convexity and approximation of nonlinear
gaussian chance constraints
Miles Lubin, Juan Pablo Vielma, and Daniel Bienstock
November 2, 2015
MIT & Columbia
motivation: chance constrained optimal power flow
Optimal power flow
minimize
p,θ
∑
ci pi
i∈G
subject to
∑
Bbn θn =
n∈B
∑
pi + wfb − db ,
pmin
≤ pi ≤ pmax
,
i
i
fmn = βmn (θm − θn ),
−
∀b ∈ B,
i∈Gb
fmax
mn
≤ fmn ≤
fmax
mn ,
∀i ∈ G,
∀{m, n} ∈ L,
∀{m, n} ∈ L,
where wfb is (uncertain) contribution from wind.
∙ Min-cost network flow governed by physical transmission
constraints
∙ Above model is “DC approximation” to nonlinear AC powerflow
laws
1
chance chance constraint model (bienstock et al., 2014)
Proposal: generators implement a proportional response policy to
random deviations in the wind forecast Ω.
pi = pi − αi Ω
∙ “Supply = Demand” always satisfied if
∑
i
αi = 1.
2
chance chance constraint model (bienstock et al., 2014)
Proposal: generators implement a proportional response policy to
random deviations in the wind forecast Ω.
pi = pi − αi Ω
∙ “Supply = Demand” always satisfied if
∑
i
αi = 1.
We then impose that the random line flows fmn stay within limits
with high probability:
P(fmn ≤ fmax
mn ) ≥ 1 − ϵ
P(fmn ≥ −fmax
mn ) ≥ 1 − ϵ
∙ Natural to treat these as soft constraints, no immediate
consequences if violated
2
gaussian chance constraints
Line flow chance constraints can be expressed as linear chance
constraints of the form
Pξ (xT ξ ≤ b) ≥ 1 − ϵ.
(1)
If ξ is jointly Gaussian with known mean and covariance and ϵ ≤ 21 ,
then (1) is representable as a single second-order cone constraint,
convex in x and b.
3
gaussian chance constraints
Line flow chance constraints can be expressed as linear chance
constraints of the form
Pξ (xT ξ ≤ b) ≥ 1 − ϵ.
(1)
If ξ is jointly Gaussian with known mean and covariance and ϵ ≤ 21 ,
then (1) is representable as a single second-order cone constraint,
convex in x and b.
Gaussian assumption can be relaxed by introducing uncertainty sets
on mean and covariance (Bienstock et al., 2014; Lubin et al., 2015)
3
improving the power flow model
Voltage-aware optimal power flow (Chertkov)
∑
minimize
ci pi
p,θ
i∈G
subject to
∑
Bbn θn =
n∈B
∑
∑
pi + wfb − db ,
∀b ∈ B,
q
qi + wfq
b − db ,
∀b ∈ B,
i∈Gb
Bbn vn =
n∈B
i∈Gb
pmin
i
qmin
i
fpmn
fqmn
(fpmn )2
∑
≤ pi ≤ pmax
,
i
∀i ∈ G,
qmax
,
i
∀i ∈ G,
≤ qi ≤
= βmn (θm − θn ),
∀{m, n} ∈ L,
= βmn (vm − vn ),
∀{m, n} ∈ L,
+
(fqmn )2
≤
2
(fmax
mn ) ,
∀{m, n} ∈ L,
4
Active and reactive power flow on the same physical lines,
transmission is limited by the norm,
2
(fpmn )2 + (fqmn )2 ≤ (fmax
mn ) ,
∀{m, n} ∈ L,
5
Active and reactive power flow on the same physical lines,
transmission is limited by the norm,
2
(fpmn )2 + (fqmn )2 ≤ (fmax
mn ) ,
∀{m, n} ∈ L,
Using a similar model to account for uncertainty in wind, we end up
with a chance constraint of the form
(
)
Pξ (aT ξ + b)2 + (cT ξ + d)2 ≤ k ≥ 1 − ϵ,
where a, b, c, d are decision variables.
5
Active and reactive power flow on the same physical lines,
transmission is limited by the norm,
2
(fpmn )2 + (fqmn )2 ≤ (fmax
mn ) ,
∀{m, n} ∈ L,
Using a similar model to account for uncertainty in wind, we end up
with a chance constraint of the form
(
)
Pξ (aT ξ + b)2 + (cT ξ + d)2 ≤ k ≥ 1 − ϵ,
where a, b, c, d are decision variables.
Is this a convex constraint??
5
Not convex for ϵ = 0.445
Satisfied with probability
P((xξ1 )2 + (yξ2 )2 ≤ 1) ≥ 1 − ϵ
0.550
0.545
0.540
0.535
0.530
0.4
0.6
0.8
1.0
1.2
x
6
Not convex for ϵ = 0.445
P((xξ1 )2 + (yξ2 )2 ≤ 1) ≥ 1 − ϵ
∙ Counterexample does not apply for smaller ϵ, but anyway let’s
look for approximations
7
summary of results
(
)
∙ Simpler constraint Pξ |aT ξ + b| ≤ k ≥ 1 − ϵ is convex
∙ Theoretically tractable by separation oracles, we give SOCP
approximation with provable approximation guarantee.
∙ Using these absolute value constraints, we obtain a
conservative approximation to the quadratic chance constraint
∙ Favorable comparison with alternative approximations via
robust optimization and CVaR (Nemirovski and Shapiro, 2007)
8
preliminaries: sϵ
Define
∫y
Sϵ := {(x, y) : x φ(t) dt ≥ 1 − ϵ} = {(x, y) : Φ(y) − Φ(x) ≥ 1 − ϵ}.
Then the set Sϵ is convex for ϵ ∈ (0, 1).
Proof: Left-hand side is log-concave.
9
4
3
2
1
0
-4
-3
-2
-1
0
Figure: Sϵ for ϵ = 0.6, 0.7, 0.8, 0.9, 0.95
10
preliminaries: s̄ϵ
Let
S̄ϵ := cl{(x, y, z) : (x/z, y/z) ∈ Sϵ , z > 0}
= cl{(x, y, z) : Φ(y/z) − Φ(x/z) ≥ 1 − ϵ, z > 0}
= cone(Sϵ × {1})
be the conic hull of Sϵ . Then S̄ϵ is convex.
11
Figure: S̄ϵ
12
convexity
Let ξ be a vector of i.i.d. standard Gaussian random variables and
0 < ϵ ≤ 21 . Then the set
{(a, b, k) ∈ Rn × R × R : P(|aT ξ + b| ≤ k) ≥ 1 − ϵ}
is convex.
13
{(a, b, k) ∈ Rn × R × R : P(|aT ξ + b| ≤ k) ≥ 1 − ϵ}
Proof:
P(|aT ξ + b| ≤ k) ≥ 1 − ϵ
iff
P(−k − b ≤ aT ξ ≤ k − b) ≥ 1 − ϵ
iff
(
P
−k − b
aT ξ
k−b
≤
≤
||a||
||a||
||a||
)
≥1−ϵ
iff
(
iff ϵ ≤
1
2
)
(−k − b, k − b, ||a||) ∈ S̄ϵ
∃ t ≥ ||a|| such that (−k − b, k − b, t) ∈ S̄ϵ
14
{(a, b, k) ∈ Rn × R × R : P(|aT ξ + b| ≤ k) ≥ 1 − ϵ} =
}
{
proja,b,k (a, b, k, t) : t ≥ ||a||, (−k − b, k − b, t) ∈ S̄ϵ
∙ Representation using standard SOC constraint plus
nonstandard “Gaussian integral” cone.
∙ Theoretically tractable via separation oracles
∙ Don’t know of any solvers which support as is
15
{(a, b, k) ∈ Rn × R × R : P(|aT ξ + b| ≤ k) ≥ 1 − ϵ} =
}
{
proja,b,k (a, b, k, t) : t ≥ ||a||, (−k − b, k − b, t) ∈ S̄ϵ
∙ Representation using standard SOC constraint plus
nonstandard “Gaussian integral” cone.
∙ Theoretically tractable via separation oracles
∙ Don’t know of any solvers which support as is
∙ Will develop polyhedral approximation of S̄ϵ
15
2-constraint outer approximation of sϵ
3.0
2.5
2.0
1.5
1.0
0.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
∙ With two linear constraints, we guarantee that chance
constraint holds with 2ϵ. (Can be made conservative.)
∙ Proof: split into two linear chance constraints
P(aT ξ + b ≤ k) ≥ 1 − ϵ, P(aT ξ + b ≥ −k) ≥ 1 − ϵ
16
3-constraint outer approximation of sϵ
3.0
2.5
2.0
1.5
1.0
0.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
∙ With three linear constraints, we guarantee that chance
constraint holds with 1.25ϵ.
∙ Proof:
17
3-constraint outer approximation of sϵ
3.0
2.5
2.0
1.5
1.0
0.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
∙ With three linear constraints, we guarantee that chance
constraint holds with 1.25ϵ.
∙ Proof: see the paper
17
approximating quadratic chance constraints
Fix ϵ <
1
2
and β ∈ (0, 1). If ∃ f1 , f2 such that
P(|aT ξ + b| ≤ f1 ) ≥ 1 − βϵ
P(|cT ξ + d| ≤ f2 ) ≥ 1 − (1 − β)ϵ
f21 + f22 ≤ k
then
(
)
P (aT ξ + b)2 + (cT ξ + d)2 ≤ k ≥ 1 − ϵ.
Proof:
18
approximating quadratic chance constraints
Fix ϵ <
1
2
and β ∈ (0, 1). If ∃ f1 , f2 such that
P(|aT ξ + b| ≤ f1 ) ≥ 1 − βϵ
P(|cT ξ + d| ≤ f2 ) ≥ 1 − (1 − β)ϵ
f21 + f22 ≤ k
then
(
)
P (aT ξ + b)2 + (cT ξ + d)2 ≤ k ≥ 1 − ϵ.
Proof: Union bound
18
Proposed a convex, tractable (via SOCP) approximation for
(
)
P (aT ξ + b)2 + (cT ξ + d)2 ≤ k ≥ 1 − ϵ.
What about other approaches?
19
Proposed a convex, tractable (via SOCP) approximation for
(
)
P (aT ξ + b)2 + (cT ξ + d)2 ≤ k ≥ 1 − ϵ.
What about other approaches?
∙ Robust optimization
∙ CVaR
19
robust approximation
Choose suitable uncertainty set U and enforce
(aT ζ + b)2 + (cT ζ + d)2 ≤ k, ∀ ζ ∈ U.
If U is ellipsoidal, tractable via SDP (Ben Tal et al., 2009).
∙ Little guidance on choosing U to enforce chance constraint
under Gaussian distribution.
∙ Naive approach: choose such that P(ξ ∈ U) = 1 − ϵ.
20
“cvar” approximation
Proposed by Nemirovski and Shapiro (2007), rewrite constraint as
[
]
E I((aT ξ + b)2 + (cT ξ + d)2 − k) ≤ ϵ,
where I(t) = 1 if t ≥ 0 and 0 otherwise. Any convex upper bound on
I yields a convex conservative approximation. “CVaR” approximation
constructs the best convex upper bound on I.
∙ Convex, but membership requires multidimensional integration
∙ Misconception: not the “best” convex approximation to the
original constraint
21
three proposed approximations
1. “Two-sided” approximation via absolute value chance
constraints (SOCP)
2. Robust optimization approximation (SDP)
3. CVaR (Multidimensional integration)
22
1.5
1.0
ct
Exa
0.5
Robust
R
CVa
0.0
ded
0.5
si
Two
1.0
1.5
1.5
1.0
0.5 0.0
0.5
1.0
1.5
Figure: P((xξ1 )2 + (yξ2 )2 ≤ 1) ≥ 1 − ϵ, ϵ = 0.5
23
Exact
Robust TwCVaR
o si
ded
0.4
0.2
0.6
0.4 Robust
0.0
0.3
0.2
0.2
0.4
0.1
0.6
0.6
0.4
0.2
0.0
0.2
0.4
Exact
0.5
0.0
0.0
0.1
CV
Two aR
side
d
0.2
0.3
0.4
0.5
0.6
Figure: P((xξ1 )2 + (yξ2 )2 ≤ 1) ≥ 1 − ϵ, ϵ = 0.05
24
conclusions
∙ No approximation dominates another, but “two-sided” is most
tractable
∙ JuMPChance modeling extension for JuMP supports these
constraints
∙ Power systems case study underway with Y. Dvorkin and L.
Roald
http://arxiv.org/abs/1507.01995
25
Thanks! Questions?
26