Data-Driven Optimization in Power Systems
Andrea Simonetto
IBM Research
DTU Summer School, June 18, 2019
What’s in hereTime-varying optimizationOptimal power flow problems (that change over time)Regularization (of optimization problems)Measurement feedback in optimization (for cyber-physicalsystems)
Motivation: ...
Recent work with a few researchers, e.g., Dr. Emiliano Dall’AneseE. Dall’Anese, AS, arXiv: 1601.07263A.S., arXiv: 1807.07032Our websites, ... [mailto: [email protected]]
A. Simonetto (IBM Research) 2 / 23
OutlineBasics
I Optimization problems, gradient descent, regularizationI Lagrangian formalism, saddle-point method, convergence
AdvancedI Time-varying optimization ideas, online gradient descent, online
saddle-pointI Measurement feedback I: idea, derivation, insightsI Optimal power flow pursuit: formulation, linear approximationI Measurement feedback II: application to OPF, convergence and
numerical results
Take-home messages
A. Simonetto (IBM Research) 3 / 23
Optimization problemsWe start with convex optimization problems of the form:
minimizex∈Rn
f (x), subject to: g(x) ≤ 0.
Immagine g(x) is not there, then the gradient method is defined as
xk = xk−1 − α∇xf (xk−1)
if f strongly smooth, then f (xk)− f ∗ ≤ O(1/k) (O(1/k2) for FGM)if f strongly convex and strongly smooth, Q-linear convergence O(%k)
‖xk − x∗‖ ≤ %‖xk−1 − x∗‖, % < 1 (α < 2/L)
But the World isn’t always nice...
A. Simonetto (IBM Research) 4 / 23
Optimization problemsWe start with convex optimization problems of the form:
minimizex∈Rn
f (x), subject to: g(x) ≤ 0.
Immagine g(x) is not there, then the gradient method is defined as
xk = xk−1 − α∇xf (xk−1)
if f strongly smooth, then f (xk)− f ∗ ≤ O(1/k) (O(1/k2) for FGM)if f strongly convex and strongly smooth, Q-linear convergence O(%k)
‖xk − x∗‖ ≤ %‖xk−1 − x∗‖, % < 1 (α < 2/L)
But the World isn’t always nice...
A. Simonetto (IBM Research) 4 / 23
Optimization problemsWe start with convex optimization problems of the form:
minimizex∈Rn
f (x), subject to: g(x) ≤ 0.
Immagine g(x) is not there, then the gradient method is defined as
xk = xk−1 − α∇xf (xk−1)
if f strongly smooth, then f (xk)− f ∗ ≤ O(1/k) (O(1/k2) for FGM)if f strongly convex and strongly smooth, Q-linear convergence O(%k)
‖xk − x∗‖ ≤ %‖xk−1 − x∗‖, % < 1 (α < 2/L)
But the World isn’t always nice...
A. Simonetto (IBM Research) 4 / 23
Optimization problemsWe start with convex optimization problems of the form:
minimizex∈Rn
f (x), subject to: g(x) ≤ 0.
Immagine g(x) is not there, then the gradient method is defined as
xk = xk−1 − α∇xf (xk−1)
if f strongly smooth, then f (xk)− f ∗ ≤ O(1/k) (O(1/k2) for FGM)if f strongly convex and strongly smooth, Q-linear convergence O(%k)
‖xk − x∗‖ ≤ %‖xk−1 − x∗‖, % < 1 (α < 2/L)
But the World isn’t always nice...
A. Simonetto (IBM Research) 4 / 23
Regularization can do a lot for youIdea: regularize the problem, solve it fast, then tune theregularization, then ...
See, e.g., F. Glineur, Yu. Nesterov, ...L. Stella, A. Themelis, P. Patrinos, arXiv: 1604.08096
Moreau’s envelope (for non-smooth):
f (x)→ infx∈Rn
{f (z) +
12γ ‖z − x‖
2}, γ > 0
Plain vanilla regularization (for non strongly-convex):
f (x)→ f (x) +εt2 ‖x‖
2
A. Simonetto (IBM Research) 5 / 23
Regularization can do a lot for youIdea: regularize the problem, solve it fast, then tune theregularization, then ...
See, e.g., F. Glineur, Yu. Nesterov, ...L. Stella, A. Themelis, P. Patrinos, arXiv: 1604.08096
Moreau’s envelope (for non-smooth):
f (x)→ infx∈Rn
{f (z) +
12γ ‖z − x‖
2}, γ > 0
Plain vanilla regularization (for non strongly-convex):
f (x)→ f (x) +εt2 ‖x‖
2
A. Simonetto (IBM Research) 5 / 23
Regularization can do a lot for youIdea: regularize the problem, solve it fast, then tune theregularization, then ...
See, e.g., F. Glineur, Yu. Nesterov, ...L. Stella, A. Themelis, P. Patrinos, arXiv: 1604.08096
Moreau’s envelope (for non-smooth):
f (x)→ infx∈Rn
{f (z) +
12γ ‖z − x‖
2}, γ > 0
Plain vanilla regularization (for non strongly-convex):
f (x)→ f (x) +εt2 ‖x‖
2
A. Simonetto (IBM Research) 5 / 23
Constrained optimization problems
minimizex∈Rn
f (x), subject to: g(x) ≤ 0.
Lagrangian formalism:
L(x,µ) := f (x) + µTg(x), x ∈ Rn,µ ∈ Rq+
Saddle-point method:
xk = xk−1 − α∇xL(xk−1,µk−1) =
xk−1 − α[∇xf (xk−1) +∇xTg(xk−1)µk−1
]µk = ΠRq
+[µk−1 + α∇µL(xk−1,µk−1)] = ΠRq
+[µk−1 + αg(xk−1)]
A. Simonetto (IBM Research) 6 / 23
Constrained optimization problems
minimizex∈Rn
f (x), subject to: g(x) ≤ 0.
Lagrangian formalism:
L(x,µ) := f (x) + µTg(x), x ∈ Rn,µ ∈ Rq+
Saddle-point method:
xk = xk−1 − α∇xL(xk−1,µk−1) =
xk−1 − α[∇xf (xk−1) +∇xTg(xk−1)µk−1
]µk = ΠRq
+[µk−1 + α∇µL(xk−1,µk−1)] = ΠRq
+[µk−1 + αg(xk−1)]
A. Simonetto (IBM Research) 6 / 23
Saddle-point convergenceSaddle-point method:
xk = xk−1 − α∇xL(xk−1,µk−1) =
xk−1 − α[∇xf (xk−1) +∇xTg(xk−1)µk−1
]µk = ΠRq
+[µk−1 + α∇µL(xk−1,µk−1)] = ΠRq
+[µk−1 + αg(xk−1)]
A. Simonetto (IBM Research) 7 / 23
Saddle-point convergenceSaddle-point method:
xk = xk−1 − α∇xL(xk−1,µk−1) =
xk−1 − α[∇xf (xk−1) +∇xTg(xk−1)µk−1
]µk = ΠRq
+[µk−1 + α∇µL(xk−1,µk−1)] = ΠRq
+[µk−1 + αg(xk−1)]
Convergence is linear (under Slater’s condition + strongconvexity/concavity and strong smoothness), for small stepsizes.Specifically, call z := [xT,µT]T, then
‖zk − z∗‖ ≤ %‖zk−1 − z∗‖, % < 1 (α < 2m/L2)
A. Simonetto (IBM Research) 7 / 23
Saddle-point convergenceSaddle-point method:
xk = xk−1 − α∇xL(xk−1,µk−1) =
xk−1 − α[∇xf (xk−1) +∇xTg(xk−1)µk−1
]µk = ΠRq
+[µk−1 + α∇µL(xk−1,µk−1)] = ΠRq
+[µk−1 + αg(xk−1)]
Assumptions do not hold, unless..
double regularization:
L(x,µ)→ L(x,µ) +εt2 ‖x‖
2 − νt2 ‖µ‖
2
What does νt imply?
A. Simonetto (IBM Research) 7 / 23
Saddle-point convergenceSaddle-point method:
xk = xk−1 − α∇xL(xk−1,µk−1) =
xk−1 − α[∇xf (xk−1) +∇xTg(xk−1)µk−1
]µk = ΠRq
+[µk−1 + α∇µL(xk−1,µk−1)] = ΠRq
+[µk−1 + αg(xk−1)]
Assumptions do not hold, unless.. double regularization:
L(x,µ)→ L(x,µ) +εt2 ‖x‖
2 − νt2 ‖µ‖
2
What does νt imply?
A. Simonetto (IBM Research) 7 / 23
OutlineBasics
I Optimization problems, gradient descent, regularizationI Lagrangian formalism, saddle-point method, convergence
AdvancedI Time-varying optimization ideas, online gradient descent, online
saddle-pointI Measurement feedback I: idea, derivation, insightsI Optimal power flow pursuit: formulation, linear approximationI Measurement feedback II: application to OPF, convergence and
numerical results
Take-home messages
A. Simonetto (IBM Research) 8 / 23
Time-varying OptimizationWe start with convex optimization problems of the form:
minimizex∈Rn
f (x; t), subject to: g(x; t) ≤ 0.
Immagine g(x) is not there, then the gradient method is defined as
xk = xk−1 − α∇xf (xk−1; tk)
If f strongly convex and strongly smooth and
‖x∗k − x∗k−1‖ ≤ δ,
‖xk − x∗k‖ ≤ %(‖xk−1 − x∗k−1‖+ δ), % < 1 (α < 2/L)
So, convergence around an error ball of size O(δ)Plenty of works: correction-only, prediction-correction, etc..see: A.S., arXiv: 1807.07032
A. Simonetto (IBM Research) 9 / 23
Time-varying OptimizationWe start with convex optimization problems of the form:
minimizex∈Rn
f (x; t), subject to: g(x; t) ≤ 0.
Immagine g(x) is not there, then the gradient method is defined as
xk = xk−1 − α∇xf (xk−1; tk)
If f strongly convex and strongly smooth and
‖x∗k − x∗k−1‖ ≤ δ,
‖xk − x∗k‖ ≤ %(‖xk−1 − x∗k−1‖+ δ), % < 1 (α < 2/L)
So, convergence around an error ball of size O(δ)Plenty of works: correction-only, prediction-correction, etc..see: A.S., arXiv: 1807.07032
A. Simonetto (IBM Research) 9 / 23
Time-varying OptimizationWe start with convex optimization problems of the form:
minimizex∈Rn
f (x; t), subject to: g(x; t) ≤ 0.
Immagine g(x) is not there, then the gradient method is defined as
xk = xk−1 − α∇xf (xk−1; tk)
If f strongly convex and strongly smooth and
‖x∗k − x∗k−1‖ ≤ δ,
‖xk − x∗k‖ ≤ %(‖xk−1 − x∗k−1‖+ δ), % < 1 (α < 2/L)
So, convergence around an error ball of size O(δ)
Plenty of works: correction-only, prediction-correction, etc..see: A.S., arXiv: 1807.07032
A. Simonetto (IBM Research) 9 / 23
Time-varying OptimizationWe start with convex optimization problems of the form:
minimizex∈Rn
f (x; t), subject to: g(x; t) ≤ 0.
Immagine g(x) is not there, then the gradient method is defined as
xk = xk−1 − α∇xf (xk−1; tk)
If f strongly convex and strongly smooth and
‖x∗k − x∗k−1‖ ≤ δ,
‖xk − x∗k‖ ≤ %(‖xk−1 − x∗k−1‖+ δ), % < 1 (α < 2/L)
So, convergence around an error ball of size O(δ)Plenty of works: correction-only, prediction-correction, etc..see: A.S., arXiv: 1807.07032
A. Simonetto (IBM Research) 9 / 23
Constrained caseWorks similarly with a doubly regularized Lagrangian
A. Simonetto (IBM Research) 10 / 23
Constrained caseWorks similarly with a doubly regularized Lagrangian
100 101 102 103
Time index
10−6
10−5
10−4
10−3
10−2
10−1
Rel
ativ
eer
ror:|f t
(zt)−f∗ t|/|f∗ t|
η = 0, ν = ε = 0
η = 0, ν = 0.1, ε = 0.01
A. Simonetto (IBM Research) 10 / 23
Constrained caseWorks similarly with a doubly regularized Lagrangian
100 101 102 103
Time index
10−6
10−5
10−4
10−3
10−2
10−1
Rel
ativ
eer
ror:|f t
(zt)−f∗ t|/|f∗ t|
η = 0, ν = ε = 0
η = 0, ν = 0.1, ε = 0.01
η = 0.5, ν = ε = 0
η = 0.5, ν = 0.1, ε = 0.01
A. Simonetto (IBM Research) 10 / 23
Physics is not convexMany cyber-physical systems have the following structure
minimizex∈Rn,y∈Rl
f (x; t) + h(y; t)
subject to: y =M(x) (Physics)x ∈ X (Engineering)
Can we linearize?Feed-forward (open loop): y =M(x) ≈ Ax+ bFeedback: y =M(x) + ω
xk = ΠX [xk − α∇xf (xk−1; t)− α(∇xy ◦ ∇yh)(xk−1)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(Axk−1 + b)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(yk−1)]
A. Simonetto (IBM Research) 11 / 23
Physics is not convexMany cyber-physical systems have the following structure
minimizex∈Rn,y∈Rl
f (x; t) + h(y; t)
subject to: y =M(x) (Physics)x ∈ X (Engineering)
Can we linearize?
Feed-forward (open loop): y =M(x) ≈ Ax+ bFeedback: y =M(x) + ω
xk = ΠX [xk − α∇xf (xk−1; t)− α(∇xy ◦ ∇yh)(xk−1)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(Axk−1 + b)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(yk−1)]
A. Simonetto (IBM Research) 11 / 23
Physics is not convexMany cyber-physical systems have the following structure
minimizex∈Rn,y∈Rl
f (x; t) + h(y; t)
subject to: y =M(x) (Physics)x ∈ X (Engineering)
Can we linearize?Feed-forward (open loop): y =M(x) ≈ Ax+ b
Feedback: y =M(x) + ω
xk = ΠX [xk − α∇xf (xk−1; t)− α(∇xy ◦ ∇yh)(xk−1)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(Axk−1 + b)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(yk−1)]
A. Simonetto (IBM Research) 11 / 23
Physics is not convexMany cyber-physical systems have the following structure
minimizex∈Rn,y∈Rl
f (x; t) + h(y; t)
subject to: y =M(x) (Physics)x ∈ X (Engineering)
Can we linearize?Feed-forward (open loop): y =M(x) ≈ Ax+ bFeedback: y =M(x) + ω
xk = ΠX [xk − α∇xf (xk−1; t)− α(∇xy ◦ ∇yh)(xk−1)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(Axk−1 + b)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(yk−1)]
A. Simonetto (IBM Research) 11 / 23
Physics is not convexMany cyber-physical systems have the following structure
minimizex∈Rn,y∈Rl
f (x; t) + h(y; t)
subject to: y =M(x) (Physics)x ∈ X (Engineering)
Can we linearize?Feed-forward (open loop): y =M(x) ≈ Ax+ bFeedback: y =M(x) + ω
xk = ΠX [xk − α∇xf (xk−1; t)− α(∇xy ◦ ∇yh)(xk−1)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(Axk−1 + b)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(yk−1)]
A. Simonetto (IBM Research) 11 / 23
Physics is not convexMany cyber-physical systems have the following structure
minimizex∈Rn,y∈Rl
f (x; t) + h(y; t)
subject to: y =M(x) (Physics)x ∈ X (Engineering)
Can we linearize?Feed-forward (open loop): y =M(x) ≈ Ax+ bFeedback: y =M(x) + ω
xk = ΠX [xk − α∇xf (xk−1; t)− α(∇xy ◦ ∇yh)(xk−1)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(Axk−1 + b)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(yk−1)]
A. Simonetto (IBM Research) 11 / 23
Physics is not convexFeedback: y =M(x) + ω
xk = ΠX [xk − α∇xf (xk−1; t)− α(∇xy ◦ ∇yh)(xk−1)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(Axk−1 + b)]
≈ ΠX [xk − α∇xf (xk−1; t)− αAT∇yh(yk−1)]
Let the physics do the job for you
First-order
algorithmx
y
E. Dall’Anese, AS, arXiv: 1601.07263A. Bernstein, E. Dall’Anese, A.S., arXiv: 1804.05159M. Colombino, J. Simpson-Porco, A. Bernstein, arXiv: 1905.07363
A. Simonetto (IBM Research) 12 / 23
Physics is not convexFeedback: y =M(x) + ω
First-order
algorithmx
y
E. Dall’Anese, AS, arXiv: 1601.07263A. Bernstein, E. Dall’Anese, A.S., arXiv: 1804.05159M. Colombino, J. Simpson-Porco, A. Bernstein, arXiv: 1905.07363
A. Simonetto (IBM Research) 12 / 23
Optimal power flow: basicsSetting (high level): distribution feeder:
1n
Node n = 1, . . . ,N:Vkn ∈ C, Ikn ∈ C, graphN
Pk`,n ∈ R,Qk
`,n ∈ Rvk := [Vk
1 , . . . ,VkN ], ik := [Ik1 , . . . , I
kN ]
Node n ∈ G:Pkn ∈ R,Qk
n ∈ R
A. Simonetto (IBM Research) 13 / 23
Optimal power flow: basicsSetting (high level): distribution feeder:
1n
PV
Node n = 1, . . . ,N:Vkn ∈ C, Ikn ∈ C, graphN
Pk`,n ∈ R,Qk
`,n ∈ Rvk := [Vk
1 , . . . ,VkN ], ik := [Ik1 , . . . , I
kN ]
Node n ∈ G:Pkn ∈ R,Qk
n ∈ R
A. Simonetto (IBM Research) 13 / 23
Optimal power flow: basicsSetting (high level): distribution feeder, AC OPF problem as
(OPFk) minimizev,i,{Pi,Qi}i∈G
hk({Vi}i∈N ) +∑i∈G
f ki (Pi,Qi)
subject to :
[Ik0ik]
=
[yk00 (yk)T
yk Yk
]︸ ︷︷ ︸
:=Yknet
[Vk0
vk
]
ViI∗i = Pi − Pk`,i + j(Qi − Qk
`,i), ∀ i ∈ GVnI∗n = −Pk
`,n − jQk`,n, ∀n ∈ N\G
Vmin ≤ |Vi| ≤ Vmax, ∀ i ∈M(Pi,Qi) ∈ Yk
i , ∀ i ∈ G ,
A. Simonetto (IBM Research) 14 / 23
Optimal power flow: linearization
ViI∗i = Pi − Pk`,i + j(Qi − Qk
`,i), ∀ i ∈ GVnI∗n = −Pk
`,n − jQk`,n, ∀n ∈ N\G
Set:s collecting net power injected, p = R(s), q = I(s)ρ = [|V1|, . . . , |VN |]T ∈ RN
Linearizing the power-flow relations as
v ≈ Hp + Jq + bρ ≈ Rp + Bq + a
To write, e.g.,: Vmin1N ≤ Rp + Bq + a ≤ Vmax1N (Measurement)
A. Simonetto (IBM Research) 15 / 23
Optimal power flow: linearization
ViI∗i = Pi − Pk`,i + j(Qi − Qk
`,i), ∀ i ∈ GVnI∗n = −Pk
`,n − jQk`,n, ∀n ∈ N\G
Set:s collecting net power injected, p = R(s), q = I(s)ρ = [|V1|, . . . , |VN |]T ∈ RN
Linearizing the power-flow relations as
v ≈ Hp + Jq + bρ ≈ Rp + Bq + a
To write, e.g.,: Vmin1N ≤ Rp + Bq + a ≤ Vmax1N (Measurement)
A. Simonetto (IBM Research) 15 / 23
Optimal power flow: formulation
(R− OPFk) minimize{ui}i∈G
∑i∈G
f ki (ui)
subject to :
gkn({ui}i∈G) ≤ 0, ∀n ∈M
gkn({ui}i∈G) ≤ 0, ∀n ∈Mui ∈ Yk
i , ∀ i ∈ Gwhere ui := [Pi,Qi]
T, f ki (ui) := f ki (ui) := f ki (ui) + hki (ui)
gkn({ui}i∈G) := Vmin − ckn −
∑i∈G
[rkn,i(Pi − Pk`,i) + bk
n,i(Qi − Qk`,i)]
gkn({ui}i∈G) :=
∑i∈G
[rkn,i(Pi − Pk`,i) + bk
n,i(Qi − Qk`,i)] + ckn − Vmax
Yk := Yk1 × . . .Yk
NG
A. Simonetto (IBM Research) 16 / 23
Optimal power flow: saddle-point problem
Lk(u,γ,µ) :=∑i∈G
f ki (Pi,Qi)+
(Pi − Pk`,i)(r
ki )T(µ− γ) + (Qi − Qk
`,i)(bki )T(µ− γ)
+ (ck)T(µ− γ) + γT1mVmin − µT1mVmax
Double smoothing:
Lkν,ε(u,γ,µ) := Lk(u,γ,µ) +
ν
2‖u‖22 −
ε
2(‖γ‖22 + ‖µ‖22)
A. Simonetto (IBM Research) 17 / 23
Optimal power flow: saddle-point problem
Lk(u,γ,µ) :=∑i∈G
f ki (Pi,Qi)+
(Pi − Pk`,i)(r
ki )T(µ− γ) + (Qi − Qk
`,i)(bki )T(µ− γ)
+ (ck)T(µ− γ) + γT1mVmin − µT1mVmax
Double smoothing:
Lkν,ε(u,γ,µ) := Lk(u,γ,µ) +
ν
2‖u‖22 −
ε
2(‖γ‖22 + ‖µ‖22)
A. Simonetto (IBM Research) 17 / 23
Saddle-point algorithmFrom feed-forward:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}, ∀ i ∈ G
γk+1n = ΠR+
{γkn + α(gk
n(uk)− εγkn)}, ∀n ∈M
µk+1n = ΠR+
{µkn + α(gk
n(uk)− εµkn)}
∀n ∈M,
To feedback-based:[S1] Collect voltage measurements {yk
n}n∈M.[S2] For all n ∈M, update dual variables as follows:
γk+1n = ΠR+
{γkn + α(Vmin − yk
n − εγkn)}
µk+1n = ΠR+
{µkn + α(yk
n − Vmax − εµkn)}
[S3] Update power setpoints at each RES i ∈ G as:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}and go to [S1].
A. Simonetto (IBM Research) 18 / 23
Saddle-point algorithmFrom feed-forward:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}, ∀ i ∈ G
γk+1n = ΠR+
{γkn + α(gk
n(uk)− εγkn)}, ∀n ∈M
µk+1n = ΠR+
{µkn + α(gk
n(uk)− εµkn)}
∀n ∈M,
To feedback-based:[S1] Collect voltage measurements {yk
n}n∈M.[S2] For all n ∈M, update dual variables as follows:
γk+1n = ΠR+
{γkn + α(Vmin − yk
n − εγkn)}
µk+1n = ΠR+
{µkn + α(yk
n − Vmax − εµkn)}
[S3] Update power setpoints at each RES i ∈ G as:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}and go to [S1].
A. Simonetto (IBM Research) 18 / 23
Saddle-point algorithmFrom feed-forward:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}, ∀ i ∈ G
γk+1n = ΠR+
{γkn + α(gk
n(uk)− εγkn)}, ∀n ∈M
µk+1n = ΠR+
{µkn + α(gk
n(uk)− εµkn)}
∀n ∈M,
To feedback-based:[S1] Collect voltage measurements {yk
n}n∈M.[S2] For all n ∈M, update dual variables as follows:
γk+1n = ΠR+
{γkn + α(Vmin − yk
n − εγkn)}
µk+1n = ΠR+
{µkn + α(yk
n − Vmax − εµkn)}
[S3] Update power setpoints at each RES i ∈ G as:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}and go to [S1].
A. Simonetto (IBM Research) 18 / 23
Saddle-point algorithmFrom feed-forward:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}, ∀ i ∈ G
γk+1n = ΠR+
{γkn + α(gk
n(uk)− εγkn)}, ∀n ∈M
µk+1n = ΠR+
{µkn + α(gk
n(uk)− εµkn)}
∀n ∈M,
To feedback-based:[S1] Collect voltage measurements {yk
n}n∈M.[S2] For all n ∈M, update dual variables as follows:
γk+1n = ΠR+
{γkn + α(Vmin − yk
n − εγkn)}
µk+1n = ΠR+
{µkn + α(yk
n − Vmax − εµkn)}
[S3] Update power setpoints at each RES i ∈ G as:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}and go to [S1].
A. Simonetto (IBM Research) 18 / 23
Saddle-point algorithmFrom feed-forward:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}, ∀ i ∈ G
γk+1n = ΠR+
{γkn + α(gk
n(uk)− εγkn)}, ∀n ∈M
µk+1n = ΠR+
{µkn + α(gk
n(uk)− εµkn)}
∀n ∈M,
To feedback-based:[S1] Collect voltage measurements {yk
n}n∈M.[S2] For all n ∈M, update dual variables as follows:
γk+1n = ΠR+
{γkn + α(Vmin − yk
n − εγkn)}
µk+1n = ΠR+
{µkn + α(yk
n − Vmax − εµkn)}
[S3] Update power setpoints at each RES i ∈ G as:
uk+1i = ΠYk
i
{uk
i − α∇uiLkν,ε(u,γ,µ)|uk
i ,γk,µk
}and go to [S1].A. Simonetto (IBM Research) 18 / 23
ConvergenceAssumptions:
1 Cost function: convex and strongly smooth over Yk with constantL
2 Slater’s condition hold3 There exist constants that upper-bound the variation in time of
the problem:
‖u∗,k+1 − u∗,k‖ ≤ σu, |gk+1n (u∗,k+1)− gk
n(u∗,k)| ≤ σd,
|gk+1n (u∗,k+1)− gk
n(u∗,k)| ≤ σd.
Which implies ‖z∗,k+1 − z∗,k‖ ≤ σz4 There exist constants that upper-bound the linearization error:
max{‖ekγ‖2, ‖ek
µ‖2} ≤ e
A. Simonetto (IBM Research) 19 / 23
ConvergenceAssumptions:
1 Cost function: convex and strongly smooth over Yk with constantL
2 Slater’s condition hold
3 There exist constants that upper-bound the variation in time ofthe problem:
‖u∗,k+1 − u∗,k‖ ≤ σu, |gk+1n (u∗,k+1)− gk
n(u∗,k)| ≤ σd,
|gk+1n (u∗,k+1)− gk
n(u∗,k)| ≤ σd.
Which implies ‖z∗,k+1 − z∗,k‖ ≤ σz4 There exist constants that upper-bound the linearization error:
max{‖ekγ‖2, ‖ek
µ‖2} ≤ e
A. Simonetto (IBM Research) 19 / 23
ConvergenceAssumptions:
1 Cost function: convex and strongly smooth over Yk with constantL
2 Slater’s condition hold3 There exist constants that upper-bound the variation in time of
the problem:
‖u∗,k+1 − u∗,k‖ ≤ σu, |gk+1n (u∗,k+1)− gk
n(u∗,k)| ≤ σd,
|gk+1n (u∗,k+1)− gk
n(u∗,k)| ≤ σd.
Which implies ‖z∗,k+1 − z∗,k‖ ≤ σz
4 There exist constants that upper-bound the linearization error:max{‖ek
γ‖2, ‖ekµ‖2} ≤ e
A. Simonetto (IBM Research) 19 / 23
ConvergenceAssumptions:
1 Cost function: convex and strongly smooth over Yk with constantL
2 Slater’s condition hold3 There exist constants that upper-bound the variation in time of
the problem:
‖u∗,k+1 − u∗,k‖ ≤ σu, |gk+1n (u∗,k+1)− gk
n(u∗,k)| ≤ σd,
|gk+1n (u∗,k+1)− gk
n(u∗,k)| ≤ σd.
Which implies ‖z∗,k+1 − z∗,k‖ ≤ σz4 There exist constants that upper-bound the linearization error:
max{‖ekγ‖2, ‖ek
µ‖2} ≤ e
A. Simonetto (IBM Research) 19 / 23
Convergence: resultTheorem.Consider the sequence {zk} := {uk,γk,µk}Let the assumptions hold.For fixed positive scalars ε, ν > 0, if the stepsize α > 0 is chosen suchthat
ρ(α) :=√
1− 2ηα+ α2L2ν,ε < 1,
that is 0 < α < 2η/L2ν,ε, then the sequence {zk} converges Q-linearly
to z∗,k := {u∗,k,γ∗,k,µ∗,k} up to the asymptotic error bound given by:
lim supk→∞
‖zk − z∗,k‖ =1
1− ρ(α)
[√2αe + σz
]
Lν,ε :=√
(L+ ν + 2G)2 + 2(G+ ε)2, G := max ‖∇g‖, η := min{ν, η}
A. Simonetto (IBM Research) 20 / 23
SimulationsReal load and solar data from Anatolia, CAPQ of inverters updated every 1sHVAC controlled every 5 minVoltage regulation and power tracking
1
23
4 5
6
7
8910
11 12 13
14
1516
17
18
19
20
21
2223
24
25
26
272829
30
31
32
33
34 3536
37
A. Simonetto (IBM Research) 21 / 23
SimulationsReal load and solar data from Anatolia, CAPQ of inverters updated every 1sHVAC controlled every 5 minVoltage regulation and power tracking
6:00 8:00 10:00 12:00 14:00 16:00 18:00 20:00
Time
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
1.06
|Vt n|
Node 2
Node 28
Node 35
A. Simonetto (IBM Research) 21 / 23
OutlineBasics
I Optimization problems, gradient descent, regularizationI Lagrangian formalism, saddle-point method, convergence
AdvancedI Time-varying optimization ideas, online gradient descent, online
saddle-pointI Measurement feedback I: idea, derivation, insightsI Optimal power flow pursuit: formulation, linear approximationI Measurement feedback II: application to OPF, convergence and
numerical results
Take-home messages
A. Simonetto (IBM Research) 22 / 23
Take-home messagesTime-varying optimization rocks!
Regularizing time-varying problems is not a bad ideaCyber-physical systems have a structure which allows you to usefeedback!
Some extra useful literature:First-order algorithms I: Adrien Taylor, PhD ThesisFirst-order algorithms II: E. K. Ryu, S. Boyd, Primer on MonotoneOperator Methods, 2016
Mailto: [email protected]
A. Simonetto (IBM Research) 23 / 23
Take-home messagesTime-varying optimization rocks!Regularizing time-varying problems is not a bad idea
Cyber-physical systems have a structure which allows you to usefeedback!
Some extra useful literature:First-order algorithms I: Adrien Taylor, PhD ThesisFirst-order algorithms II: E. K. Ryu, S. Boyd, Primer on MonotoneOperator Methods, 2016
Mailto: [email protected]
A. Simonetto (IBM Research) 23 / 23
Take-home messagesTime-varying optimization rocks!Regularizing time-varying problems is not a bad ideaCyber-physical systems have a structure which allows you to usefeedback!
Some extra useful literature:First-order algorithms I: Adrien Taylor, PhD ThesisFirst-order algorithms II: E. K. Ryu, S. Boyd, Primer on MonotoneOperator Methods, 2016
Mailto: [email protected]
A. Simonetto (IBM Research) 23 / 23
Take-home messagesTime-varying optimization rocks!Regularizing time-varying problems is not a bad ideaCyber-physical systems have a structure which allows you to usefeedback!
Some extra useful literature:First-order algorithms I: Adrien Taylor, PhD ThesisFirst-order algorithms II: E. K. Ryu, S. Boyd, Primer on MonotoneOperator Methods, 2016
Mailto: [email protected]
A. Simonetto (IBM Research) 23 / 23