Robust Power System Stability Assessment with Extensions to Inertia Control

Robust
Power System Stability
Assessment with
Extensions to
Inertia Control

Outline
▪ Introduction to Transient Stability AssessmentandEnergy
Methods
▪
Robust Stability Certificate
▪
Robust ResiliencyCertificate
▪
Certificates including RemedialActions: Inertia/Damping
Control
5

Power Blackouts
▪Statistics:
Frequency: ≈1hr/year⟹ economicdamage:≈100B$/year
▪
6

Power Blackouts
▪Statistics:
Frequency: ≈1hr/year⟹ economicdamage:≈100B$/year
▪
▪Challenges and opportunities:
Newalgorithms for betterdecision-making
6

Dynamic Security Assessment
▪ Security = ability to
▪ withstand disturbances
Security Assessment:
▶ Screen contingency list
every 15 mins
▶ Prepare contingency plans
for critical scenarios.
State of the art:
▪
Smallsubset of scenarios selected by engineeringjudgement.
▪
No uncertainty due to modeling error and renewableoutput.
7

Dynamic Security Assessment
▪ Security = ability to
▪ withstand disturbances
Security Assessment:
▶ Screen contingency list
every 15 mins
▶ Prepare contingency plans
for critical scenarios.
State of the art:
▪
Smallsubset of scenarios selected by engineeringjudgement.
▪
No uncertainty due to modeling error and renewableoutput.
Can we do better?
7

Security certiﬁcates
С0
▪ Security region: non-convex, NP-hard characterization
8

С0
С1
Security region: non-convex, NP-hard characterization
▪
Security certiﬁcates: tractable suﬃcient conditions
8

С0
С1
С2
Security region: non-convex, NP-hard characterization
▪
Security certiﬁcates: tractable suﬃcient conditions
▪
Strategy: certify security of most of scenarioswith conservative
conditions, usesimulations for fewreally dangerous scenarios
8

Transient Stability Analysis
▪ Goal: Certify convergence to post-fault equilibrium
9

Dynamic simulations are hard:
▶ DAE system with about 10k degrees of freedom
▶ Faster than real-time simulations are still a challenge
9

▪ Direct (Energy) methods:
▶ Developed since 1930s ( graphical “equal area” criterion).
▶ Used by CAISO, TEPCO, ...
▶ Generally conservative, fail to certify many safe events
▶ Based on NP-hard algorithms, poor scalability
▶ Relying on simulations of fault-on dynamics
▶ Applicable to limited choice of models.
9

▪ Direct (Energy) methods:
▶ Developed since 1930s ( graphical “equal area” criterion).
▶ Used by CAISO, TEPCO, ...
▶ Generally conservative, fail to certify many safe events
▶ Based on NP-hard algorithms, poor scalability
▶ Relying on simulations of fault-on dynamics
▶ Applicable to limited choice of models.
Canwedobetter?
9

Swing equation
Mδ̈ +Dδ̇ +κsin(Eδ) −P =0
10

Swing equation
▪ δ ∈ ℝn
- generatorrotor anglesin rotating reference frame
10

Swing equation
n
δ ∈ ℝ - generatorrotor anglesin rotating reference frame
n×n
▪ M = 𝐝𝐢𝐚𝐠[m] ∈ ℝ - turbine inertias
10

Swing equation
▪ δ
M
∈
=
ℝ
𝐝
n
𝐢𝐚
-
𝐠
g
[m
en
]
e
∈
rat
ℝ
or
n×
r
n
otoranglesin rotating referenceframe
- turbine inertias
n×n
▪ D = 𝐝𝐢𝐚𝐠[d] ∈ ℝ - governordroop settings
10

Swing equation
▪ δ
M
∈
=
ℝ
𝐝
n
𝐢𝐚
-
𝐠
g
[m
en
]
e
∈
rat
ℝ
or
n×
r
n
- turbine inertias
n×n
▪ κ ∈ ℝn×m
- coupling (susceptance)matrix
10

Swing equation
▪ δ
M
∈
=
ℝ
𝐝
n
𝐢𝐚
-
𝐠
g
[m
en
]
e
∈
rat
ℝ
or
n×
r
n
- turbine inertias
n×n
▪ κ ∈ ℝn×m
m×n
▪ E ∈ ℝ - incidencematrix of the powernetwork
10

Swing equation
▪ δ
M
∈
=
ℝ
𝐝
n
𝐢𝐚
-
𝐠
g
[m
en
]
e
∈
rat
ℝ
or
n×
r
n
- turbine inertias
n×n
▪ κ ∈ ℝn×m
m×n
▪ E ∈ ℝ - incidencematrix of the powernetwork
▪ P ∈ ℝn
- mechanicaltorques ongenerators
10

Energy function
E =1
2
δ̇T
Mδ̇ −∑ κkj cos δkj −PT
δ.
11

Energy function
E =1
2
δ̇T
δ.
▪ Kinetic energyof the turbines
11

Energy function
E =1
2
δ̇T
δ.
Kinetic energyof the turbines
▪
Inductive energyof thelines
11

Energy function
E =1
2
δ̇T
δ.
Kinetic energyof the turbines
▪
Inductive energyof thelines
▪
Work extracted from steam
11

Energy function
E =1
2
δ̇T
δ.
▪ Kinetic energyof the turbines
▪ Inductive energyof thelines
▪ Work extracted from steam
▪ Total energydecaysin time. 0
5
−5 −5
0
−3
−4
5
−2
−1
1
0
2
4
3
X: 2.513
Y: 0.7854
Z: 0.4943
11

Energy method
Stable Equilibrium Point
Unstable Equilibrium Point
!
?
▪ If E(δ(0), δ(0))<ECUEP, then δ → δ∗ ast→ ∞
12

Energy method
!
?
CUEP ∗If E(δ(0), δ(0)) < E , then δ → δ ast→ ∞
▪ Fast transient stability certiﬁcate
12

Energy method
!
?
Computing ECUEP is anNP-hard problem
12

Energy method
!
?
CUEPComputing E isanNP-hardproblem
▪ Certiﬁcates aregenerallyconservative
12

Conservativeness
▪ Fortypical generatorlossfault:
▶ All generators decelerate and contribute to power balancing.
13

Conservativeness
▶
∗Extra energy E −E is distributed “uniformly” across many
generators and lines.
13

Conservativeness
▶
▪ generators and lines.
Closest Unstable Equilibria:
▶ Has very concentrated energy distribution.
▶ All extra energy in one line for radial systems.
13

Conservativeness
▶
▪ generators and lines.
Closest Unstable Equilibria:
▶ Has very concentrated energy distribution.
▶ All extra energy in one line for radial systems.
▪ Twoprocessescompete:
▶ Energy decays due to damping
▶ Energy is transfered due to nonlinearity
▶ Enough energy to get to CUEP, not enough time to transfer it
13

Outline
Methods
▪
▪
▪
Control
14

Modeling Approach
▪ Non-linear swingequation
k k k k ∑
j∈𝒩k
m δ̈ + d δ + asin(δ−δ )=Pkj k j k (1)
mkδ̈k +dkδ̇k +
j
∑∈ 𝒩k
akj(sin(δkj)−sin(δ∗
kj))= 0 (2)
ẋ =Ax −BF(Cx) (3)
F(Cx) standsfor the non-linearfunction sin(δ)−sin(δ ∗
kj kj)
▪ Structure-preserving model: A andB donot correspondto the
areindependentof the operating pointPk
15

Bounding nonlinearity
▪ Sector bou
∗
nd
2
onnonlin
∗
earityfor polyt
∗
ope 𝒫 ∶ {δ,
∗
δ̇
2
∶ |δkj| <π
2
}
▶ g(δkj
1
−
−si
δ
nγ
kj)≤(δkj−δkj)(sinδkj −sinδkj) ≤(δkj −δkj)
▶ g =π/2−γ
▶ 0<|δkj| <γ< π/2
16

Stability certiﬁcate
▪ If:
AT̄P +PĀ +
(1−
4
g)2
CT
C +PBBT
P ⪯ 0 (4)
▪ there exists aquadratic Lyapunovfunction V =xT
Px that is
decreasingwheneverx(t)∈ 𝒫.
17

Stability region
−6 0 2 4 6
−6
8
θ
θ˙6
4
2
0
−2
−4
Flow−in bou ndary
SEP
Flow−out bo undary
-2-4
18

Stability region
−6
−6
0 2 4 6 8
θ
θ˙6
4
2
0
−2
−4
Flow−in bou ndary
SEP
-2-4
▪ Stability certiﬁed only aslong astrajectory staysin 𝒫
18

Stability region
−6
−6
0 2 4 6 8
θ
θ˙6
4
2
0
−2
−4
Flow−in bou ndary
SEP
-2-4
Stability certiﬁed only aslong astrajectory staysin𝒫
▪ Levelset of Lyapunovfunction shouldnot intersectthe
18

Systemisstableaslong asV(x(0)) <Vmin with
Vmin =
x
m
∈∂𝒫
inout
V(x)
▪ ConvexOptimization Problem
19

Systemis stableaslong asV(x(0)) <Vmin with
Vmin =
x
m
∈∂𝒫
inout
V(x)
▪ Minimization over∼nhyperplanes
kj
π
2
δ =± ,
̇±δkj ≥0
▪ ConvexOptimization Problem
19

Incorporating uncertainty
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
−2.5
1.5
1
0.5
0
−0.5
−1
−1.5
−2
2
2.5
(-2,2)
(2,-2)
θ
θ˙
▪ Canguaranteestability in the presenceof uncertaintyabout
▪ stableequilibrium points
If δ ∈ Δ(γ) wecancheckstability with asimple condition
▪
Allows uncertainty in operatingconditions
20

Outline
Methods
▪
▪
▪
Control
21

Robust Resiliency Certiﬁcate
Fault: LineTripping
ij u,v▪ Remove from coupling matrix {a } line {a }
ẋF =AxF −BF(CxF) +BD {u,v} sinδFuv
22

Fault: LineTripping
▪ Find P to fulﬁll following LMI condition:
̄T ̄A P +PA +
(1− g)2
4 {u,v}CT
C +PBBT
P + μPBD DT
{u,v}
T
B P ⪯0
(5)
22

Fault: LineTripping
▪ Find P to fulﬁll following LMI condition:
̄T ̄A P +PA +
(1− g)2
4 {u,v}CT
C +PBBT
P + μPBD DT
{u,v}
T
B P ⪯0
(5)
▪ If τclearing ≤ μVmin, the system is guaranteed to remainstable.
22

FFrom LMI proof it wasshownthat: V(x ) ≤ 1
μ
F clearing▪ Fault-cleared state x (τ )isstill in the set R .
τProof:
: ﬁrst time the fault-on trajectory meetsboundary segments.
V(xF(τ)) −V(xF(0)) =
∫0
τ
V(xF(t))dt≤
μ
τ
τ ≥ μVmin
V(xF(0)) = 0 and V(xF(τ)) =Vmin
Forall τclearing ≤μVmin, the trajectory remainsin set R (stable).
23

Fault-On Dynamics
▪ Opportunity for completely simulation-free critical clearingtime
estimation
24

▪ Forasinglefault:
AT̄P +PĀ +
(1−
4
g)2
CT
C +PBBT
P +μPBD{u,v}DT
{u,v}BT
P ⪯0(6)
Weextend it for multiple faults:
25

AT̄P +PĀ +
(1−
4
g)2
CT
C +PBBT
P +μPBD{u,v}DT
{u,v}BT
P ⪯0(6)
{u,v}
T
{u,v}▪ Eq. 6 will hold for any D so that D ⪰ D D .
25

AT̄P +PĀ +
(1−
4
g)2
CT
C +PBBT
P +μPBD{u,v}DT
{u,v}BT
P ⪯0(6)
▪ {u,v}
T
{u,v}Eq. 6 will hold for any D so that D ⪰ D D .
T
u,v∈E {u,v} {u,v}▪ Here we select D = ∑ D D = I|E|×|E|
̄T ̄A P +PA +
(1− g)2
4
CT
C +(1+μ)PBBT
P ≤ 0,
25

Remedial action schemes
Canwestabilize the systemwith fast response?
26

▪
Virtual inertia anddampingfor wind andsolarpower.
26

▪
▪
HVDC lines, FACTS:adjustable susceptance.
26

▪
▪
HVDC lines, FACTS:adjustable susceptance.
▪
Storage, demandresponse,etc.
26

Outline
Methods
▪
▪
▪
Control
27

Resiliency Certificate with Inertia Control
Assumption: There isafault for which wecannot find a P
T
▪ Goal: increasethe volumeof the ellipsoid x Px bydetermining
the minimum additional inertia anddampingcoefficients m, d.
⎡
⎢
⎣
T ̃A(m, d) P +PA(m, d)+
(1− g)2
4
T ̃C C PB(m,d)
T ̃B(m, d)P −I
⎤
⎥
⎦
≤ 0, (7)
▪ Problem isbilinear:
▶ Determine optimal m, dand P at the same time
28

Problem reformulation
1) Rewrite matrices A, B, C removingthe slackbus⇒ eliminate
zero eigenvalue
T
2) Introduce rescaling factor Q =Λ Λ, with Q, Λ diagonal, sothat:
ẋ = Ax− BΛ−1
ΛF(Cx) (8)
⎢
⎣
̄⎡ T ̄A P +PA +
(1− g)2
4
T
C QC PB ⎤
⎥
BT
P −Q ⎦
≤0, (9)
▪ Reformulation isexact for any Q >0. Let solverfreely
determine Q.
29

1) Find apositive deﬁnite matrix P satisfying the LMI for basecase.
30

1) Find apositive deﬁnite matrix P satisfying the L
τ
M
cle
I
ar
f
i
o
ng
rbasecase.
2) Calculate the minimum value Vmin andlet μ =
Vmin
.
30

τ
M
cle
I
ar
f
i
o
ng
rbasecase.
Vmin
.
deﬁnite matrix P =P(m, d)such that3) Set the upperboundsfor inertia anddampingandﬁnd apositive
⎡
⎢
⎣
T ̃ ̃ ̄A(m, d) P + PA( +
(1− g)2
4
T ̃C QC P B(m,d)
̄ T ̃B(m, d)P −Q
⎤
⎥
⎦
≤ 0,
(10)
and
P ≤P(m,d). (11)
30

τ
M
cle
I
ar
f
i
o
ng
rbasecase.
Vmin
.
⎡
⎢
⎣
T ̃ ̃ ̄A(m, d) P + PA( +
(1− g)2
4
T ̃C QC P B(m,d)
̄ T ̃B(m, d)P −Q
⎤
⎥
⎦
≤ 0,
(10)
and
P ≤P(m,d). (11)
4) Forﬁxed P minimize mand d
30

τ
M
cle
I
ar
f
i
o
ng
rbasecase.
Vmin
.
⎡
⎢
⎣
T ̃ ̃ ̄A(m, d) P + PA( +
(1− g)2
4
T ̃C QC P B(m,d)
̄ T ̃B(m, d)P −Q
⎤
⎥
⎦
≤ 0,
(10)
and
P ≤P(m,d). (11)
4) Forﬁxed P minimize mand d
5) Alternate betweensteps4) and 5)
30

clearing At the clearingtime τclearing, the fault iscleared and
the inertia anddampingaretuned backto their initial values.
▪ SinceP(m, d)satisﬁesthe LMI (10), andweselectedμ=
τclearing
Vmin
T
0 0 minwe can prove that x P(m,d)x < V .
T
0
T
00 0 min▪ Together with P ≤ P(m,d) leads to x Px ≤ x P(m,d)x < V .
▪ Conclusion: fault-cleared state staysinside the regionof
attraction.
31

Algorithm for Identifying Inertia Setpoints
1. Set maximumvaluesfor virtual inertia anddampingmi, di for
eachgenerator i
2. Find P = P(mi, di)
3. Forﬁxed P minimize M =diag(mi) andD =diag(di)
∗ ∗
i i4. Alternate betweensteps2-3 until convergence to m, d .
▪ Translate the boundsm∗
i, d∗
ito constraints about the injected
power, i.e. power, energy, andramp rate.
32

Numerical Example
▪ Unstable case
▪ newm, dduring the fault, τclearing = 200ms
33

Numerical Example
▪ Higher inertia resultsto lowergrowth rate of the Lyapunov
function
34

Conclusions
▪ Powerinterruptions areextremely costly
35

Conclusions
Powerinterruptions areextremely costly
▪
Secureoperation ischallenging
35

Conclusions
Powerinterruptions areextremely costly
▪▪ Secureoperation ischallenging
Next generationsecurity assessmenttoolset
▶ Offline construction of certificates
▶ Tools from nonlinear and convex analysis
▶ Rigorous security and stability certificates
▶ Incorporation of Remedial Actions in the security certificates
35

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