Article
Reference
Adaptive robust vulnerability analysis of power systems under uncertainty: a multilevel OPF-based optimization approach
ABEDI, Amin, HESAMZADEH, Mohammad Reza, ROMERIO-GIUDICI, Franco
Abstract
With the growing level of uncertainties in today's power systems, the vulnerability analysis of a power system with uncertain parameters becomes a must. This paper proposes a two-stage adaptive robust optimization (ARO) model for the vulnerability analysis of power systems. The main goal is to immunize the solutions against all possible realizations of the modeled uncertainty. In doing so, the uncertainties are defined by some pre-determined intervals defined around the expected values of uncertain parameters. In our model, there are a set of first-stage decisions made before the uncertainty is revealed (attacker decision) and a set of second-stage decisions made after the realization of uncertainties (defender decision). This setup is formulated as a mixed-integer trilevel nonlinear program (MITNLP). Then, we recast the proposed trilevel program to a single-level mixed-integer linear program (MILP), applying the strong duality theorem (SDT) and appropriate linearization approaches. The efficient off-the-shelf solvers can guarantee the global optimum of our final MILP model. We also prove a lemma which makes our model [...]
ABEDI, Amin, HESAMZADEH, Mohammad Reza, ROMERIO-GIUDICI, Franco. Adaptive robust vulnerability analysis of power systems under uncertainty: a multilevel OPF-based optimization approach. International Journal of Electrical Power & Energy Systems , 2022, vol. 134, no. 107432, p. 1-13
DOI : 10.1016/j.ijepes.2021.107432
Available at:
http://archive-ouverte.unige.ch/unige:153430
Disclaimer: layout of this document may differ from the published version.
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⎪⎪
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α
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⃒⩽
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d;∀ i ∈ D /9B0
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∑
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ijB
ij−
∑i,j∈N
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z
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ijB
ij+ ω
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(∑
i∈N
( λ
i+ α
i)
̃Pd
i|i∈D+
∑i∈G
γ
ĩPg
i+
∑i,j∈N
( φ
ij− φ
ij) S
maxij+
∑i∈N
( ω
i− ω
i) θ
maxi )/6?0
Maximize
Z,̃Pg,̃Pd, λi,μij,αi,ωi, ωi,γi,φij,φij
(∑
i∈N
( λ
i+ α
i)
̃Pd
i|i∈D+
∑i∈G
γ
ĩPg
i+
∑i,j∈N
( φ
ij− φ
ij) S
maxij+
∑i∈N
( ω
i− ω
i) θ
maxi )/9<0
04
' 1
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( = 6,...,
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(̃
Pg
i,̃Pd
i)
= Max {
∑i∈N
(
λ
pi+ α
pi)Pd
̃ i|i∈D+
∑
i∈G
γ
pĩPg
i+
∑i,j∈N
(
φ
pij− φ
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S
maxij+
∑i∈N
(
ω
pi− ω
pi)θ
maxi ,p =
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̃Pd
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2
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= Pg
i+
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g−i/<?0
̃
Pd
i= Pd
i+
̂Pd
iβ
d+i− Pd
̂ iβ
d−i/<:0 β
+ ,β
− ,β
+ ,β
−∈ {;,6}. * A β
()+= 6 β
()−= 6 β
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()−. * - /<?0 /<:0
λ̃*|∈α
̃*|∈γ
̃*λ
β
+λ
β
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β
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β
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β
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β
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β
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β
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(/<90. 2
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ij⩽T
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ij/D;0
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i∈G
( β
g+i+ β
g−i)⩽ DR
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g+i+ β
g−i ⩽1;∀ i ∈ G /D=0
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d;∀ i ∈ D /D>0
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d+i+ β
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Z,̃Pg,̃Pd,λi,μij,αi ,ωi,ωi,γi,φij,φij
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
∑
i∈N
Pd
i|i∈D( λ
i+ α
i) +
∑i∈N
( Pd
̂ i|i∈DT
1i− Pd
̂ i|i∈DT
2i) +
∑i∈N
( Pd
̂ i|i∈DT
3i−
̂Pd
i|i∈DT
4i) +
∑i∈G
γ
iPg
i|i∈G+
∑i∈G
(
̂Pg
i|i∈GT
5i−
̂Pg
i|i∈GT
6i) +
∑i,j∈N
( φ
ij− φ
ij) S
maxij+
∑i∈N
( ω
i− ω
i) θ
maxi⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
/D:0 8
0.5
×
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(1 − z
ij) = NPO
;∀ i
,j ∈ N /DB0
z
ij= z
ji;∀ i
,j ∈ N /=;0
∑
i∈G
( β
g+i+ β
g−i)⩽ DR
g;∀ i ∈ G /=60
β
g+i+ β
g−i ⩽1;∀ i ∈ G /=90
∑
i∈D
( β
d+i+ β
d−i)⩽ DR
d;∀ i ∈ D /=<0
β
d+i+ β
d−i ⩽1;∀ i ∈ D /=D0
− λ
i+ μ
ij+ φ
ij+ φ
ij=
0;∀ i
,j ∈ N /==0 λ
i+ γ
i⩽0;∀ i ∈ G /=>0
∑
i,j∈N
|S(ij)=i
B
ijT
ij−
∑i,j∈N
|R(ij)=i
B
ijT
ij+ ω
i+ ω
i=
0;∀ i
,j ∈ N /=?0
α
i+ λ
i⩽1;∀ i ∈ D /=:0 T
ij= μ
ij− H
ij,− B
1z
ij⩽T
ij⩽B
1z
ij,− B
1( 1− z
ij)⩽
H
ij⩽B
1( 1− z
ij);
∀ i
,j ∈ N /=B0
T
1i= λ
i− H
1i,− B
2β
d+i ⩽T
1i⩽B
2β
d+i ,− B
2( 1− β
d+i )⩽
H
1i⩽B
2( 1− β
d+i );
∀ i ∈ D />;0 T
2i= λ
i− H
2i,− B
2β
d−i ⩽T
2i⩽B
2β
d−i ,− B
2( 1− β
d−i )⩽
H
2i⩽B
2( 1− β
d−i );
∀ i ∈ D />60 T
3i= α
i− H
3i,− B
3β
d+i ⩽T
3i⩽B
3β
d+i ,− B
3( 1− β
d+i )⩽
H
3i⩽B
3( 1− β
d+i );
∀ i ∈ D />90 T
4i= α
i− H
4i,− B
3β
d−i ⩽T
4i⩽B
3β
d−i ,− B
3( 1− β
d−i )⩽
H
4i⩽B
3( 1− β
d−i );
∀ i ∈ D /><0 T
5i= γ
i− H
5i,− B
4β
g+i ⩽T
5i⩽B
4β
g+i ,− B
4( 1− β
gi+)⩽
H
5i⩽B
4( 1− β
gi+);
∀ i ∈ G />D0 T
6i= γ
i− H
6i,− B
4β
g−i ⩽T
6i⩽B
4β
g−i ,− B
4( 1− β
g−i )⩽
H
6i⩽B
4( 1− β
g−i );
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MITNLP model which is an NP-hard problem converges to the global optimum [22].
5. Numerical results 5.1. The IEEE RTS
The IEEE RTS consisting of 24 buses, 32 generator units, and 38 branches (lines and transformers) is used in this paper. Fig. 3 shows the
single-line diagram of single area IEEE RTS.
Table 1presents the generating unit characteristics. Other detailed data of this test system can be found in [60].
All the simulations have been performed using a processor at 2.4 GHz with a total RAM of 32 GB. We also implement our model using the GAMS modeling language environment and the solver CPLEX [61]. In all simulations, the CPLEX relative optimality criterion was set at 0.001.
Table 2 shows the size and complexity of our model for the IEEE RTS
comparing with the previous literature.
Table 3
Results for the IEEE RTS without uncertainty in comparison with previous literature.
NPO References [25,27,31]* This work
Critical lines load shedding (MW) Critical lines load shedding (MW)
1 – 0 – 0
2 16–19,20–23 309 16–19,20–23 309
3 7–8,15–21,16–17 387 7–8,15–21,16–17 387
4 3–24,12–23,13–23,14–16 516 3–24,12–23,13–23,14–16 516
5 12–23,13–23,15–21,16–17,20–23 872 12–23,13–23,15–21,16–17,20–23 872
6 11–13,12–13,12–23,15–21,16–17, 20–23 1198 11–13,12–13,12–23,15–21,16–17,20–23 1198
*Two parallel lines on the same tower are considered as two independent lines in references [25,27].
Fig. 4. (a) Comparison of the optimal solutions (LS) with different levels of load uncertainty (DRd =0 (no uncertainty) to DRd =17 (the most conservative case)), (b) Maximum difference of LS in comparison with “no uncertainty” case when DRd =17.
Fig. 5. (a) Comparison of the optimal solutions (LS) with different levels of generation uncertainty (DRg =0 (no uncertainty) to DRg =14 (the most conservative case)), (b) Maximum difference of LS in comparison with “no uncertainty” i.e. when DRg =14.
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11
load bus number 15 leads to a system which is not
“N-1
”secure.
5.2. Iran’s 400-kV power system
Iran’s 400-kV power system as a realistic test system is used in this subsection. This system is comprised of 28 generators, 52 buses, and 99 transmission lines as shown in Fig. 9. The solid assets are existing assets and the dashed assets are candidate assets that are planned for the expanded network
[63]. The detailed data associated with this powersystem can be found in [63,64].
In this subsection, DR
dand DR
gtake values in the range of zero (no uncertainty) to the total number of generation units and load buses which are 28 and 48, respectively. Furthermore, we set the range of generation and load variations to be
̂Pgi=α
giPgi,∀i∈G and ̂Pdi =α
diPdi,∀i∈D, respectively. We assume
α
giand α
diare fixed at 0.2 and 0.05 for all generation and load units, respectively. Moreover, the impact of the uncertainties on the vulnerability analysis is investigated for both
existing and expanded networks.
Fig. 10 and Fig. 11 show the LS as a function of NPO in the existing
and expanded networks, respectively, considering the uncertain pa- rameters. As expected, the expanded network operates more reliable and robust than the existing network. For instance, when there is no un- certainty, the expanded network is N-1 secure, and the total possible LS i.e., 10390 MW occurs with more transmission line outages as compared to the existing network. Moreover, for a given NPO, the LS in the expanded network is lower than the one for the existing one.
Fig. 11 also highlights that the expanded network might be no longer
N-1 secure when there are uncertain load and/or generation units.
Finally, the proposed set of critical lines is also different. For example, when NPO
=1, line 1–2 is a critical transmission line for the existing network. However, considering the uncertainty and the most conser- vative case, the critical transmission line will change to line 30
–31.
Fig. 9. Modified Iran’s 400-kV power system from [63], the solid assets are existing assets and the dashed assets are planned to be added.
Fig. 10. Comparison of the optimal solutions with different levels of uncer- tainty in the existing network (DRd =DRg =0 (no uncertainty) to DRg =28 and DRd =48 (the most conservative case)).
Fig. 11.Comparison of the optimal solutions with different levels of uncer- tainty in the expanded network (DRd =DRg =0 (no uncertainty) to DRg =28 and DRd =48 (the most conservative case)).
12 6. Conclusions
This study aims to propose a three-level optimization problem to analyze the power system vulnerability in the context of system uncer- tainty. A robust optimization approach has been proposed. In our pro- posed model, the ULP represents the attacker, the MLP models the worst- case uncertainty level, and finally, the LLP represents the defender. The proposed model is a MITNLP, which is hard to solve. Applying the SDT of the linear programs, the LLP is replaced by its dual program. Then the original MITNLP problem is recast to a max-max-max problem, which can be presented as a single-level MINLP. The nonlinear terms of the single-level MINLP model are linearized using appropriate linearization approaches. We also observe two properties of our MITNLP model and prove a lemma that improves the computational performance of our proposed final MILP model. Our final MILP model has been applied to the IEEE RTS and modified Iran’s power system, and the simulation results are carefully studied. Our simulation results show that the vulnerability analysis without considering uncertainties leads to opti- mistic results. Moreover, increasing the level of uncertainty in our case studies leads to higher levels of LS and different critical lines in com- parison with no uncertainty case. An interesting future research project can be dedicated to investigating how our final model can be adjusted to model the asymmetric uncertainties accurately.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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