control of uncertain power systems via convex optimization · control of uncertain power systems...
TRANSCRIPT
![Page 1: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/1.jpg)
Control of Uncertain Power Systemsvia
Convex Optimization
Eilyan Bitar
School of Electrical and Computer Engineering
CERTS Annual Review Meeting, Ithaca, Aug. 4, 2015
![Page 2: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/2.jpg)
The Problem of Variability in Power Systems
Existing operational tools and electricity market designs not equipped toefficiently accommodate variability in renewable supply at scale.
1 Predominant approach to economic dispatch makes inefficient use ofinformation
2 Existing market designs reflect this inefficiency of operation and rely onad hoc schemes to allocate the cost of variability (e.g., A/S costs) tomarket participants
H. Madsen et al. (2011). Forecasting Wind and Solar Power Production.2
![Page 3: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/3.jpg)
Broad Project Objectives
(A) Develop computational tools to manage uncertainty in power systemoperations
• data driven
• computational scalable
• provable performance guarantees
(B) Design novel market systems that:
• provide a competitive medium through which variable power producerscan sell their supply on equal footing with conventional power producers.
• efficiently reflect and allocate the cost of uncertainty owing to variablerenewable generation.
3
![Page 4: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/4.jpg)
Talk Outline
1 Power system model
2 Economic Dispatch as Stochastic Control
3 Policy approximation
4 Constraint approximation
5 Numerical Experiment
4
![Page 5: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/5.jpg)
The Power System Model
The power system is modeled as a partially-observed linear stochasticsystem:
xt+1 = Axt + But + Gwt
yt = Cxt + Hwt
for all t = 0, . . . ,T .
• xt ∈ Rnx (state) generator operating points, energy storage levels, etc.• ut ∈ Rnu (control) generator and storage power injections• yt ∈ Rny (measurements) partial noisy observation of system state• wt ∈ Rnw (exogenous random process) wind, solar, demand, etc.
Notation• x t = (x0, . . . , xt) ∈ Rnx(t+1) – history of process until time t• x = (x0, . . . , xT−1) ∈ RnxT – history of a enitre process
5
![Page 6: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/6.jpg)
The Power System Model
The power system is subjected to linear constraints
Fxxt + Fuut + Fwwt ≤ g,
which we enforce probabilistically as
Prob {Fxxt + Fuut + Fwwt ≤ g} ≥ 1− ε
for all t = 0, . . . ,T , where ε ∈ (0, 1].
• Uncertainty in renewable supply, demand, network line outages
• Transmission network constraints subject to linearized power flow
• Resource constraints – storage, generator ramping, flexible demand
• Doesn’t capture AC power flow or unit commitment decisions
Model Features
6
![Page 7: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/7.jpg)
The Stochastic Economic Dispatch (SED) Problem
Compute a causal output-feedback dispatch policy π = (µ0, . . . , µT−1)
ut = µt(y0, . . . , yt)
that solves chance constrained stochastic control problem:
(P) minimizeπ
Eπ
[x ′T QxT +
T−1∑t=0
x ′tQxt + u′tRut
]subject to Prob {Fxxt + Fuut + Fwwt ≤ g} ≥ 1− ε
xt+1 = Axt + But + Gwt
yt = Cxt + Hwt
ut = µt(yt), t = 0, . . . ,T .
We assume convex quadratic costs, Q,R � 0.7
![Page 8: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/8.jpg)
Why is Problem P Difficult to Solve?
In general, problem P is difficult to solve...
• Infinite-dimensional in its optimization variables• Nonconvex in its feasible region• Requires specification of a prior probability distribution
We reformulate problem P as a finite-dimensional convex program that:
• is computationally scalable,• requires minimal distributional information,• admits computable performance guarantees,• and enables the systematic trade off between computational burden andperformance.
8
![Page 9: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/9.jpg)
Why is Problem P Difficult to Solve?
In general, problem P is difficult to solve...
• Infinite-dimensional in its optimization variables• Nonconvex in its feasible region• Requires specification of a prior probability distribution
We reformulate problem P as a finite-dimensional convex program that:
• is computationally scalable,• requires minimal distributional information,• admits computable performance guarantees,• and enables the systematic trade off between computational burden andperformance.
8
![Page 10: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/10.jpg)
Policy Approximation
9
![Page 11: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/11.jpg)
An Output Transformation
Define the purified observation process zt = yt − yt , where
xt+1 = Axt + But and yt = Cxt
The purified output zt can be thought of as an output prediction error.
1 {zt} generates the same amount of information as {yt}
σ(z0, . . . , zt) = σ(y0, . . . , yt).
2 {zt} is independent of the control process {ut} and satisfies
zt = Ltwt ,
where the matrix Lt is easily constructed from problem data. Wewrite the entire purified output vector as
z = Lw.
Lemma (Kailath ’68, Ben-Tal et al. ’09, Hadjiyiannis et al. ’10)
10
![Page 12: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/12.jpg)
An Output Transformation
Define the purified observation process zt = yt − yt , where
xt+1 = Axt + But and yt = Cxt
The purified output zt can be thought of as an output prediction error.
1 {zt} generates the same amount of information as {yt}
σ(z0, . . . , zt) = σ(y0, . . . , yt).
2 {zt} is independent of the control process {ut} and satisfies
zt = Ltwt ,
where the matrix Lt is easily constructed from problem data. Wewrite the entire purified output vector as
z = Lw.
Lemma (Kailath ’68, Ben-Tal et al. ’09, Hadjiyiannis et al. ’10)
10
![Page 13: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/13.jpg)
Purified Output-Feedback
Reparameterize the causal feedback policies in the purified observation {zt}:
(P) minimizeπ
Eπ
[x ′T QxT +
T−1∑t=0
x ′tQxt + u′tRut
]subject to Prob {Fxxt + Fuut + Fwwt ≤ g} ≥ 1− ε
xt+1 = Axt + But + Gwt
zt = Ltwt
ut = µt(z t), t = 0, . . . ,T .
• Eliminates dependency of observations on control inputs
• This is without loss of optimality, i.e. problem P equivalent to P
• Problem P still intractable however....
11
![Page 14: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/14.jpg)
A Finite-Dimensional Approximation of the Policy Space
Restrict the space of admissible control policies to those representable as finitelinear combinations of a preselected basis functions φt = (φ1
t , . . . , φntt )′,
ut = Ktφt(z t) =
|K1t
|
φ1t (z t) + · · · +
|Kntt
|
φntt (z t)
where Kt ∈ Rnu×nt is matrix of weighting coefficients.
We write the entire input vector as u = Kφ(z), whereu0
u1...
uT−1
=
K0 0 · · · 0
0 K1...
.... . . 0
0 · · · 0 KT−1
φ0(z0)φ1(z1)
...φT−1(zT−1)
12
![Page 15: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/15.jpg)
A Finite-Dimensional Approximation of the Policy Space
Restrict the space of admissible control policies to those representable as finitelinear combinations of a preselected basis functions φt = (φ1
t , . . . , φntt )′,
ut = Ktφt(z t) =
|K1t
|
φ1t (z t) + · · · +
|Kntt
|
φntt (z t)
where Kt ∈ Rnu×nt is matrix of weighting coefficients.
We write the entire input vector as u = Kφ(z), whereu0
u1...
uT−1
=
K0 0 · · · 0
0 K1...
.... . . 0
0 · · · 0 KT−1
φ0(z0)φ1(z1)
...φT−1(zT−1)
12
![Page 16: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/16.jpg)
Reduction of P to a Chance Constrained Program
Given a restriction to dispatch policies of the form
u = Kφ(z), where z = Lw,
problem P reduces to a finite-dimensional chance constrained program
(CCP) minimizeK
tr(MφK ′VK) + tr(NφUK)
subject to Prob {EtKφ(z) + Ftw ≤ gt} ≥ 1− ε
t = 0, . . . ,T
K = diag(K0, . . . ,KT−1)
Moment matrices given by Mφ := E [φ(z)φ(z)′] and Nφ := E [φ(z)w′].
Proposition
Matrices V � 0, U , and {Et ,Ft , gt} are computed from the primitive data.13
![Page 17: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/17.jpg)
Reduction to a Chance Constrained Program
(CCP) minimizeK
tr(MφK ′VK) + tr(NφUK)
subject to Prob {EtKφ(z) + Ftw ≤ gt} ≥ 1− ε
t = 0, . . . ,T
K = diag(K0, . . . ,KT−1)
• CCP is a inner approximation of the original problem P
• Objective is convex
• Feasible region is nonconvex, in general
There is a rich literature1 on convex approximations of chance constraints• convex inner approximations• randomized convex approximations
1. Nemirovski et al. (2006). Convex approximations of chance constrained programs
14
![Page 18: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/18.jpg)
Chance Constraint Approximation
15
![Page 19: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/19.jpg)
A Scenario Constrained Program
Define the Scenario Constrained Program induced by CCPε as
(SCPN ) minimizeK
tr(MφK ′VK) + tr(NφUK)
subject to EtKφ(Lwi) + Ftwi ≤ gt
t = 0, . . . ,T
i = 1, . . . ,N
where (w1, . . . ,wN ) are N i.i.d. samples of the random vector w.
Definition
SCPN is a random convex quadratic program (QP)• However, solutions to SCPN are random and may not be feasible for CCPε
• How large does N have to be?16
![Page 20: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/20.jpg)
A Scenario Constrained Program
Define the Scenario Constrained Program induced by CCPε as
(SCPN ) minimizeK
tr(MφK ′VK) + tr(NφUK)
subject to EtKφ(Lwi) + Ftwi ≤ gt
t = 0, . . . ,T
i = 1, . . . ,N
where (w1, . . . ,wN ) are N i.i.d. samples of the random vector w.
Definition
SCPN is a random convex quadratic program (QP)• However, solutions to SCPN are random and may not be feasible for CCPε
• How large does N have to be?16
![Page 21: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/21.jpg)
Bound on the Number of Scenarios
The following result is an immediate consequence of Campi et al. (2008).∗
Fix a choice of basis φ and a probability level ε ∈ (0, 1). If
N ≥ 1ε
(ln 1β
+ 1 + nu · card(φ)),
then an optimal solution to SCPN will be feasible for CCPε with probabilitygreater than or equal to 1− β.
Theorem
∗M.C. Campi et al. (2008).The exact feasibility of randomized solutions of uncertain convexprograms.
17
![Page 22: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/22.jpg)
Discussion of SCPN
How does SCPN fair as a surrogate the original stochastic control problem P?
Minimal Distributional Assumptions• Results hold for any distribution on w• Ability to procure independent samples of w
Polynomial-Time Complexity• SCPN is a convex QP• It has dimension = poly(ny,nu ,T), if card(φ) ≤ poly(ny,T)• Not exponential in the horizon T ....
Fidelity• Optimal solution to SCPN is feasible for P with probability 1− β• Richness of basis φ controls fidelity of approximation• Similar techniques can be applied to obtain dual lower bounds for P
18
![Page 23: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/23.jpg)
Discussion of SCPN
How does SCPN fair as a surrogate the original stochastic control problem P?
Minimal Distributional Assumptions• Results hold for any distribution on w• Ability to procure independent samples of w
Polynomial-Time Complexity• SCPN is a convex QP• It has dimension = poly(ny,nu ,T), if card(φ) ≤ poly(ny,T)• Not exponential in the horizon T ....
Fidelity• Optimal solution to SCPN is feasible for P with probability 1− β• Richness of basis φ controls fidelity of approximation• Similar techniques can be applied to obtain dual lower bounds for P
18
![Page 24: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/24.jpg)
Discussion of SCPN
How does SCPN fair as a surrogate the original stochastic control problem P?
Minimal Distributional Assumptions• Results hold for any distribution on w• Ability to procure independent samples of w
Polynomial-Time Complexity• SCPN is a convex QP• It has dimension = poly(ny,nu ,T), if card(φ) ≤ poly(ny,T)• Not exponential in the horizon T ....
Fidelity• Optimal solution to SCPN is feasible for P with probability 1− β• Richness of basis φ controls fidelity of approximation• Similar techniques can be applied to obtain dual lower bounds for P
18
![Page 25: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/25.jpg)
Numerical Experiment
19
![Page 26: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/26.jpg)
Test Case Description
We consider a modified IEEE 14-bus power system dispatched over T = 12hours.
• Constrained DC power flow
• Dispatchable thermal generation
• Energy storage
• Wind generation (30% penetration)
System Features
Wind power data acquired from NREL Eastern Wind Integration andTransmission Study (EWITS).
20
![Page 27: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/27.jpg)
Experiment Description
• Constraint probability, ε = 0.1
• Approximation confidence, β = 0.001• Basis, φ = {set of all d-degree monomials} in z
- d = 1 (affine control policies)- d = 2 (quadratic control policies)
Parameters
1 Fix the degree d
2 Set the sample size N = N (ε, β, φ), where
N (ε, β, φ) = 1ε
(ln 1β
+ 1 + nu
(T · ny
d
))3 Solve 100 instances of SCPN
Procedure
21
![Page 28: Control of Uncertain Power Systems via Convex Optimization · Control of Uncertain Power Systems via Convex Optimization EilyanBitar SchoolofElectricalandComputerEngineering CERTSAnnualReviewMeeting,](https://reader033.vdocuments.site/reader033/viewer/2022060510/5f26f1c360f32b767d004385/html5/thumbnails/28.jpg)
Empirical Results
Optimal cost distribution of SCPN vs. d
3
4
5
6
7
8
9
10
11
12
x 104
1 2
d
There is a 36% reduction in average cost in moving from affine (d = 1) toquadratic (d = 2) policies.
22