Online Feedback Optimizationwith Applications to Power SystemsFlorian DörflerETH ZürichEuropean Control Conference 2020

Acknowledgements

Adrian Hauswirth Saverio BolognaniLukas

Ortmann

CURRICULUM VITAE

Name: Irina Subotić Birth: 22.03.1993, Belgrade, Serbia e-mail: suboticirina@gmail.com

EDUCATION

WORK EXPERIENCE summer 2015 summer 2011, 2013

“Life Activities Advancement Center” • Speech Recognition project

o Software IKARlab o Forensic analysis of speech, speech recognition

• E-speech Therapist o Communication between a patient and a therapist using AdobeMediaInteractivServer o Design of a data base (MySQL DataBase) for patients’ electronic cards

2013-2016 Undergraduate teaching student assistant at Faculty of Electrical Engineering

• Checking preparedness of students for laboratory exercise • Explaining apparatus and demonstrational experiments to students

2011-2016 Tutoring

• Tutored high school pupils/students mathematics, physics • Preparation for faculty admission exams

AWARDS AND SCHOLARSHIPS

Sep 2016- present

Swiss Federal Institute of Technology in Zurich • Master of Science in Robotics, Systems and Control • GPA: - out of 6.0

2012- 2016 School of Electrical Engineering, University of Belgrade, Serbia • Bachelor of Science in Electrical Engineering and Computing Degree • Signals and Systems department • Focused on Systems Control and Signal Processing • GPA: 9.39 out of 10.0

2008-2012 Mathematical Grammar School, Belgrade

• Secondary School Leaving Diploma • Completed experimental final two years of Elementary School at Mathematical Grammar

School, Belgrade, GPA: 5.00 out of 5.00 • 4 years of high school, focused on Mathematic, Physics and Computer Science, GPA 5.00 out

of 5.00 • Honors:

o “Vuk Karadžić diploma“ for overall success in education

• The “Dositeja” reward for studying abroad in the year of 2016 • Scholar of the City of Belgrade in the years of 2012-2016, 2009 and 2010 • The “Dositeja” reward for the extraordinary success in the year of 2009 and 2010 • Second award on the National Physics competition, in Subotica 2010 • Third award on the National Physics competition, in Belgrade 2009

Irina Subotić Gabriela Hug Miguel Picallo Verena Häberle

1 / 31

feedforwardoptimization

Optimization System

d estimate

u

w

y

complex specifications & decisionoptimal, constrained, & multivariablestrong requirementsprecise model, full state, disturbanceestimate, & computationally intensive

vs. feedbackcontrol

Controller Systemr +u

y

w

−

simple feedback policiessuboptimal, unconstrained, & SISOforgiving nature of feedbackmeasurement driven, robust touncertainty, fast & agile response

→ typically complementary methods are combined via time-scale separation

Optimization Controller Systemr +

u

y

−

offline & feedforward∣∣∣ real-time & feedback

2 / 31

Example: power system balancingoffline optimization: dispatch basedon forecasts of loads & renewables

0

50

100

150

200

0 10 20 30 40 50 60 70 80 90 100

mar

gina

l cos

ts in

€/M

Wh

Capacity in GW

RenewablesNuclear energyLigniteHard coalNatural gasFuel oil

online control based on frequencyFrequency

ControlPower

System50Hz +

u

y

frequency measurement

−

re-schedule set-point to mitigate severeforecasting errors (redispatch, reserve, etc.)

more uncertainty & fluctuations→ infeasible& inefficient to separate optimization & control

50 Hz

51 49

generation

load

control [Milano, 2018]

Re-scheduling costs Germany [mio. €]

!!"#$%

!&' !&(

%%(# %%"&

%!()%!*!

!"## !"#! !"#$ !"#% !"#& !"#' !"#( !"#)

[Bundesnetzagentur, Monitoringbericht 2011-2019]

3 / 31

Synopsis & proposal for control architecturepower grid: separate decision layers hit limits under increasing uncertaintysimilar observations in other large-scale & uncertain control systems:process control systems & queuing/routing/infrastructure networks

proposal: open︸ ︷︷ ︸with inputs & outputs

and online︸ ︷︷ ︸iterative & non-batch

optimization algorithm as feedback︸ ︷︷ ︸real-time interconnected

control

optimizationalgorithm

e.g.,

u = −∇φ(y, u)

dynamicalsystem

x = f (x, u, w)y = h(x, u, w)

actuationu

measurementy

operationalconstraints

u ∈ U

disturbance w

4 / 31

Historical roots & conceptually related workprocess control: reducing the effect of uncertainty in sucessive optimizationOptimizing Control [Garcia & Morari, 1981/84], Self-Optimizing Control [Skogestad, 2000], ModifierAdaptation [Marchetti et. al, 2009], Real-Time Optimization [Bonvin, ed., 2017], . . .

extremum-seeking: derivative-free but hard for high dimensions & constraints[Leblanc, 1922], . . . [Wittenmark & Urquhart, 1995], . . . [Krstić & Wang, 2000], . . . , [Feiling et al., 2018]

MPC with anytime guarantees (though for dynamic optimization): real-time MPC[Zeilinger et al. 2009], real-time iteration [Diel et al. 2005], [Feller & Ebenbauer 2017], etc.

optimal routing, queuing, & congestion control in communication networks:e.g., TCP/IP [Kelly et al., 1998/2001], [Low, Paganini, & Doyle 2002], [Srikant 2012], [Low 2017], . . .

optimization algorithms as dynamic systems: much early work [Arrow et al., 1958],[Brockett, 1991], [Bloch et al., 1992], [Helmke & Moore, 1994], . . . & recent revival [Holding & Lestas,2014], [Cherukuri et al., 2017], [Lessard et al., 2016], [Wilson et al., 2016], [Wibisono et al, 2016], . . .

recent system theory approaches inspired by output regulation [Lawrence et al. 2018]& robust control methods [Nelson et al. 2017], [Colombino et al. 2018]

5 / 31

Theory literature inspired by power systemslots of recent theory development stimulated by power systems problems

[Simpson-Porco et al., 2013], [Bolognaniet al, 2015], [Dall’Anese & Simmonetto,2016], [Hauswirth et al., 2016], [Gan &Low, 2016], [Tang & Low, 2017], . . .

1

A Survey of Distributed Optimization and ControlAlgorithms for Electric Power Systems

Daniel K. Molzahn,⇤ Member, IEEE, Florian Dorfler,† Member, IEEE, Henrik Sandberg,‡ Member, IEEE,Steven H. Low,§ Fellow, IEEE, Sambuddha Chakrabarti,¶ Student Member, IEEE,

Ross Baldick,¶ Fellow, IEEE, and Javad Lavaei,⇤⇤ Member, IEEE

Abstract—Historically, centrally computed algorithms havebeen the primary means of power system optimization and con-trol. With increasing penetrations of distributed energy resourcesrequiring optimization and control of power systems with manycontrollable devices, distributed algorithms have been the subjectof significant research interest. This paper surveys the literatureof distributed algorithms with applications to optimization andcontrol of power systems. In particular, this paper reviewsdistributed algorithms for offline solution of optimal power flow(OPF) problems as well as online algorithms for real-time solutionof OPF, optimal frequency control, optimal voltage control, andoptimal wide-area control problems.

Index Terms—Distributed optimization, online optimization,electric power systems

I. INTRODUCTION

CENTRALIZED computation has been the primary waythat optimization and control algorithms have been ap-

plied to electric power systems. Notably, independent systemoperators (ISOs) seek a minimum cost generation dispatchfor large-scale transmission systems by solving an optimalpower flow (OPF) problem. (See [1]–[8] for related litera-ture reviews.) Other control objectives, such as maintainingscheduled power interchanges, are achieved via an AutomaticGeneration Control (AGC) signal that is sent to the generatorsthat provide regulation services.

These optimization and control problems are formulatedusing network parameters, such as line impedances, systemtopology, and flow limits; generator parameters, such as costfunctions and output limits; and load parameters, such as anestimate of the expected load demands. The ISO collects allthe necessary parameters and performs a central computationto solve the corresponding optimization and control problems.

With increasing penetrations of distributed energy resources(e.g., rooftop PV generation, battery energy storage, plug-invehicles with vehicle-to-grid capabilities, controllable loads

⇤: Argonne National Laboratory, Energy Systems Division, Lemont, IL,USA, dmolzahn@anl.gov. Support from the U.S. Department of En-ergy, Office of Electricity Delivery and Energy Reliability under contractDE-AC02-06CH11357.†: Swiss Federal Institute of Technology (ETH), Automatic Control Labora-tory, Zurich, Switzerland, dorfler@control.ee.ethz.ch‡: KTH Royal Institute of Technology, Department of Automatic Control,Stockholm, Sweden, hsan@kth.se§: California Institute of Technology, Department of Electrical Engineering,Pasadena, CA, USA, slow@caltech.edu¶: University of Texas at Austin, Department of Electrical and Computer En-gineering, Austin, TX, USA, sambuddha.chakrabarti@gmail.com,baldick@ece.utexas.edu. Support from NSF grant ECCS-1406894.⇤⇤: University of California, Berkeley, Department of Industrial Engineeringand Operations Research, Berkeley, CA, USA, lavaei@berkeley.edu

providing demand response resources, etc.), the centralizedparadigm most prevalent in current power systems will poten-tially be augmented with distributed optimization algorithms.Rather than collecting all problem parameters and performinga central calculation, distributed algorithms are computedby many agents that obtain certain problem parameters viacommunication with a limited set of neighbors. Depending onthe specifics of the distributed algorithm and the application ofinterest, these agents may represent individual buses or largeportions of a power system.

Distributed algorithms have several potential advantagesover centralized approaches. The computing agents only haveto share limited amounts of information with a subset ofthe other agents. This can improve cybersecurity and reducethe expense of the necessary communication infrastructure.Distributed algorithms also have advantages in robustness withrespect to failure of individual agents. Further, with the abilityto perform parallel computations, distributed algorithms havethe potential to be computationally superior to centralizedalgorithms, both in terms of solution speed and the maxi-mum problem size that can be addressed. Finally, distributedalgorithms also have the potential to respect privacy of data,measurements, cost functions, and constraints, which becomesincreasingly important in a distributed generation scenario.

This paper surveys the literature of distributed algorithmswith applications to power system optimization and control.This paper first considers distributed optimization algorithmsfor solving OPF problems in offline applications. Many dis-tributed optimization techniques have been developed con-currently with new representations of the physical modelsdescribing power flow physics (i.e., the relationship betweenthe complex voltage phasors and the power injections). Thecharacteristics of a power flow model can have a large impacton the theoretical and practical aspects of an optimizationformulation. Accordingly, the offline OPF section of thissurvey is segmented into sections based on the power flowmodel considered by each distributed optimization algorithm.This paper then focuses on online algorithms applied toOPF, optimal voltage control, and optimal frequency controlproblems for real-time purposes.

Note that algorithms related to those reviewed here havefound a wide variety of power system applications in dis-tributed optimization and control. See, for instance, surveyson the large and growing literature relevant to distributedoptimization of electric vehicle charging schedules [9] anddemand response applications [10] as well as work on dis-tributed solution of multi-period formulations for model pre-

Steven Low

Enrique Mallada

John Simpson-Porco

Changhong Zhao

Claudio De Persis

Nima Monshizadeh

Arjan Van der SchaftMarcello Colombino

Emiliano Dall’Anese

Sairaj Dhople

Andrey Bernstein

Krishnamurthy Dvijotham

Andrea Simonetto

Na Li

Sergio Grammatico

Yue Chen

Florian Dörfler

Saverio Bolognani

Sandro ZampieriJorge Cortez

Henrik Sandberg

Karl Johansson

Ioannis Lestas

Andre Jokic

early adoption: KKT control [Jokic et al, 2009]

literature kick-started ∼ 2013 by groups fromCaltech, UCSB, UMN, Padova, KTH, & Groningen

changing focus: distributed & simple→ centralized & complex models/methods

implemented in microgrids (NREL, DTU, EPFL, . . . )& conceptually also in transactive control pilots (PNNL)

6 / 31

Overview

algorithms & closed-loop stability analysisprojected gradient flows on manifoldsrobust implementation aspectspower system case studies throughout

7 / 31

ALGORITHMS & CLOSED-LOOP

STABILITY ANALYSIS

Stylized optimization problem & algorithmsimple optimization problem

minimizey,u

φ(y, u)

subject to y = h(u)

u ∈ U

cont.-time projected gradient flow

u = ΠgU

(−∇φ

(h(u), u

))= Πg

U

(−[∂h∂u

I]∇φ(y, u)

)∣∣∣y=h(u)

Fact: a regular† solution u : [0,∞]→Xconverges to critical points if φ has Lip-schitz gradient & compact sublevel sets.

projected dynamical system

x ∈ ΠgX [f ](x) , arg min

v∈TxX‖v − f(x)‖g(x)

I domain XI vector field fI metric gI tangent cone TX

all sufficiently regular†

† regularity conditions made precise later8 / 31

Algorithm in closed-loop with LTI dynamicsoptimization problem

minimizey,u

φ(y, u)

subject to y = Hiou+Riow

u ∈ U

→ open & scaled projected gradient flow

u = ΠU(−ε[HTio I

]∇φ(y, u)

)

LTI dynamicsx = Ax+Bu+ Ew

y = Cx+Du+ Fw

const. disturbance w & steady-statemaps

x = −A−1B︸ ︷︷ ︸His

u −A−1E︸ ︷︷ ︸Rds

w

y =(D − CA−1B

)︸ ︷︷ ︸Hio

u +(F − CA−1E

)︸ ︷︷ ︸Rdo

w

ε∫

Uu

B∫

w E A

∇u φ D F

HTio∇y φ

yC

+ x

++

+++

− ++−

9 / 31

Stability, feasibility, & asymptotic optimality

Theorem: Assume thatregularity of cost function φ: compact sublevel sets & `-Lipschitz gradient

LTI system asymptotically stable: ∃ τ > 0 , ∃P � 0 : PA+ATP � −2τP

sufficient time-scale separation (small gain): 0 < ε < ε? , 2τcond(P )

· 1`‖Hio‖

Then the closed-loop system is stable and globally converges to the criticalpoints of the optimization problem while remaining feasible at all times.

Proof: LaSalle/Lyapunov analysis via singular perturbation [Saberi & Khalil ’84]

Ψδ(u, e) = δ · eTP e︸ ︷︷ ︸LTI Lyapunov function

+ (1− δ) · φ(h(u), u

)︸ ︷︷ ︸

objective function

with parameter δ∈(0, 1) & steady-state error coordinate e=x−Hisu−Rdsw

→ derivative Ψδ(u, e) is non-increasing if ε ≤ ε? and for optimal choice of δ

10 / 31

Example: optimal frequency control

dynamic LTI power system modelpower balancing objective

control generation set-points

unmeasured load disturbances

measurements: frequency + constraint variables (injections & flows)

I linearized swing dynamicsI 1st-order turbine-governorI primary frequency droopI DC power flow approximation

optimization problem

→ objective: φ(y, u) =cost(u)︸ ︷︷ ︸economic generation

+ 12‖max{0, y − y}‖2Ξ + 1

2‖max{0, y − y}‖2Ξ︸ ︷︷ ︸

operational limits (line flows, frequency, . . . )

→ constraints: actuation u ∈ U & steady-state map y = Hiou+Rdow

→ control u = ΠU (. . .∇φ) ≡ super-charged Automatic Generation Control

11 / 31

Test case: contingencies in IEEE118 systemevents: generator outage at 100 s & double line tripping at 200 s

0 50 100 150 200 250 3000

2

4

6

Time [s]

Power Generation (Gen 37) [p.u.]

Setpoint Output

12 / 31

How conservative is ε < ε? ?still stable for ε = 2 ε?

−5

0

5

·10−2 Frequency Deviation from f0 [Hz]

System Frequency

0 5 10 15 20

0

1

2

3

Time [s]

Line Power Flow Magnitudes [p.u.]

23→26 90→26 flow limit other lines

unstable for ε = 10 ε?

−2

0

2

4

Frequency Deviation from f0 [Hz]

System Frequency

0 5 10 15 20

0

2

4

Time [s]

Line Power Flow Magnitudes [p.u.]

23→26 90→26 flow limit other lines

Note: conservativeness problem dependent & depends, e.g., on penalty scalings13 / 31

Highlights & comparison of approachWeak assumptions on plant

internal stability→ no observability / controllability→ no passivity or primal-dual structure

measurements & steady-state I / O map→ no knowledge of disturbances→ no full state measurement→ no dynamic model

Weak assumptions on costLipschitz gradient + properness

→ no (strict/strong) convexity required

Parsimonious but powerful setuppotentially conservative bound, but

→ minimal assumptions onoptimization problem & plant

robust & extendable proof→ nonlinear dynamics→ time-varying disturbances→ general algorithms

take-away: open online optimizationalgorithms can be applied in feedback

→ Hauswirth, Bolognani, Hug & Dörfler (2020)“Timescale Separation in Autonomous Optimization”→ Menta, Hauswirth, Bolognani, Hug & Dörfler (2018)

“Stability of Dynamic Feedback Optimizationwith Applications to Power Systems”

14 / 31

Nonlinear systems & general algorithmsgeneral system dynamics x = f(x, u) with steady-state map x = h(u)

incremental Lyapunov functionW (x, u) w.r.t error coordinate x− h(u)

W (x, u) ≤ −γ ‖x− h(u)‖2 ‖∇uW (x, u)‖ ≤ ζ ‖x− h(u)‖

variable-metric Q(u) ∈ Sn+ gradient flow

u = −Q(u)−1∇φ(u)

examples: Newton method Q(u)=∇2φ(u)

or mirror descent Q(u)=∇2ψ(∇ψ(u)−1)

stability condition: ζ`γ·supu‖Q(u)−1‖ < 1

Similar results for algorithms with memory:momentum methods (e.g., heavy-ball)

(exp. stable) primal-dual saddle flows

non-examples: bounded-metricor Lipschitz assumption violated

0 10 20 30 40 500

5

10

15

20Cost Value

Dynamic IC

Algebraic IC

0 20 40 60 80 10010

-10

10-5

100

105

1010

Cost Value

Dynamic IC

Algebraic IC

co

st

valu

e

algebraic plant

dynamic plant

algebraic plant

dynamic plant

discontinuous subgradient

Nesterov acceleration

15 / 31

Highly nonlinear & dynamic test case

Nordic system: case study known forvoltage collapse (South Sweden ’83)

(static) voltage collapse: sequenceof events→ saddle-node bifurcation

high-fidelity model of Nordic systemI RAMSES + Python + MATLABI state: heavily loaded system & large

power transfers: north→ centralI load buses with Load Tap ChangersI generators equipped with Automatic

Voltage Regulators, Over ExcitationLimiters, & speed governor control

g15

g11

g20

g19

g16

g17

g18

g2

g6

g7

g14

g13

g8

g12

g4

g5

g10 g3g1

g9

4011

4012

1011

1012 1014

1013

10221021

2031

cs

404640434044

40324031

4022 4021

4071

4072

4041

1042

10451041

4063

40611043

1044

4047

4051

40454062

400 kV

220 kV

130 kV synchronous condenserCS

NORTH

CENTRAL

EQUIV.

SOUTH

4042

2032

41

1 5

3

2

51

47

42

61

62

63

4

43 46

3132

22

11 13

12

72

71

16 / 31

Voltage collapse

event: 250 MW load rampfrom t = 500 s to t = 800 s

unfortunate control response:non-coordinated + saturation

I extra demand is balanced byprimary frequency control

I cascade of activation ofover-excitation limiters

I load tap changers increasepower demand at load buses

bifurcation: voltage collapse

very hard to mitigate viaconventional controllers

→ apply feedback optimizationto coordinate set-pointsof Automatic Voltage Controllers

17 / 31

Voltage collapse averted !distance-to-collapse objective : φ = −log det

(power flow Jacobian

)

18 / 31

PROJECTED GRADIENT

FLOWS ON MANIFOLDS

Motivation: steady-state AC power flowstationary model

Ohm’s Law Current Law

AC power

AC power flow equations

(all variables and parameters are -valued)imagine constraints slicing this set⇒ nonlinear, non-convex, disconnected

additionally the parameters are ±20%

uncertain . . . this is only the steady state!

graphical illustration of AC power flow

[Hiskens, 2001]

[Molzahn, 2016]19 / 31

Key insights on physical equality constraint

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

vdc

idc

m

iI

v

LI

CI GI

RI

τm

θ, ω

vf

v

if

τe

is

Lθ

Mrf

rs rs

v

iTLT

CG GqC v

RTiI

AC power flow is complex but takesthe form of a smooth manifold

→ local tangent plane approximations,local invertibility, & generic LICQ

→ regularity (algorithmic flexibility)

→ Hauswirth, Bolognani, Hug, & Dörfler (2015)“Fast power system analysis via implicitlinearization of the power flow manifold”

→ Bolognani & Dörfler (2018)“Generic Existence of Unique LagrangeMultipliers in AC Optimal Power Flow”

AC power flow is attractive steadystate for ambient physical dynamics

→ physics enforce feasibility even fornon-exact (e.g., discrete) updates

→ robustness (algorithm & model)

→ Gross, Arghir, & Dörfler (2018)“On the steady-state behavior ofa nonlinear power systemmodel”

20 / 31

Feedback optimization on the manifold

challenging specificationson closed-loop trajectories:1. stay on manifold at all times2. satisfy constraints at all times3. converge to optimal solution

feedbackoptimizationalgorithm

x = ΠgX (−gradφ(x))

physical steady-statepower system

(AC power flow)

Sk + wk =∑

`1

zk`

∗Vk(V∗k − V∗` )

renewablesloads w

generationsetpoints

measurements

prototypical optimal power flow

minimize φ(x)

subject to x ∈ X =M∩K

φ : Rn → R objective functionM⊂ Rn AC power flow manifoldK ⊂ Rn operational constraints

v

TxX

X

projection of trajectory on feasible cone21 / 31

Simple low-dimensional case studies . . .. . . can have simple feasible sets . . . or can have really complex sets

v0 = 1

slack bus generator

qG ∈ [q, q]

vref = 1

load

pL(t)

pG

1j

θ0 = 0

0-2

0.5

3

v

1

2

pG-pL

0

qG

10

2 -1

application demands sophisticated level of generality !22 / 31

Projected dynamical systems on irregular domains

Theorem: Consider a Carathéodory solutionx : [0,∞)→ X of the initial value problem

x = ΠgX (−gradφ(x)) , x(0) = x0 ∈ X .

If φ has compact sublevel sets on X , then x(t)

converges to the set of critical points of φ on X .

Hidden assumption: existence, uniqueness, &completeness of Carathéodory solution x(t) ∈ Xin absence of convexity, Euclidean space, . . . ? X =

{x : ‖x‖22 = 1 , ‖x‖1 ≤

√2}

regularity conditions constraint set vector field metric manifoldexistence of Krasovski loc. compact loc. bounded bounded C1

existence of Carathéodory Clarke regular C0 C0 C1

uniqueness of solutions prox regular C0,1 C0,1 C1,1

→ Hauswirth, Bolognani, & Dörfler (2018)“Projected Dynamical Systems on Irregular

Non-Euclidean Domains for Nonlinear Optimization”

→ Hauswirth, Bolognani, Hug, & Dörfler (2016)“Projected gradient descent on Riemanniann manifolds with

applications to online power system optimization”23 / 31

ROBUST IMPLEMENTATION ASPECTS

Robust implementation of projectionsprojection & integrator→ windup→ robust anti-windup approximation→ saturation often “for free” by physics

K

∫PU

k(·, u) x = f(x, ·)

+

−

u

PU (u)

−+

u = ΠU [k(x, ·)](u)K → ∞

disturbance→ time-varyingdomain

f(x)

ΠtX f(x)

X (t)

X (t + δ)

I temporal tangentcone & vector field

I ensure suff. regularity& tracking certificates

→ Hauswirth, Dörfler, & Teel (2020)“Anti-Windup Approximations of Oblique Projected Dynamical

Systems for Feedback-based Optimization”

handling uncertainty when enforcingnon-input constraints : x ∈ X or y ∈ Y

I cannot measure state x directly→ Kalman filtering: estimation& separation

I cannot enforce constraints on y=h(u)by projection (not actuated & h(·)unknown)

→ soft penalty or dualization + grad flows(inaccurate, violations, & strong assumptions)

→ project on1st order predictionof y=h(u)

y+ ≈ h(u)︸︷︷︸measured

+ ε ∂h∂u︸︷︷︸

steady-stateI/O sensitivity

w︸︷︷︸feasible descent

direction

⇒ global convergence to critical points

→ Häberle, Hauswirth, Ortmann, Bolognani, & Dörfler (2020)“Enforcing Output Constraints in Feedback-based Optimization”

→ Hauswirth, Subotić, Bolognani, Hug, & Dörfler (2018)“Time-varying Projected Dynamical Systems with Applications. . . ”24 / 31

Tracking performance under disturbances

G1

G2 C1

C3 C2

W

S

Generator

Synchronous Condensor

Solar

Wind

G

C

S

W

5.3.2 30 Bus Power Flow Test Case

To investigate the capabilities of our new scheme under time-varying generation limits and fluc-tuating load conditions, we consider a power system setup based on the IEEE 30 bus power flowtest case, where wind and solar generation has been added, similar to the one adopted in [13].The grid topology is shown in Figure 5.4, where the controllable units along with their gener-ation limits and the operational constraints of the associated buses are listed in Table 5.3. Inparticular, the upper power generation limit of the solar and the wind farm are time-varying, dueto the fluctuating nature of the corresponding primary sources. This results in a time-varyingconstraint set U of the controllable variables. Additional operational constraints that need to besatisfied include line current limits for different branches. The total generation cost � we aim tominimize is composed of the costs of each generator in [$/h], given as aip

2i +bipi, where ai, bi > 0

are constant cost-coefficients provided in Table 5.3. The marginal operating cost of the solar andthe wind farm is set to zero.

We simulate 24 hours of operation and run Algorithm 1, where the controller receives fieldmeasurements of the system state z every minute. The demand profile is shown in Figure 5.4,which exhibits an abrupt demand reduction of approximately 20% between 20:30 and 21:30 atseveral system buses.

G1

G2 C1

C3 C2S

Generator

�������������� � �

Solar

Wind

G

C

S

W

W

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2470

80

90

100

110

120

Figure 5.4: Modified IEEE 30 bus power flow test case.

Unit Type Set-points ai bi pgi pgi

qgi qgi

vi vi

G1 Gen.1 V✓, v 0.1 0.9 75 0 50 -50 1.06 0.94freq. ctrl.

G2 Gen.2 PV p, v 0.04 0.5 60 0 50 -50 1.06 0.94C1 Cond.1 PV v 0 0 0 0 50 -50 1.06 0.94C2 Cond.2 PV v 0 0 0 0 50 -50 1.06 0.94C3 Cond.3 PV v 0 0 0 0 50 -50 1.06 0.94S Solar PQ p, q 0 0 ps(t) 0 50 -50 1.06 0.94W Wind PQ p, q 0 0 pw(t) 0 50 -50 1.06 0.94

line 1-2 line 6-8 line 12-15

Table 5.3: Cost coefficients a and b in [$/MW2h] and [$/MWh], respectively. Active powergeneration limits in [MW] and reactive power generation limits in [MVAr], and bus voltagelimits in [p.u.]. The system base power is fixed to 100MVA.

41

net demand: load, wind, & solar (discontinuous)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-50

0

50

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-50

0

50

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.95

1

1.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.95

1

1.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

100

200

300

(a) Simulation results of controlled 30 buspower system with exact Jacobian matrixru,yF (u, y, w).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

100

200

300

(b) Simulation results of controlled 30 buspower system with constant approximation ofthe Jacobian matrix.

Figure 5.5: Simulation results of controlled 30 bus power system for the exact Jacobian matrixru,yF (u, y, w) and a constant approximation thereof. The dashed lines represent the constraintsand the colors are the same as in Table 5.3.

43

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-50

0

50

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-50

0

50

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.95

1

1.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.95

1

1.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

100

200

300

(a) Simulation results of controlled 30 buspower system with exact Jacobian matrixru,yF (u, y, w).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

100

200

300

(b) Simulation results of controlled 30 buspower system with constant approximation ofthe Jacobian matrix.

Figure 5.5: Simulation results of controlled 30 bus power system for the exact Jacobian matrixru,yF (u, y, w) and a constant approximation thereof. The dashed lines represent the constraintsand the colors are the same as in Table 5.3.

43

25 / 31

Optimality despite disturbances&uncertainty

transient trajectory feasibilitypractically exact tracking ofground-truth optimizer(omniscient & no computation delay)

robustness to model mismatch(asymptotic optimality under wrong model)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-50

0

50

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-50

0

50

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.95

1

1.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.95

1

1.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

100

200

300

(a) Simulation results of controlled 30 buspower system with exact Jacobian matrixru,yF (u, y, w).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

100

200

300

(b) Simulation results of controlled 30 buspower system with constant approximation ofthe Jacobian matrix.

Figure 5.5: Simulation results of controlled 30 bus power system for the exact Jacobian matrixru,yF (u, y, w) and a constant approximation thereof. The dashed lines represent the constraintsand the colors are the same as in Table 5.3.

43

offline optimization feedback optimizationmodel uncertainty feasible ? φ− φ∗ ‖v − v∗‖ feasible ? φ− φ∗ ‖v − v∗‖loads ±40% no 94.6 0.03 yes 0.0 0.0line params ±20% yes 0.19 0.01 yes 0.01 0.0032 line failures no -0.12 0.06 yes 0.19 0.007

conclusion: simple algorithm performs extremely well & robust26 / 31

EXPERIMENTS

Experimental case study @ DTU

EV

SE

1

EV

SE

2

EV

SE

3

EV

SE

4

EV

SE

5

EV

SE

6

EV

SE

7

EV

SE

8

Busbar A

Busbar B

Busbar A

Busbar B

630 kVA100 kVA

Bat

tery

Ext

. 117

-5

Cab

le C

2

Cab

le C

1

CE

E

Ext

. 117

-2

Chg

. pos

t

Cab

le D

1

PV

NE

VIC

Busbar A Busbar B Busbar C

150 kVA

100 kVA

Gaia

Flexhouse

PV

Cable B1

Cable B2

Busbar B Busbar B

Busbar A 200 kVA

Static load

Diesel

CEE

Aircon

Cable A1

Cable A2

PV

Busbar B

Busbar A Busbar B

Cable F1

Flexhouse 2 Flexhouse 3

Cable E1

Cable E2

CEE

CHP

Heatpump 1

BoosterHeater

Cable F1

Crossbar switch

Load conv.

SYSLABbreaker overview

Building 716 Building 715 Building 319

Building 117

Ship

Shore

Mach.set

Con

tain

er 1

Con

tain

er 2

Con

tain

er 3

I

I

I

PCC

v1 v2 v3R1, L1 R2, L2 R3, L3

p1, q1 p2, q2 p3, q3

PV1 PV2 Battery

±8 kVAr

Static load

±6 kVAr±6 kVAr 0 kVAr

10 kW0 kVAr0 kW −15 kW

Voltage[p.u.]

10.99

1.061.05

0.95

21min experiment with eventsI t = 3min: control turned ONI t ∈ [11, 14]min: Pbatt = 0 kW

base-line controllersdecentralizednonlinearproportionaldroop control(IEEE1547.2018)

vi

qiqmaxi

qmini

vmin vmax

qi(t+ 1) = fi(vi(t))

1

comparison of three controllersI decentralized controlI feedforward optimizationI feedback optimization

→ Ortmann, Hauswirth, Caduff, Dörfler, & Bolognani (2020)“Experimental Validation of Feedback Optimization

in Power Distribution Grids”

27 / 31

Decentralized feedback controldecentralized nonlinear proportional droop control

0.97

1

1.031.051.07

−5

0

5

0 5 10 15 20

0.970.980.99

1

Time [min]0 5 10 15 20

0

1

2

0.970.980.99

1

Volta

ge[p.u.]

0

1

2

Reac

tivePo

wer[kV

Ar]Battery

PV2

PV1

constraint violations due to local control saturation & lack of coordination

28 / 31

Successive feedforward optimizationcentralized, omniscient, & successively updated at high sampling rate

0.97

1

1.031.051.07

−5

0

5

0 5 10 15 20

0.970.980.99

1

Time [min]0 5 10 15 20

−2

−1

0

0.970.980.99

1

Volta

ge[p.u.]

−2

−1

0

Reac

tivePo

wer[kV

Ar]Battery

PV2

PV1

performs well but persistent constraint violation due to model uncertainty

29 / 31

Feedback optimizationprimal-dual flow with 10 s sampling time requiring only model I/O sensitivity∇h (or an estimate)

0.97

1

1.031.051.07

−5

0

5

0 5 10 15 20

0.970.980.99

1

Time [min]0 5 10 15 20

−6

−4

−2

0

0.970.980.99

1

Volta

ge[p.u.]

−4

−2

0

Reac

tivePo

wer[kV

Ar]Battery

PV2

PV1

excellent performance & model-free(!) since ∇h(u) approximated by[

1 1 11 1 11 1 1

]30 / 31

CONCLUSIONS

ConclusionsSummary

open & online feedback optimization algorithms as controllers

approach: projected dynamical systems & time-scale separation

unified framework: broad class of systems, algorithms, & programs

illustrated throughout with non-trivial power systems case studies

Ongoing work & open directions

analysis: robustness, performance, stochasticity, sampled-data

algorithms: 0th-order, sensitivity estimation, distributed, minmax

power systems: more experiments, virtual power plant extensions

further app’s: seeking optimality in uncertain & constrained systems

It works much better than it should ! We still need to fully grasp why ?

31 / 31

Thanks !

Florian Dörflerhttp://control.ee.ethz.ch/~floriand

[link] to related publications