from the probability mass distribution with values λn¨li¨Xf,i
řNl
l“1λn¨li¨Xf,i
and l “ 1, .., Nl. Set Xf,i “ 0, sample a load profile (Lf)
accordingly to t. Set the failed line replacement to 100 % and save Xf and Lf
for f. Update % from full replacement using TTE(e ` 1)-TTE(e) and normal replacement speed. Then, go to point 5.
3 Sample the severe weather duration (Te) and intensity (Ng, ∆w), compute the
increased total failure rate λptq using Eq.6.8. Sample time-to-failure using the HPP, and by inputting λptq and the interval [t t ` Te]. If at least one failure
event is sampled, set t equal to the next failure occurrence and proceed to step 4, otherwise go to step 5.
4 Set f “ f ` 1, sample one failed line using the probability mass function
λptq¨li¨Xf,i
řNl
l“1λptq¨li¨Xf,i
and l “ 1, .., Nl. Sample a load profile (Lf) accordingly to t. Set
the failed line replacement to 100 % and save the lines states vector Xf and load
profile Lf for failure f. Update lines % from full replacement, using the reduced
repairing speed computed as in Eq. 6.11. If the severe weather failures are all evaluated go to step 5, otherwise set t equal to the occurrence time of the next failure and repeat point 4.
5 If t ą Tsim or e is the last event in the TTE list, stop simulation. Otherwise, set
e “ e ` 1 and t “ T T Epeq. If it is a ‘normal’ failure, go to point 2 otherwise go to point 3.
The first unit is used to produce a set of Xf and Lf to be used as vectorised input
to a previously trained ANN. The ANN input is an pNl` NLq ˆ F matrix of load
profiles and state vectors, where F is the total number of failures faced by the grid in 1 year. A total of Ns independent histories (Ns S-MC) are simulated until convergence
of the ErENSs is obtained.
Fig.6.3 displays a simplified version of the methods used within the framework. The procedure used to simulate Ns independent grid years is displayed in the top panel on
the left-hand side. The method run by first selecting the OPF solver or the efficient method based on the ANN emulator (i.e. presented in Fig.6.2). If the (time costly) S-MC optimal power flow solver is selected, the procedure displayed in the top panel on the right-hand side run Ns times (the diagram has been adapted from [23]). If the
efficient method is selected, the procedure displayed in Fig.6.2 run Ns times.
The bottom panel in Fig.6.3 summarises the overall work flow for the analysis. This is used to present, from an intuitive point of view, how the efficient resilience assessment
INPUTS: -Loads -State Vectors Train ANN OUTPUT: -Load curtailed
Overall Work Flow for the Analysis
Sensitivity and Generalised UQ
Simulate Ns Years (ANN)
Use the efficient assessment method (in Fig. 3) & Validate S-MC(ANN) against S-MC(OPF)
Impreicse Resilience Propagate Imprecision Global Sensitivity To Reval Imprecision Key Drivers Characterise Imprecision Impreicse weather
parameters Impreicse grid parameters
Simulate Ns Years (OPF)
S-MC
OPF Sequential Time Mone CarloDraw failed line (norm. weather) Assess Load cut (OPF) Assess replacement speed & begin restoration (norm. weather)
Go to next event
t=0
is t<Tmax?
Sample extr. weather intensity and duration
Increase failure rate Gener. time to failure
(HPP, extr. weather)
Draw failed line (extr. weather) Assess Load cut (OPF) Assess replacement speed & begin restoration (extr. weather)
Are all the failures during the extreme weather
duration analysed? Gener. weather events (NHPP) &
time to failure (norm weather (HPP)
An extreme weather event occur first?
end start Compute ENS
S-MC
OPF/ANN is Y < Ns years? Simulate Ns Years (OPF/ANN) Y=1 Y=Y+1 no yes Save: Loads (L) State Vectors (X) Load curtailed (Lcut) Energy not sup (ENS)Compute Expeted energynot supplied E[ENS]=[ENS(1)+...+ENS(Ns)]/Ns
end start
For each level of imprecision (membership): min/max E[ENS] constrained by the Hyper-parameter rectangle p p1 pn membership members. E[ENS] no imprecision small impreicsion Sensiti vi ty Imprecise (weather-grid) parameters Assess the reduction in the E[ENS] imprecision when the imprecision in parameters is reduced
membership
Figure 6.3: A diagram for the overall work flow of the analysis (in the bottom panel). The procedure to compute the ErEN Ss using the S-MC (in top panel on the left-hand side) and the (computationally demanding) algorithm adapted from [23] (in the top panel on the right-hand side).
method is embedded within an advanced uncertainty quantification framework to assess effects of imprecision (i.e. introduced in Section 6.5). The analysis starts by simulating Ns grid years using the S-MC OPF solver. Then, the ANN is trained as explained in
Section 6.3 and its results are validated against the original model (S-M OPF). Once the emulator is validated, the imprecision affecting the poorly known parameters of the grid and weather models is characterised using Credal sets. To conclude, the effect of imprecision on the grid resilience is quantified using advanced uncertainty propagation methods and global sensitivity analysis.