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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
An optimization-on-manifold approachto the design of distributed feedback control
in smart grids
Saverio Bolognani, Florian Dörfler
Automatic Control LaboratoryETH Zürich
ECC 2016 WorkshopDistributed and Stochastic Optimization: Theory and Applications
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Future electric power distribution grids
FUTURE ELECTRIC POWER DISTRIBUTION GRIDS
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Future electric power distribution grids
Power distribution grids
transmissiongrid
distributiongrid
TraditionalPower
Generation
I Distribution grid: the“capillary system” ofpower networks
I It delivers power fromthe transmission gridto the consumers.
I Very little sensing,monitoring, actuation.
I The “easy” part of thegrid: conventionallyfit-and-forget design.
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Future electric power distribution grids
New challengesI Distributed microgenerators (conventional and renewable sources)I Electric mobility (large flexible demand, spatio-temporal patterns).
41GW75%
Germany17 August 2014 wind
solarhydro
biomass
Distribution grid
solar
wind
hydro + biomass
Installed renewable generationGermany 2013
24 GW
15 GW
Transmission grid
6 GW
2015 2020
200k
400k
600k
800k
PHEV
BEV
SwitzerlandVISION 2020
Electricityconsumption
Buildings40.9%
Industry31.3%
Transportation27.8%
Energy consumptionby sector(2010)
73.9%
25.9%
Primary fuelconsumption
Electric VehicleFast charging
120KWTeslasupercharger
4KWDomestic
consumer
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Future electric power distribution grids
Distribution grid congestion
Operation of the grid close or above the physical limits, due tosimultaneous and uncoordinated power demand/generation.
→ lower e�ciency, blackouts→ curtailment of renewable generation
→ bottleneck to electric mobility
Fit-and-forget→ unsustainable grid reinforcement
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Future electric power distribution grids
Distribution Grid
Transmission Grid
x x xx
Distribution Grid
Transmission Grid
CONT
ROL
LAYE
R
Curtailment Reduced hosting capacity
Higher renewable generationLarger hosting capacity
overvoltage
renewable generation
controlled
undervoltage
power demand
uncontrolled
uncontrolled
controlled
distributiongrid
control
controlcontrol
control
I Virtual grid reinforcementI same infrastructureI more sensors and intelligenceI controlled grid = larger capacity
I Transparent control layerI invisible to the usersI modular design
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
OVERVIEW
1. A feedback control approach2. A tractable model for control design3. Control design example
I Reactive power control for voltage regulation4. Next step
I Optimization on the power flow manifold
5. Conclusions
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A feedback control approach
A FEEDBACK CONTROL APPROACH
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A feedback control approach
Distribution grid model
active power ph
reactive power qh
voltage magnitude vh
voltage angle θh h
0
microgenerator load
supply point
Grid equations
diag(u)Yu = s
whereI uh = vhejθh complex voltagesI sh = ph + jqh complex powers
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A feedback control approach
Distribution grid model
active power ph
reactive power qh
voltage magnitude vh
voltage angle θh h
0
microgenerator load
supply point
Actuation
I Tap changer / voltage regulators – supply point voltage v0I Reactive power compensators – reactive power qh
I static compensatorsI power inverters of the microgenerators (when available)
I Active power management – active power phI smart building control, storage and deferrable loadsI generator curtailment and load shedding
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A feedback control approach
Distribution grid model
active power ph
reactive power qh
voltage magnitude vh
voltage angle θh h
0
microgenerator load
supply point
Sensing
I Power meters – active power ph and reactive power qhI Voltage meters – nodal voltage vhI Phasor measurement units (PMU) – voltage magnitude vh and angle θh
(PQube @ UC Berkeley, GridBox in Zürich/Bern, Smart Grid Campus @ EPFL)I Line currents, transformer loading, . . .
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A feedback control approach
A control framework
grid sensing
grid actuation
Power distributionnetwork
plant
state x
power demands
power generation
Control objective
Drive the system to a state x∗ =[v∗ θ∗ p∗ q∗
]subject to
I soft constraints x∗ = argminx J(x)
I hard constraints x ∈ XI chance constraints P [x 6∈ X ] < ε
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A feedback control approach
Feedforward control
grid sensing
grid actuation
Power distributionnetwork
plant
state x
power demands
power generation
OPF
Conventional approach: Optimal Power Flow
I Similar to power transmission grid OPFI Motivated by encouraging results on OPF convexification
(Lavaei (2012), Farivar (2013), . . . )I Requires full disturbance knowledge - full communicationI Heavily model basedI Requires co-design of grid control and users’ behavior
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A feedback control approach
Feedback control
grid sensing
grid actuation
Power distributionnetwork
plant
state x
power demands
power generation
FEEDBACK
input
disturbance
output
Control theory answer
I Robustness against parametric uncertainty/unmodeled disturbanceI Time varying demand/generation becomes disturbanceI Model-free designI Explored so far only for limited cases (e.g. purely local VAR control)I Allows modular design of grid control and users’ behavior
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A feedback control approach
A similar scenario: frequency control
frequencyPowernetwork
plant
state x
power demands
power generation
FEEDBACK
input
disturbance
output
primary control
secondary control
In the transmission grid, feedback is used for frequency regulationI Frequency deviation as a implicit signal for power unbalanceI Purely local proportional control: primary droop controlI Integral control: secondary frequency regulation
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A tractable model for control design
A TRACTABLE MODEL FOR CONTROL DESIGN
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A tractable model for control design
Power flow manifoldI Grid state x =
[v θ p q
]I Set of all states that satisfy the grid equations diag(u)Yu = s
→ power flow manifold M := {x | F(x) = 0}I Regular submanifold of dimension 2n (6n if three-phase)
10.5
p2
0-0.5
0
0.5
q2
1
1.2
1
0.8
0.6
0.4
v2
node 2node 1
v1 = 1, θ1 = 0
y = 0.4− 0.8j
v2, θ2p2, q2p1, q1
v21g− v1v2 cos(θ1 − θ2)g− v1v2 sin(θ1 − θ2)b = p1
−v21b + v1v2 cos(θ1 − θ2)b− v1v2 sin(θ1 − θ2)g = q1
v22g− v1v2 cos(θ2 − θ1)g− v1v2 sin(θ2 − θ1)b = p2
−v22b + v1v2 cos(θ2 − θ1)b− v1v2 sin(θ2 − θ1)g = q2
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A tractable model for control design
Power flow manifold approximation
1.5
1
0.5
q2
0
-0.5
-11.51
0.5
p2
0-0.5
-1
1.2
1
1.4
0.8
0.6
v2
Best linear approximant
Ax∗(x − x∗) = 0
Ax∗ :=∂F(x)
∂x
∣∣∣∣x=x∗
Tangent plane at a nominalpower flow solution x∗ ∈M
Example x∗: no-load solution
I Implicit – No input/outputs (not a disadvantage)I Sparse – The matrix Ax∗ has the sparsity pattern of the grid graphI Structure preserving – Elements of Ax∗ depend on local parameters
→ Bolognani & Dörfler, Allerton (2015)→ Source code on github
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
A tractable model for control design
Power flow manifold approximation
2
1
0
�2
-1
-21.4
1.2
v2
1
0.8
0.6
0.5
-1
-0.5
0
1
1.5
p2
Standard modelsAdding assumption one obtains
I linear coupled power flowI DC power flowI rectangular DC flow
1.510.5
q2
0-0.5-121
p2
0
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
-1
v2
Nonlinear coordinate transf.
xh = xh(xh),∂xh∂xh
= 1
Di�erent manifold curvature!I vh → v2h : LinDistFlow
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
CONTROL DESIGN EXAMPLES
REACTIVE POWER CONTROL FOR VOLTAGE REGULATION
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Problem statement
active power ph
reactive power qh
voltage magnitude vh
voltage angle θh h
0
microgenerator load
supply point
I Inputs: reactive power qh of microgeneratorsI Outputs: voltage measurement vh at the microgeneratorsI Control objective:
I Soft constraints: minimize J(x) = vTLv (voltage drops)I Hard constraints: guarantee Vmin ≤ vh ≤ Vmax at all sensors
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Control design for soft constraint
power flow manifold
linear approximant
1. Modeling assumption
2. Control specs3. Proj on linear manifold4. Derive feedback law
Modeling assumption: constant R/X ratio ρ.
Ax∗(x − x∗) = 0 becomes (around the no-load state)
[ρL −L−L −ρL
∣∣∣∣ −I 00 −I
]vθpq
= 0
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Control design for soft constraint
power flow manifold
linear approximant
1. Modeling assumption2. Control specs
3. Proj on linear manifold4. Derive feedback law
Control specification: Distributed and asynchronous.
Minimal update δq{qh ← qh + δqk ← qk − δ
→ Communication graph Gcomm describing all possible updates (pairs h, k).
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Control design for soft constraint
power flow manifold
linear approximant
search direction
x
1. Modeling assumption2. Control specs3. Proj on linear manifold
4. Derive feedback law
Search directions: By projecting each possible direction δq on the linearmanifold ker Ax∗ , we obtain feasible search directions in the state space.
δx =
− 1
1+ρ2 L†δq− ρ
1+ρ2 L†δq0δq
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Control design for soft constraint
power flow manifold
linear approximant
search direction
x
Gradient of cost function
x + tδx1. Modeling assumption2. Control specs3. Proj on linear manifold4. Derive feedback law
Optimal step length: Given a search direction δx, we determine the steplength that minimizes the cost function J(x) = vTLv.
∇J(x) =
2Lv000
∇J(x + tδx)Tδx = 0 ⇒ t = (1 + ρ2)vTδq
δqTL†δq
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Control design for soft constraint
power flow manifold
linear approximant
search direction
x
Gradient of cost function
x + tδx1. Modeling assumption2. Control specs3. Proj on linear manifold4. Derive feedback law
Because the model is sparse and structure preserving. . .
t = (1 + ρ2)vTδq
δqTL†δq= (1 + ρ2)
vh − vkXhk
Gossip-like feedback law
{qh ← qh + (1 + ρ2) vh−vk
Xhk
qk ← qk − (1 + ρ2) vh−vkXhk
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Convergence and performance analysis
microgenerator voltage vh
microgenerator reactive power qh
Power distributionnetwork
plant
state x
power demands
power generation
FEEDBACK
input
disturbance
output
I Asynchronous distributedfeedback control
I no demand or generation measurementI limited model knowledgeI no power flow solverI alternation of sensing and actuation.
{qh ← qh + (1 + ρ2) vh−vk
Xhk
qk ← qk − (1 + ρ2) vh−vkXhk
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Convergence and performance analysis
0 10 20 30 40 50 60 70iteration
E [
J(x)
- J o
pt ]
1
0.1
0.01 56
58
60
62
100 150 200 250 300 350 400iteration
loss
es
[kW
]
→ Bolognani & Zampieri, IEEE TAC (2013)
I Extension to J(x) = uTLu (power losses), if θ can be measured (PMUs).I Proof of mean square convergence (with randomized async updates).I Explicit bound on the exponential rate of convergence.I Analysis of the dynamic performance (disturbance rejection).I Optimal communication graph: Gcomm ≈ Ggrid.
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Communication co-design
0
microgenerator load
supply point Gcomm
Ggrid
GgridSparsity of thepower system
GcommSparsity of the
communication graph
Fundamental design problem: implications of the communicationarchitecture on the control performance.
Joint design vs. separation results
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Communication co-design
Powerline communication10.0.0.1
18.0.1.2
18.0.1.3
... ...
...General purpose network
Wireless Multi-area
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Control design with hard constraints
microgenerator voltage vh
microgenerator reactive power qh
Power distributionnetwork
plant
state x
power demands
power generation
FEEDBACK
input
disturbance
output
v ≤ vh ≤ vqh≤ qh ≤ qh
I Power losses minimizationI Hard constraints on inputs and outputs.I Construct Lagrangian→ Saddle point algorithm
L(q, λ, η) = J(q) + λT(v − v) + ηT(qh − q)
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Control design with hard constraints
microgenerator voltage vh
microgenerator reactive power qh
Power distributionnetwork
plant
state x
power demands
power generation
FEEDBACK
input
disturbance
output
v ≤ vh ≤ vqh≤ qh ≤ qh
λh ← [λh + α(vh − v)]≥0
ηh ←[ηh + β(qh − qh)
]≥0
q← q− γ∇J(q)− λ−Lη
−Lη Di�usion term that requires nearest-neighbor communication.
→ Bolognani, Carli, Cavraro & Zampieri, IEEE TAC (2015)
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Simulations and comparison
0 10 20 30 40 50
Volt
ag
e [
p.u
.]
1
1.05
1.1
React
ive p
ow
er
[p.u
.]
0
1v1
v2
|q1| |q2|
Modified IEEE 123 Distribution Test Feeder → githubLight load + 2 microgenerators→ overvoltage
2 sets of constraints:
{voltage limits vh ≤ vmax reactive power qh ≤ qh
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Simulations and comparison
0 10 20 30 40 50
Volt
ag
e [
p.u
.]
1
1.05
1.1
React
ive p
ow
er
[p.u
.]
0
1
steady state error
saturation
vi
qi
qmaxi
−qmaxi
vmin vmax
qh(t) = −f (vh(t))
Fully decentralized, proportional controller.
Latest grid code drafts, Vovos (2007), Turitsyn(2011), Aliprantis (2013), ...
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Simulations and comparison
0 10 20 30 40 50
Volt
ag
e [
p.u
.]
1
1.02
1.04
1.06
1.08
React
ive p
ow
er
[p.u
.]
0
0.5
1
steady state error
saturation
vi
δqi
vmin
vmax
qh(t + 1) = qh(t)− f (vh(t))
Fully decentralized, integral controller.
Zero steady error without saturation limits. Li (2014)
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Simulations and comparison
0 10 20 30 40 50
Volt
ag
e [
p.u
.]
0.98
1
1.02
1.04
1.06
React
ive p
ow
er
[p.u
.]
0
0.5
1
no steady state error
reactive power sharing
λh ← [λh + α(vh − v)]≥0
ηh ←[ηh + β(qh − qh)
]≥0
q← q− γ∇J(q)− λ −Lη
Networked feedback control (neighbor-to-neighbor async communication)
→ Cavraro, Bolognani, Carli & Zampieri, IEEE CDC (2016)
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Control design examples
Reactive power control for voltage regulation
Chance constraints on the state
Chance-constrained decision
mininput δ
J(δ)
subject to Prob [x /∈ Xc] < ε
I Xc can encodeI under/over voltage limitsI power injection limitsI voltage stability region→ Bolognani & Zampieri, IEEE TPS(2015)
I A stochastic model for thedisturbance is available
I Via linear approximant→deterministic polytope constraints → Bolognani & Dörfler, PSCC (2016)
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Next step
NEXT STEP
OPTIMIZATION ON THE POWER FLOW MANIFOLD
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Next step
Optimization on the power flow manifold
power flow manifold
linear approximant
x(t)
Gradient of cost function
Projected gradient
x
Continuous time trajectory on the manifold:1. ∇J(x): gradient of the cost function (soft constraints) in ambient space2. Πx∇J(x): projection of the gradient on the linear approximant in x3. Evolve according to x = −γΠx∇J(x)
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Next step
Staying on the power flow manifold
x =
[xexoxendo
]Exogenous variablesInputs/disturbances that areimposed on the model.Reactive power injection qi
Endogenous variablesDetermined by the physics ofthe grid.Voltage vi
Iterative algorithm: at each step1. Compute Πx∇J(x) (sparse Ax(t) ⇒ distributed algorithm)2. Actuate system based on δx = −γΠx∇J (exogeneous variables / inputs)3. Retraction step x(t + 1) = Rx(t)(δx) ⇒ x(t + 1) ∈M.
From iterative optimization algorithm to feedback control on manifolds.
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Next step
Hard constraints on exogenous variables
Feasible input region
I Can be enforced via saturation of the corresponding coordinatesI Primal feasibility at all timesI The resulting feasible input region is invariant with respect to the
retraction.I We can saturate δx = −γΠx∇J(x) because
x + δ(x) ∈ F ⇒ x(t + 1) = Rx(t)(δx) ∈ F
→ Geometric Projected Dynamical Systems
I Extension of results on existence and uniqueness of executions forhybrid automata to manifolds
I Guarantees of no Zeno execution
Ongoing work with Adrian Hauswirth, Gabriela Hug, Florian Dörfler.
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Next step
Hard constraints on endogeneous variables
Operational constraints
I Barrier functions not suitable:I Backtracking line search is not possible in closed loopI Primal feasibility cannot be guaranteed during tracking
I Time-varying penalty functions not suitable:I Persistent feedback control for tracking
I Can be tackled via dualization / Lagrangian approach.I The corresponding operational constraints are satisfied at steady
state, despite model uncertainty.→ Saddle/primal-dual algorithm on manifolds
Ongoing work with Adrian Hauswirth, Gabriela Hug, Florian Dörfler.
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Next step
Optimization on the power flow manifold
0 100 200 300 400
0
2
4
6
8a
ctive
po
we
r
0 100 200 300 4000.9
0.95
1
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vo
lta
ge
0 100 200 300 400-4
-3
-2
-1
0
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rea
ctive
po
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p1
p2
v1
v2
q1
q2
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Conclusions
CONCLUSIONS
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Conclusions
Conclusions
A power system problem for control theory tools!
I A tractable modelI implicit linearI sparseI structure preserving
I Output feedbackin power systems
I model-freeI robustI limited measurement
I Networked controlI co-design?
I Feedback control on thepower flow manifold
I exploit the physics ofthe system in the loop
DistributionGrid
FeedbackControl
Real-timemeasurementsmicro-PMUvoltage meas.line currents
Controlsignals
reactive powertap changers
voltage regulators
Power demandPower generation
Operatinggrid state
operationalconstraints
feasible power region(uncontrolled grid)
feasible power region(virtual reinforcement)
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An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani
Conclusions
Thanks!
Saverio [email protected]
This work is licensed under the Creative Commons Attribution 4.0 Intl License.