convex relaxations for optimization of power grids under ... · 9 dtu electrical engineering convex...

Convex Relaxations for Optimization of Power Grids

under Uncertainty

International Conference on Future Electric Power Systems and the Energy

Transition, Champery, Switzerland

Spyros Chatzivasileiadis

Acknowledgements (i.e. the one who did the work)

Andreas VenzkeMSc. ETHfrom June 2017: PhD at DTU

2 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017

OPF under uncertainty: necessary for power systemoperation and planning

• Optimal Power Flow:

• operation: determining control setpoints• market clearing• planning

•“We live in an uncertain world”:

• wind• solar• load, e.g. electric vehicles• . . .• investment decisions

Explicitly accounting for uncertainties → more informed (=better?)decisions about power system operation and planning


Approaches for OPF under uncertainty

• AC-OPF is a non-convex non-linearproblem → difficult to solve

• Uncertain variables → additionalcomplexity

x

Costf(x)

Two approaches

DC-OPF → linearize the powerflow equations and the chanceconstraints

Iterative → use AC-OPF, linearizethe chance constraints around anoperating point, and re-solve untilconvergence


Linear vs. Iterative Chance Constrained OPF

Linear OPF

• Pros:

• Faster• Scalable

• Cons:

• No losses• No reactive power flows• Approximate

Iterative non-linear OPF

• Pros:

• Losses considered• Reactive power considered• Scalable

• Cons:

• Non-convex → might betrapped in a local minimum

• 1Vrakopoulou, Margellos, Lygeros,Andersson, TPWRS 20132Roald, Oldewurtel, Krause, Andersson,Powertech 20133Bienstock, Chertkov, Harnett, SIAMReview 20144Lubin, Dvorkin, Backhaus, TPWRS, 2016

1Zhang, Li, TPWRS, 20112Gugilam, Dall’Anese, Chen, Dhople,Giannakis, TSG 20163Baker, Dall’Anese, Summers, NAPS 2016

4Schmidli, Roald, Chatzivasileiadis,

Andersson, PES GM 2016


Approach in this work

• Integrate the chance constraints in an AC Optimal Power Flow withconvex relaxations

• Pros:

• Losses considered• Reactive power considered → capabilities for reactive power control• Can consider large uncertainty deviations• Convex → can find global optimum

• Cons:

• Scalable?

• First steps taken in Vrakopoulou et al, 2013. Here we extend this work inseveral ways.

0M. Vrakopoulou, M. Katsampani, K. Margellos, J. Lygeros, G. Andersson. “Probabilisticsecurity-constrained AC optimal power flow”. In: IEEE PowerTech (POWERTECH). Grenoble,France, 20126 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017

Focus of this work

• Derive a tractable formulation

• For two types of uncertainty sets: rectangular and Gaussian

• Ensure we obtain a zero relaxation gap (our solution is exact for theoriginal problem)

• Investigate the conditions under which we obtain a zero relaxation gap


Outline

• Introduction

•Motivation: Convex vs. Non-Convex Problem and SDP

• SDP-based AC-OPF

• Managing Uncertainty in the OPF

• Rectangular Uncertainty Set

• Test Cases

• Loss Penalty Factor for Zero Relaxation Gap

• Gaussian Uncertainty Set

• Conclusions


Why convex?

• Assume that the difference in thecost function of a local minimumversus a global minimum is 2%

• The total electric energy cost inthe US is ≈ 400 Billion$/year

• 2% amounts to 8 billion US$ ineconomic losses per year x

Costf(x)

• Similar can be the case when we minimize control effort → most controlefforts have an associated cost impact (use of resources, investmentcosts, replacement costs)

• Convex problems guarantee that we find a global minimum → convexifythe OPF problem


Convexifying the Optimal Power Flow problem(OPF)

• Convex relaxations transform theOPF to a convex Semi-DefiniteProgram (SDP)

• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem1

x

Costf(x)

Convex Relaxation

1Javad Lavaei and Steven H Low. “Zero duality gap in optimal power flow problem”. In:IEEE Transactions on Power Systems 27.1 (2012), pp. 92–10710 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017



• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem1

x

Costf(x)f(x)

Convex Relaxation




• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem1 x

Costf(x)f(x)

Convex Relaxation


What is Semidefinite Programming? (SDP)

• SDP is the “generalized” form of an LP (linear program)

Linear Programming Semidefinite Programming

min cT · x

subject to:

ai · x = bi, i = 1, . . . ,m

x ≥0, x ∈ Rn

minC •X :=∑i

∑j

CijXij

subject to:

Ai •X = bi, i = 1, . . . ,m

X �0

• LP: Optimization variables in the form of a vector x.

• SDP: Optim. variables in the form of a positive semidefinite matrix X.


Example: Feasible space of SDP vs LP variables

LP SDP

x1 ≥ 0

x2 ≥ 0X =

[x2 x1x1 1

]� 0⇒ x2 − x21 ≥ 0

x2

x1

x2

x1

• In SDP we can express quadratic constraints, e.g. x21 or x1x2

• optimization variables need not be strictly non-negative

• LP is a special case of SDP


Transforming the AC-OPF to an SDP

• Power is a quadratic function of voltage, e.g.: Pij = f(V 2i , V

2j , ViVj)

• Let W = V V T and express P = f(W ). In that case, P is an affinefunction of W .

• If W � 0 and rank(W ) = 1:

W can be expressed as a product of vectors and we can recover thesolution V to our original problem

• However the rank-1 constraint is non-convex. . .


Applying convex relaxations with SDP

x

f(x) f(Y ∗) ≤ f(x∗)

x

f(x)

f(x∗) = f(Y ∗)

rank(Y ∗) = 1

EXACT: W = V V T

⇓RELAX: W � 0

(((((((rank(W ) = 1

• For the objective functions,it holds EXACT ≥ RELAX

• The RELAX problem is anSDP problem!

• If W ∗ happens also to berank-1, then EXACT =RELAX!


Convex Relaxation of OPF

. . . for each node k and line lm:

Minimize Generation Cost∑k∈G

{ck2(Tr{YkW}+ PDk)2+

ck1(Tr{YkW}+ PDk) + ck0}s. t. Active Power Balance Pmin

k ≤ Tr{YkW} ≤ Pmaxk

Reactive Power Balance Qmink ≤ Tr{YkW} ≤ Qmax

k

Bus Voltages (V mink )2 ≤ Tr{MkW} ≤ (V max

k )2

Active Branch Flow − Pmaxlm ≤ Tr{YlmW} ≤ Pmax

lm

Semi-Definiteness of W W � 0

Rank Constraint (((((((hhhhhhhrank(W ) = 1 ⇒ Convex Relaxation


Notes on the Convex Relaxation

• Relaxation gap: Difference between the solution oforiginal non-convex, non-linear OPF and the SDP

x

Costf(x)f(x)

• If rank(W ) = 1 or 2: solution to original OPF problem can be recovered→ global optimum

• If rank(W ) ≥ 3: the solution W has no physical meaning (but still it is alower bound)

• Molzahn2 derives a heuristic rule: if the ratio of the 2nd to the 3rdeigenvalue of W is larger than 105 → we obtain rank-2.

2Daniel K Molzahn et al. “Implementation of a large-scale optimal power flow solver based onsemidefinite programming”. In: IEEE Transactions on Power Systems 28.4 (2013),pp. 3987–3998.16 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017

Introducing Uncertainty

• Include wind in-feeds with forecast value P fWi

and forecast error ∆PWi :

PWi = P fWi−∆PWi

PWi

Probability

•W (∆PW ): Solution depending on the forecast error

• How can we approximate the dependency on the uncertainty W (∆PW )?

• Use an affine policy:

W (∆PW ) = W0 +

nw∑i=1

∆PWiBi (1)

with 2n× 2n decision matrix Bi for each uncertainty ∆PWi


Affine Policy

With the affine policy we include the following control policies:

• Generator droop d control

∆PWi

Pg

Pg(∆PWi)

Tr{YkBi} = dk(1 + γi) ∀k ∈ G, ∀i ∈ W (2)

γi accounts for non-linear change in losses

⇒ loss penalty with weight µ in objective:

minimize Gen. Cost + µ

nw∑i

γi (3)

• Generator voltage set-point (AVR) control

• Reactive power capabilities of wind farms (power factor)


Uncertainty Sets - Rectangular & Gaussian

How can we model the uncertainty - distribution of forecast errors ∆PWi?

PW1

PW2

P fW1−∆Pmax

W1P fW1

P fW1

+ ∆PmaxW1

P fW2

P fW2

+∆Pmax

W2

P fW2−

∆PmaxW2

W0

Rectangular uncertainty set: Generalnon-Gaussian distributions. Upperand lower bounds are known a-priori.

PW1

PW2

P fW1−∆Pmax

W1P fW1

P fW1

+ ∆PmaxW1

P fW2

P fW2

+∆Pmax

W2

P fW2−

∆PmaxW2

W0

Ellipsoid uncertainty set:Multivariate Gaussian distributionwith known standard deviation andconfidence interval ε.


Formulation for Rectangular Uncertainty Set

PW1

PW2

P fW1−∆Pmax

W1P fW1

P fW1

+ ∆PmaxW1

P fW2

P fW2

+∆Pmax

W2

P fW2−

∆PmaxW2

W0

B1 B2

B3 B4

• It suffices to enforce the chance constraints at the vertices v of theuncertainty set3.

3Kostas Margellos, Paul Goulart, and John Lygeros. “On the road between robust optimizationand the scenario approach for chance constrained optimization problems”. In: IEEE Transactionson Automatic Control 59.8 (2014), pp. 2258–2263.20 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017

Modification of Affine Policy

PWi

System change

W0

W0 −∆PmaxWi

Bi

W0 + ∆PmaxWi

Bi

Bi Bli

Bdi

P fWi−∆Pmax

WiP fWi

P fWi

+ ∆PmaxWi

Modification of affine policy: The linearization between the borders of theconfidence interval is split up into two parts starting from the operatingpoint W0. Red line indicates true system behavior and the dashed lines theapproximation made with the corresponding affine policy.21 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017

Test System

2

8 7 6

3

9

41

G3G2

G1

W2L1L2

5L3 W1

Modified IEEE 9-bus system with wind farms W1 and W2

•W1 with ± 50 MW deviation inside confidence interval

•W2 with ± 40 MW deviation inside confidence interval

• SDP-Solver: MOSEK

• Coded with Julia (open-source)


Simulations – compare the SDP-based OPF formulation:

• with chance constraints based on the affine policy

• with a linearized version of the chance constraints based on PTDFs: lineflows are calculated based on the PTDFs and the correspondingmaximum uncertainty deviations


Simulation Results: Chance-constraints based onPTDFs

Generator droops d1 = [0.5 0.25 0.25 0 -1 0 0 0 0]Generator droops d2 = [0.5 0.25 0.25 0 0 0 -1 0 0]

Generator cost 3384.55 $h

Eigenvalue ratio ρ(W0) = 1.2 × 106

• we satisfy theconditions toobtain the globaloptimum

# Gen VG PG QG V ∗G P∗

G Q∗G

[p.u.] [MW] [Mvar] [p.u.] [MW] [Mvar]

G1 1.09 67.01 9.33 1.09 63.26 13.60G2 1.10 96.79 1.28 1.10 94.44 -4.85G3 1.06 63.54 -43.43 1.06 61.19 -46.78W1 — 50.00 11.34 — 100.00 22.68W2 — 40.00 0.15 — 0.00 0.00∑

— 317.34 -21.33 — 318.89 -15.36

# Branch from to Plm P∗lm Qlm Q∗

lm[MW] [MW] [Mvar] [Mvar]

3 5 6 41.18 67.63 -25.76 -24.00

Maximum voltage [p.u.] V max 1.100 (V max)∗ 1.103

• constraints are notsatisfied


Simulation Results: Chance Constraints based onAffine Policy for Rectangular Uncertainty Set

Generator droops d1 = [0.5 0.25 0.25 0 -1 0 0 0 0]Generator droops d2 = [0.5 0.25 0.25 0 0 0 -1 0 0]

Weight power loss µ = 0.4 $hMW

Generator cost 3378.73 $h

Eigenvalue ratios ρ(W0) = 6.4 × 106

ρ∗(W0 + ∆Pmax1 B1) = 2.5 × 105

ρ∗(W0 + ∆Pmax2 B2) = 2.4 × 105

ρ∗(W0 + ∆Pmax3 B3) = 2.7 × 106

ρ∗(W0 + ∆Pmax4 B4) = 1.9 × 106

• we satisfy theconditions toobtain the globaloptimum

# Gen VG PG QG V ∗G P∗

G Q∗G

[p.u.] [MW] [Mvar] [p.u.] [MW] [Mvar]

G1 1.10 64.70 8.09 1.07 60.96 31.00G2 1.09 97.21 -12.17 1.10 95.34 32.70G3 1.08 65.43 -32.98 0.97 63.56 -80.45W1 — 50.00 11.45 — 100.00 22.94W2 — 40.00 1.39 — 0.00 0.00∑

— 317.34 -24.23 — 319.86 6.18

# Branch from to Plm P∗lm Qlm Q∗

lm[MW] [MW] [Mvar] [Mvar]

3 5 6 42.87 67.50 -24.07 -35.04

Maximum voltage [p.u.] V max 1.100 (V max)∗ 1.100

• all constraints aresatisfied

• we find the trueglobal minimum


Affine Policy

• Generator droop d control

∆PWi

Pg

Pg(∆PWi)

Tr{YkBi} = dk(1 + γi) ∀k ∈ G, ∀i ∈ W (4)

γi accounts for non-linear change in losses

⇒ loss penalty with weight µ in objective:

minimize Gen. Cost + µ

nw∑i

γi (5)

• Penalty factor to minimize the non-linear change in losses plays andimportant role for the zero relaxation gap


Objective value and eigenvalue ratios

... as a function of the droop penalty µ:

Objective value and eigenvalue ratios... as a function of the droop penalty µ:

0 50 100 150 200 250 300

102

106

1010

Eig

enva

lue

ratios⇢

⇢(W0)

⇢(W0 + ⇣1B1)

⇢(W0 + ⇣2B2)

⇢(W0 + ⇣3B3)

⇢(W0 + ⇣4B4)

Limit

0 50 100 150 200 250 3002,140

2,160

2,180

2,200

2,220

Cost

$ h

Generation Cost

0 50 100 150 200 250 300�100

�50

0

Weight for droop penalty µ (p. u.)

Cost

$ h

Droop penalty

) In the region from µ ⇡ 60 � 130 we obtain zero relaxation gap.

February 7, 2017 1 / 1

In the region from µ ≈ 60− 130 we obtain zero relaxation gap.


Formulation for Gaussian Uncertainty Set

PW1

PW2

P fW1−∆Pmax

W1P fW1

P fW1

+ ∆PmaxW1

P fW2

P fW2

+∆Pmax

W2

P fW2−

∆PmaxW2

W0

• The constraints must hold for the outer border of the ellipse

• Using theoretical results on chance constraints4 analytical reformulationof the scalar chance constraints

• Safe approximation of positive semi-definite constraint

• Obtain zero relaxation gap at the operating point and at the four circledpoints, but not necessarily for the worst-case scenario

4Arkadi Nemirovski. “On safe tractable approximations of chance constraints”. In: EuropeanJournal of Operational Research 219.3 (2012), pp. 707–718.28 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017

Formulation for Gaussian Uncertainty Set

PW1

PW2

P fW1−∆Pmax

W1P fW1

P fW1

+ ∆PmaxW1

P fW2

P fW2

+∆Pmax

W2

P fW2−

∆PmaxW2

W0

Bl2

Bu1Bl

1

Bu2

• The constraints must hold for the outer border of the ellipse

• Using theoretical results on chance constraints4 analytical reformulationof the scalar chance constraints

• Safe approximation of positive semi-definite constraint

• Obtain zero relaxation gap at the operating point and at the four circledpoints, but not necessarily for the worst-case scenario4Arkadi Nemirovski. “On safe tractable approximations of chance constraints”. In: European

Journal of Operational Research 219.3 (2012), pp. 707–718.28 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017

Ongoing Work

• Investigating the conditions to obtain zero relaxation gap

• Investigating how to achieve scalability

• Collaboration with MOSEK and faculty from the computer science andmath department at DTU

• Extending this formulation to combined AC and HVDC grids

• HVDC lines can offer corrective control


Conclusions

• Presented a chance-constrained AC-OPF formulation with convexrelaxations

• developed a tractable formulation for rectangular and gaussianuncertainty sets• can reach global optimality• can accomodate large uncertainty deviations and reactive power

control

• Next steps of this work include (open questions):

• How to make this approach more scalable?• What are the conditions to obtain zero relaxation gap? Which of these

conditions depend on the model and which on the numerics/solver?• What are the deviations that are approximated well enough with

linearized chance-constraints (e.g. PTDFs) and for which cases do weneed a more exact representation• Inclusion of HVDC grids


Thank you!

[email protected]

A. Venzke, L. Halilbasic, U. Markovic, G. Hug, S. Chatzivasileiadis. Convex Relaxations ofChance-Constrained Optimal Power Flow. Submitted. 2017. [Online]:arxiv.org/abs/1702.08372

A. Venzke et al, Convex Relaxations of Chance-Constrained Optimal Power Flow for AC andHVDC grids, in preparation


arxiv.org/abs/1702.08372

convex relaxations for optimization of power grids under ... · 9 dtu electrical engineering convex...

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