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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 approach to the design of distributed feedback control in smart grids

Saverio Bolognani, Florian Dörfler Automatic Control Laboratory ETH Zürich

ECC 2016 Workshop Distributed and Stochastic Optimization: Theory and Applications

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

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

transmission grid distribution grid

Traditional Power Generation

I Distribution grid: the

“capillary system” of power networks

I It delivers power from

the transmission grid to the consumers.

I Very little sensing,

monitoring, actuation.

I The “easy” part of the

grid: conventionally fit-and-forget design.

An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani Future electric power distribution grids

New challenges

I Distributed microgenerators (conventional and renewable sources) I Electric mobility (large flexible demand, spatio-temporal patterns).

41GW 75%

Germany 17 August 2014

wind solar hydro biomass

Distribution grid solar wind hydro + biomass Installed renewable generation Germany 2013 24 GW 15 GW Transmission grid 6 GW

2015 2020 200k 400k 600k 800k PHEV BEV

Switzerland VISION 2020

Electricity consumption Buildings 40.9% Industry 31.3% Transportation 27.8% Energy consumption by sector (2010) 73.9% 25.9% Primary fuel consumption

Electric Vehicle Fast charging

120KW Tesla supercharger 4KW Domestic consumer

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 to simultaneous and uncoordinated power demand/generation. ! lower eciency, blackouts ! curtailment of renewable generation ! bottleneck to electric mobility Fit-and-forget ! unsustainable grid reinforcement

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 x x

Distribution Grid Transmission Grid CONTROL LAYER

Curtailment Reduced hosting capacity Higher renewable generation Larger hosting capacity

• vervoltage

renewable generation controlled undervoltage power demand uncontrolled uncontrolled controlled

distribution grid

control control control control

I Virtual grid reinforcement

I same infrastructure I more sensors and intelligence I controlled grid = larger capacity

I Transparent control layer

I invisible to the users I modular design An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani

OVERVIEW

• 1. A feedback control approach
• 2. A tractable model for control design
• 3. Control design example

I Reactive power control for voltage regulation

• 4. Next step

I Optimization on the power flow manifold

• 5. Conclusions

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

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 microgenerator load supply point

Grid equations diag(u)Yu = s where

I uh = vhejθh complex voltages I sh = ph + jqh complex powers

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 microgenerator load supply point

Actuation

I Tap changer / voltage regulators – supply point voltage v0 I Reactive power compensators – reactive power qh

I static compensators I power inverters of the microgenerators (when available)

I Active power management – active power ph

I smart building control, storage and deferrable loads I generator curtailment and load shedding 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 microgenerator load supply point

Sensing

I Power meters – active power ph and reactive power qh I Voltage meters – nodal voltage vh I Phasor measurement units (PMU) – voltage magnitude vh and angle ✓h

(PQube @ UC Berkeley, GridBox in Zürich/Bern, Smart Grid Campus @ EPFL)

I Line currents, transformer loading, ...

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 distribution network

plant state x

power demands power generation

Control objective Drive the system to a state x∗ = ⇥ v∗ ✓∗ p∗ q∗⇤ subject to

I soft constraints

x∗ = argminx J(x)

I hard constraints

x 2 X

I chance constraints

P [x 62 X] < ✏

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 distribution network

plant state x

power demands power generation

OPF

Conventional approach: Optimal Power Flow

I Similar to power transmission grid OPF I Motivated by encouraging results on OPF convexification

(Lavaei (2012), Farivar (2013), ...)

I Requires full disturbance knowledge - full communication I Heavily model based I Requires co-design of grid control and users’ behavior

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 distribution network

plant state x

power demands power generation

FEED BACK input disturbance

• utput

Control theory answer

I Robustness against parametric uncertainty/unmodeled disturbance I Time varying demand/generation becomes disturbance I Model-free design I Explored so far only for limited cases (e.g. purely local VAR control) I Allows modular design of grid control and users’ behavior

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

frequency

Power network

plant state x

power demands power generation

FEED BACK input disturbance

• utput

primary control secondary control

In the transmission grid, feedback is used for frequency regulation

I Frequency deviation as a implicit signal for power unbalance I Purely local proportional control: primary droop control I Integral control: secondary frequency regulation

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

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

I Grid state x =

⇥ v ✓ 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) 1 0.5 p2

• 0.5

0.5 q2 1 1.2 1 0.8 0.6 0.4 v 2

node 2 node 1

v1 = 1, θ1 = 0 y = 0.4 − 0.8j v2, θ2 p2, q2 p1, q1

v2

1g − v1v2 cos(θ1 − θ2)g − v1v2 sin(θ1 − θ2)b = p1

−v2

1b + v1v2 cos(θ1 − θ2)b − v1v2 sin(θ1 − θ2)g = q1

v2

2g − v1v2 cos(θ2 − θ1)g − v1v2 sin(θ2 − θ1)b = p2

−v2

2b + v1v2 cos(θ2 − θ1)b − v1v2 sin(θ2 − θ1)g = q2 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

• 0.5
• 1

1.5 1 0.5 p2

• 0.5
• 1

1.2 1 1.4 0.8 0.6 v 2

Best linear approximant Ax∗(x x∗) = 0 Ax∗ := @F(x) @x

• x=x∗

Tangent plane at a nominal power flow solution x∗ 2 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 graph I Structure preserving – Elements of Ax∗ depend on local parameters

! Bolognani & Dörfler, Allerton (2015) ! Source code on github

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

2

• 1
• 2

1.4 1.2 v 2 1 0.8 0.6 0.5

• 1
• 0.5

1 1.5 p2

Standard models Adding assumption one obtains

I linear coupled power flow I DC power flow I rectangular DC flow

1.5 1 0.5 q2

• 0.5
• 1

2 1 p2 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

• 1

v 2

Nonlinear coordinate transf. ˜ xh = ˜ xh(xh), @˜ xh @xh = 1 Dierent manifold curvature!

I vh ! v2 h : LinDistFlow

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

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 microgenerator load supply point I Inputs: reactive power qh of microgenerators I Outputs: voltage measurement vh at the microgenerators I Control objective:

I Soft constraints: minimize J(x) = vTLv

(voltage drops)

I Hard constraints: guarantee Vmin  vh  Vmax

at all sensors

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

Modeling assumption: constant R/X ratio ⇢. Ax∗(x x∗) = 0 becomes (around the no-load state)  ⇢L L L ⇢L

• I

I

• 2

6 6 4 v ✓ p q 3 7 7 5 = 0

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 specs

Control specification: Distributed and asynchronous. Minimal update q ⇢qh qh + qk qk ! Communication graph Gcomm describing all possible updates (pairs h, k).

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 assumption
• 2. Control specs
• 3. Proj on linear manifold

Search directions: By projecting each possible direction q on the linear manifold ker Ax∗, we obtain feasible search directions in the state space. x = 2 6 6 4

• 1

1+ρ2 L†q

• ρ

1+ρ2 L†q

q 3 7 7 5

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 + tδx

• 1. Modeling assumption
• 2. Control specs
• 3. Proj on linear manifold
• 4. Derive feedback law

Optimal step length: Given a search direction x, we determine the step length that minimizes the cost function J(x) = vTLv. rJ(x) = 2 6 6 4 2Lv 3 7 7 5 rJ(x + tx)Tx = 0 ) t = (1 + ⇢2) vTq qTL†q

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 + tδx

• 1. Modeling assumption
• 2. Control specs
• 3. Proj on linear manifold
• 4. Derive feedback law

Because the model is sparse and structure preserving... t = (1 + ⇢2) vTq qTL†q = (1 + ⇢2)vh vk Xhk Gossip-like feedback law ( qh qh + (1 + ⇢2) vh−vk

Xhk

qk qk (1 + ⇢2) vh−vk

Xhk

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 distribution network

plant state x

power demands power generation

FEED BACK input disturbance

• utput

I Asynchronous distributed

feedback control

I no demand or generation measurement I limited model knowledge I no power flow solver I alternation of sensing and actuation.

( qh qh + (1 + ⇢2) vh−vk

Xhk

qk qk (1 + ⇢2) vh−vk

Xhk

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

10 20 30 40 50 60 70 iteration E [ J(x) - Jopt ] 1 0.1 0.01 56 58 60 62 100 150 200 250 300 350 400 iteration losses [kW]

! Bolognani & Zampieri, IEEE TAC (2013)

I Extension to J(x) = uTLu (power losses), if ✓ 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.

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

microgenerator load supply point

Gcomm Ggrid Ggrid

Sparsity of the power system

Gcomm

Sparsity of the communication graph Fundamental design problem: implications of the communication architecture on the control performance. Joint design vs. separation results

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 communication

10.0.0.1 18.0.1.2 18.0.1.3 ... ... ...

General purpose network Wireless Multi-area

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 distribution network

plant state x

power demands power generation

FEED BACK input disturbance

• utput

v ≤ vh ≤ v qh ≤ qh ≤ qh

I Power losses minimization I Hard constraints on inputs and outputs. I Construct Lagrangian ! Saddle point algorithm

L(q, , ⌘) = J(q) + T(v v) + ⌘T(qh q)

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 distribution network

plant state x

power demands power generation

FEED BACK input disturbance

• utput

v ≤ vh ≤ v qh ≤ qh ≤ qh h [h + ↵(vh v)]≥0 ⌘h ⇥ ⌘h + (qh qh) ⇤

≥0

q q rJ(q) ˜ L⌘

˜ Lη

Diusion term that requires nearest-neighbor communication. ! Bolognani, Carli, Cavraro & Zampieri, IEEE TAC (2015)

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

10 20 30 40 50

Voltage [p.u.]

1 1.05 1.1

Reactive power [p.u.]

1

v1 v2 |q1| |q2| Modified IEEE 123 Distribution Test Feeder ! github Light load + 2 microgenerators ! overvoltage 2 sets of constraints: ( voltage limits vh  v max reactive power qh  qh

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

10 20 30 40 50

Voltage [p.u.]

1 1.05 1.1

Reactive power [p.u.]

1

steady state error saturation

vi qi

qmax

i

−qmax

i

vmin vmax

qh(t) = f(vh(t)) Fully decentralized, proportional controller. Latest grid code drafts, Vovos (2007), Turitsyn (2011), Aliprantis (2013), ...

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

10 20 30 40 50

Voltage [p.u.]

1 1.02 1.04 1.06 1.08

Reactive power [p.u.]

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)

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

10 20 30 40 50

Voltage [p.u.]

0.98 1 1.02 1.04 1.06

Reactive power [p.u.]

0.5 1

no steady state error reactive power sharing

h [h + ↵(vh v)]≥0 ⌘h ⇥ ⌘h + (qh qh) ⇤

≥0

q q rJ(q) ˜ L⌘ Networked feedback control (neighbor-to-neighbor async communication) ! Cavraro, Bolognani, Carli & Zampieri, IEEE CDC (2016)

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 min

input δ

J() subject to Prob [x / 2 Xc] < ✏

I Xc can encode

I under/over voltage limits I power injection limits I voltage stability region

! Bolognani & Zampieri, IEEE TPS (2015)

I A stochastic model for the

disturbance is available

I Via linear approximant !

deterministic polytope constraints

! Bolognani & Dörfler, PSCC (2016)

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

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. rJ(x): gradient of the cost function (soft constraints) in ambient space
• 2. ΠxrJ(x): projection of the gradient on the linear approximant in x
• 3. Evolve according to ˙

x = ΠxrJ(x)

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 =  xexo xendo

• Exogenous variables

Inputs/disturbances that are imposed on the model.

Reactive power injection qi

Endogenous variables Determined by the physics of the grid.

Voltage vi

Iterative algorithm: at each step

• 1. Compute ΠxrJ(x) (sparse Ax(t) ) distributed algorithm)
• 2. Actuate system based on x = ΠxrJ (exogeneous variables / inputs)
• 3. Retraction step x(t + 1) = Rx(t)(x)

) x(t + 1) 2 M. From iterative optimization algorithm to feedback control on manifolds.

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 coordinates I Primal feasibility at all times I The resulting feasible input region is invariant with respect to the

retraction.

I We can saturate δx = γΠxrJ(x) because

x + δ(x) 2 F ) x(t + 1) = Rx(t)(δx) 2 F

! Geometric Projected Dynamical Systems

I Extension of results on existence and uniqueness of executions for

hybrid automata to manifolds

I Guarantees of no Zeno execution

Ongoing work with Adrian Hauswirth, Gabriela Hug, Florian Dörfler.

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 loop I 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.

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

100 200 300 400 2 4 6 8 active power 100 200 300 400 0.9 0.95 1 1.05 1.1 voltage 100 200 300 400

• 4
• 3
• 2
• 1

1 reactive power

p1 p2 v1 v2 q1 q2

An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani Conclusions

CONCLUSIONS

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 model

I implicit linear I sparse I structure preserving

I Output feedback

in power systems

I model-free I robust I limited measurement

I Networked control

I co-design?

I Feedback control on the

power flow manifold

I exploit the physics of

the system in the loop Distribution Grid Feedback Control

Real-time measurements

micro-PMU voltage meas. line currents

Control signals

reactive power tap changers voltage regulators

Power demand Power generation Operating grid state

• perational

constraints feasible power region (uncontrolled grid) feasible power region (virtual reinforcement)

An optimization-on-manifold approach to the design of distributed feedback control in smart grids Saverio Bolognani Conclusions

Thanks!

Saverio Bolognani

bsaverio@ethz.ch This work is licensed under the Creative Commons Attribution 4.0 Intl License.