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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
3 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
4 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
5 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
5 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
7 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
8 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
9 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
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)f(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
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)f(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
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.
11 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
12 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
12 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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. . .
13 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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. . .
13 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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. . .
13 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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!
14 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
15 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
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
17 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
17 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
17 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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)
18 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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)
18 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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)
18 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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)
18 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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.
19 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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.
19 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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.
19 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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)
22 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
23 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
24 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
25 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
26 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
26 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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.
27 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
• 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
29 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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
30 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
Thank you!
spchatz@elektro.dtu.dk
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
31 DTU Electrical Engineering Convex Relaxations for Optimization of Power Grids under Uncertainty Feb 8, 2017
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