Robust Optimal Power Flow with Uncertain
Renewables
Sean Harnett, Daniel Bienstock, Misha Chertkov
Columbia University, LANL

Dimacs Workshop on Energy Infrastructure

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLCIGRE -International Conference on Large High Voltage
Electric Systems ’09
Large unexpected fluctuations in wind power can cause
additional flows through the transmission system (grid)
Large power deviations in renewables must be balanced byother sources, which may be far away
Flow reversals may be observed – control difficult
A solution – expand transmission capacity! Difficult(expensive), takes a long timeProblems already observed when renewable penetration high

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLCIGRE -International Conference on Large High Voltage
Electric Systems ’09“Fluctuations” – 15-minute timespanDue to turbulence (“storm cut-off”)
Variation of the same order of magnitude as mean
Most problematic when renewable penetration starts toexceed 20 − 30%
Many countries are getting into this regime

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Optimal power flow (economic dispatch, tertiary control)

Used periodically to handle the next time window(e.g. 15 minutes, one hour)
Choose generator outputs
Minimize cost (quadratic)
Satisfy demands, meet generator and network constraints
Constant load (demand) estimates for the time window
Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLOPF:
min c(p)(a quadratic)s.t.
Bθ = p − d|yij (θi − θj )| ≤ uij(1)
for each line ij

Pgmin ≤ pg ≤ Pgmax(2)for each bus g(3)Notation:p = vector of generations ∈ Rn ,n×n

B ∈R
∀i, j :

d = vector of loads ∈ Rn,(bus susceptance matrix)
ij ∈ E (set of lines)
 P −yij ,y,i =jBij =kjk;{k,j}∈E
0,otherwise

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

min c(p)(a quadratic)s.t.
Bθ = p − d|yij (θi − θj )| ≤ uij
minPg

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables≤ pg ≤max
Pg

for each line ij
for each bus g

Columbia University, LANL

min c(p)(a quadratic)s.t.
Bθ = p − d|yij (θi − θj )| ≤ uij
minPg≤ pg ≤for each line ij

max
Pgfor each bus g

How does OPF handle short-term fluctuations in demand (d)?
Frequency control:
Automatic control: primary, secondary
Generator output varies up or down proportionally to aggregatechangeHow does OPF handle short-term fluctuations in renewable output?
Answer: Same mechanism, now used to handle aggregate wind powerchangeBienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLWind model?

Need to model variation in wind power between dispatchesWind at farm attached to bus i of the form µi + wi – Weibull
distribution?

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLWind model
From CIGRE report, aggregated over Germany:

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Experiment
Bonneville Power Administration data, Northwest US
data on wind fluctuations at planned farms
with standard OPF, 7 lines exceed limit ≥ 8% of the time

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Line limits and line tripping
If power flow in a line exceeds its limit, the line becomes compromised
and may ’trip’. But process is complex and time-averaged:
Thermal limit is most common
Thermal limit may be in terms of terminal equipment, not line itself
Wind strength and wind direction contributes to line temperature
In medium-length lines (∼ 100 miles) the line limit is due to voltage
drop, not thermal reasons
In long lines, it is due to phase angle change (stability), not thermal
reasons
In 2003 U.S. blackout event, many critical lines tripped due tothermal reasons, but well short of their line limit

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLLine trip modelsummary: exceeding limit for too long is bad, but complicatedwant: ”fraction time a line exceeds its limit is small”
proxy: prob(violation on line i) <  for each line i

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Goalssimple control
aware of limits
not too conservative
computationally practicable

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLControlFor each generator i, two parameters:pi = mean output
αi = response parameter
Real-time output of generator i:pi = p i − α iX

∆ωjjwhere ∆ωj = change in output of renewable j (from mean).
Xαi = 1i∼ primary + secondary control
Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Set up
control

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Computing line flows

wind power at bus i: µi + wi
DC approximationBθ = p − d P+(µ + w − α i∈G wi )
θ = B + (p̄ − d + µ) + B + (I − αe T )w
flow is a linear combination of bus power injections:f ij = yij (θi − θj )Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLComputing line flows
f ij = yij(Bi+ − Bj+ )T (p̄ − d + µ) + (Ai − Aj )T w

,

A = B + (I − αe T )Given distribution of wind can calculate moments of line flows:
E f ij = yij (Bi+ − Bj+ )T (p̄ − d + µ)
Pvar (f ij ) := sij2 ≥ yij2 k (Aik − Ajk )2 σk2(assuming independence)and higher moments if necessary
Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Chance constraints to deterministic constraints

chance constraint: P(f ij > fijmax ) < ij and P(f ij < −fijmax ) < ijfrom moments of f ij , can get conservative approximations using e.g.
Chebyshev’s inequality

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Chance constraints to deterministic constraints

chance constraint: P(f ij > fijmax ) < ij and P(f ij < −fijmax ) < ijfrom moments of f ij , can get conservative approximations using e.g.
Chebyshev’s inequality
for Gaussian wind, can do better, since f ij is Gaussian :|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Formulation:
Choose mean generator outputs and control to minimize expected cost,with the probability of line overloads kept small.
min E[c(p)]p,αXs.t.
αi = 1, α ≥ 0i∈GBδ = α, δn = 0X
XX
pi +µi =dii∈Gi∈Wi∈D

f ij = yij (θi − θj ),Bθ = p + µ − d, θn = 0X
sij2 ≥ yij2σk2 (Bik+ − Bjk+ − δi + δj )2k∈W|f ij | + sij φ−1 (1 − ij ) ≤ fijmax
Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Data errors?
sij2 ≥ yij2X

σk2 (Bik+ − Bjk+ − δi + δj )2k∈W|f ij | + sij φ−1 (1 − ij ) ≤ fijmax(the f ij implicitly incorporate the µi )Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Data errors?
sij2 ≥ yij2X

σk2 (Bik+ − Bjk+ − δi + δj )2k∈W|f ij | + sij φ−1 (1 − ij ) ≤ fijmax(the f ij implicitly incorporate the µi )What if the µi or the σk are incorrect? ... What happens toProb(f ij > uij )?

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLLet the correct parameters be µ̃i , σ̃i for each farm i.

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLLet the correct parameters be µ̃i , σ̃i for each farm i.
Theorem: Suppose there are parameters M > 0, V > 0 such that|µ̄i − µi | < Mµi and | σ̄i2 − σi | < V σifor all i. Then:

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLLet the correct parameters be µ̃i , σ̃i for each farm i.
Theorem: Suppose there are parameters M > 0, V > 0 such that|µ̄i − µi | < Mµi and | σ̄i2 − σi | < V σifor all i. Then:
Prob(fij > fijmax ) < ij + O(V ) + O(M)Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLLet the correct parameters be µ̃i , σ̃i for each farm i.
Theorem: Suppose there are parameters M > 0, V > 0 such that|µ̄i − µi | < Mµi and | σ̄i2 − σi | < V σifor all i. Then:
Prob(fij > fijmax ) < ij + O(V ) + O(M)
Here, the O() “hides” some constants dependent on e.g. reactances

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLLet the correct parameters be µ̃i , σ̃i for each farm i.
Theorem: Suppose there are parameters M > 0, V > 0 such that|µ̄i − µi | < Mµi and | σ̄i2 − σi | < V σifor all i. Then:Prob(fij > fijmax ) < ij + O(V ) + O(M)
Here, the O() “hides” some constants dependent on e.g. reactances
Can we guarantee that Prob(fij > fijmax ) is small even under data errors?

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLPolyhedral data error model:|σ̃i − σi | ≤ γi ∀i,

X |σ̃i − σi |iγi

≤ Γ.

Ellipsoidal data error model:(σ̃ − σ)T A(σ̃ − σ) ≤ b
Here A  0 and b > 0 are parameters.

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

chance constraints

Nominal case:

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

chance constraints

Nominal case:|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

chance constraints

Nominal case:|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax→ a conic constraint

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

chance constraints

Nominal case:|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax→ a conic constraint
Robust case:maxE |E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax( E : data error model)

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

chance constraints

Nominal case:|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax→ a conic constraint
Robust case:maxE |E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax( E : data error model) how to solve?

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

chance constraints

Nominal case:|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax→ a conic constraint
Robust case:maxE |E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax( E : data error model) how to solve?
Theorem. The robust problem is a convex optimization problem and canbe solved in polynomial time in the polyhedral and ellipsoidal data cases.

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

chance constraints

Nominal case:|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax→ a conic constraint
Robust case:maxE |E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax( E : data error model) how to solve?
Theorem. The robust problem is a convex optimization problem and canbe solved in polynomial time in the polyhedral and ellipsoidal data cases.
An “ambiguous chance-constrained problem”

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Toy example
What if no line limits?
2 What if tight limit on line connecting generators?
1

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Answer 1
What if no line limits?

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Answer 2
What if small limit on line connecting generators?

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Experiment
How much wind penetration can we handle?
And how much money does this save?
39-bus New England system from MATPOWER

Bienstock, Harnett, Chertkov

30% penetration, CC-OPF cost 264,000

Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Experiment’standard’ OPF solution with 10% buffer on line limits
feasible only up to 5% penetration (right)

Cost 1,275,000 – almost 5 times greater than chance-constrained
Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Big cases
Polish system - winter 2003-04 evening peak

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Big cases
Polish 2003-2004 winter peak case2746 buses, 3514 branches, 8 wind sources
5% penetration and σ = .3µ each source

CPLEX: the optimization problem has
36625 variables
38507 constraints, 6242 conic constraints
128538 nonzeros, 87 dense columns
Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Big cases
Polish 2003-2004 winter peak case2746 buses, 3514 branches, 8 wind sources
5% penetration and σ = .3µ each source

CPLEX: the optimization problem has36625 variables
38507 constraints, 6242 conic constraints
128538 nonzeros, 87 dense columns
Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Big cases

CPLEX:
total time on 16 threads = 3393 seconds”optimization status 6”
solution is wildly infeasible
Gurobi:time: 31.1 seconds”Numerical trouble encountered”

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Cutting-plane method
overview

Cutting-plane algorithm:
remove all conic constraints
repeat until convergence:solve linearly constrained problem
if no conic constraints violated: return
find separating hyperplane for maximum violation
add linear constraint to problem

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Cutting-plane method
Candidate solution violates conic constraint
10864
2-3-2

1-1

2

3-2-4

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Cutting-plane method
Separate: find a linear constraint also violated
108642-3-21-1

23-2-4

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Cutting-plane method
Solve again with linear constraint
108642-3-2

1-1

23-2-4

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Cutting-plane method
New solution still violates conic constraint
108642-3-2

1-1

23-2-4

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Cutting-plane method
Separate again
108642-3-2

1-1

23-2-4

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Cutting-plane method
We might end up with many linear constraints
108642-3-2

1-1

23-2-4

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Cutting-plane method... which approximate the conic constraint1086
42-3-21-1

23-2-4

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

conic constraint:
qx12 + x22 + ... + xk2 = kxk2 ≤ ycandidate solution:(x ∗ , y ∗ )cutting-plane (linear constraint):kx ∗ k2 +x ∗Tx ∗T x
∗
(x−x)
=
≤ykx ∗ k2
kx ∗ k2

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Polish 2003-2004 case
CPLEX: “opt status 6”
Gurobi: “numerical trouble”

Example run of cutting-plane algorithm:
Iteration147
10
12Max rel. error
1.2e-11.3e-3
1.9e-31.0e-48.9e-7

Objective
7.0933e6
7.0934e6
7.0934e6
7.0964e6
7.0965e6

Total running time: 32.9 seconds
Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Back to motivating example
BPA casestandard OPF: cost 235603, 7 lines unsafe ≥ 8% of the time
CC-OPF: cost 237297, every line safe ≥ 98% of the time
run time = 9.5 seconds (one cutting plane!)

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL

Back to motivating example
BPA casestandard OPF: cost 235603, 7 lines unsafe ≥ 8% of the time
CC-OPF: cost 237297, every line safe ≥ 98% of the time
run time = 9.5 seconds (one cutting plane!)

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANLConclusion

Our chance-constrained optimal power flow:
safely accounts for variability in wind power between dispatches
uses a simple control which is easily integrable into existing systemis fast enough to be useful at the appropriate time scale

Bienstock, Harnett, Chertkov
Robust Optimal Power Flow with Uncertain Renewables

Columbia University, LANL