Multi-time-step Chance Constrained Generation
Re-dispatch
Scott Backhaus, Russell Bent, Daniel Bienstock, Misha
Chertkov
Columbia University, LANLFERC Software Converence 2014

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Organization
1

Review: chance-constrained OPF

2Extension: Robustness

3

Extension: Multi-time-step formulation

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Review of past work: chance-constrained DC OPF
CIGRE ’09: large unexpected fluctuations in wind power cancause 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 time
Problems already observed when renewable penetration highBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLDC-OPF:
min c(p)(a quadratic)s.t.
Bθ = p − d(1)|βij (θi − θj )| ≤ uij
Pgmin ≤ pg ≤ Pgmax

for each line ij(2)for each generator g(3)

Notation:p = vector of generations ∈ Rn ,B ∈ Rn×n ,d = vector of loads ∈ Rn(bus susceptance matrix)

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Proposed Control Mechanism

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Proposed Control Mechanism
F = set of renewable sources, e.g. wind farms∆ωj = real-time fluctuation in output of renewable j(deviation from mean).

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Proposed Control Mechanism
F = set of renewable sources, e.g. wind farms∆ωj = real-time fluctuation in output of renewable j(deviation from mean).
For each generator i, two parameters:pi = mean output
αi = response parameter (“participation factor”)
Real-time output of generator i:X
pi = p i − α i
∆ωj
j

where
X

αi = 1,α≥0i∼ primary + secondary control,
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Proposed Control Mechanism
F = set of renewable sources, e.g. wind farms∆ωj = real-time fluctuation in output of renewable j(deviation from mean).
For each generator i, two parameters:pi = mean output
αi = response parameter (“participation factor”)
Real-time output of generator i:X
pi = p i − α i
∆ωj
j

where
X

αi = 1,α≥0i∼ primary + secondary control, extends existing practice
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Modeling risk: 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
IEEE Standard 738 computes line temperature as a function of
power flow and numerous exogenous parameters (wind,temperature, humidity, air pressure, date, time of day, latitude andlongitude, ...)In 2003 U.S. blackout event, many critical lines tripped due tothermal reasons, but well short of their line limit

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Modeling risk: line limits and line tripping

summary: exceeding limit for too long is bad, but precise modeldifficultwant: ”fraction time a line exceeds its limit is small”
proxy: prob(violation on line pq) < pq

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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 = βij (θi − θj )Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Computing line flows
f ij = βij

(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 = βij (Bi+ − Bj+ )T (p̄ − d + µ)P
var (f ij ) := sij2 ≥ βij2 k (Aik − Ajk )2 σk2(assuming independence)and higher moments if necessaryBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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,α
X
s.t.
αi = 1, α ≥ 0i∈GBδ = α, δn = 0
X
X
X
pi +
µi =
di
i∈G

i∈F

i∈D

f ij = βij (θi − θj ),Bθ = p + µ − d, θn = 0
X
sij2 ≥ βij2
σk2 (Bik+ − Bjk+ − δi + δj )2k∈F|f ij | + sij φ−1 (1 − ij ) ≤ fijmax
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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,α
X
s.t.
αi = 1, α ≥ 0i∈GBδ = α, δn = 0
X
X
X
pi +
µi =
di
i∈G

i∈F

i∈D

f ij = βij (θi − θj ),Bθ = p + µ − d, θn = 0
X
sij2 ≥ βij2
σk2 (Bik+ − Bjk+ − δi + δj )2k∈F|f ij | + sij φ−1 (1 − ij ) ≤ fijmax
A convex optimization problem.
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Summary of computational experience

Polish 2003-2004 winter peak caseBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Summary of computational experience

Polish 2003-2004 winter peak case2746 buses, 3514 branches, 8 wind sources

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Summary of computational experience

Polish 2003-2004 winter peak case2746 buses, 3514 branches, 8 wind sources
36625 variables
38507 constraints, 6242 conic constraints
128538 nonzeros, 87 dense columns

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Summary of computational experience

Polish 2003-2004 winter peak case2746 buses, 3514 branches, 8 wind sources
36625 variables
38507 constraints, 6242 conic constraints
128538 nonzeros, 87 dense columns

Solvers (Cplex, Gurobi) cannot solve problem

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Summary of computational experience

Polish 2003-2004 winter peak case2746 buses, 3514 branches, 8 wind sources
36625 variables
38507 constraints, 6242 conic constraints
128538 nonzeros, 87 dense columns

Solvers (Cplex, Gurobi) cannot solve problem
Specialized cutting-plane algorithm solves in ∼ 30 seconds onnormal computer

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLMathConic constraint:

v
u n
uX
x2
z ≥ t
i

i=1

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLMathConic constraint:

v
u n
uX
x2
z ≥ t
i

i=1

x̃ ∈ Rn :Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatcha given vector

Columbia University, LANLMathConic constraint:

v
u n
uX
x2
z ≥ t
i

i=1

x̃ ∈ Rn :a given vector

First-order approximation:Pn
i=1 x̃i xi
z ≥ qPn2i=1 x̃iBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLMathConic constraint:

v
u n
uX
x2
z ≥ t
i

i=1

x̃ ∈ Rn :a given vector

First-order approximation:Pn
i=1 x̃i xi
z ≥ qPn2i=1 x̃i
Soon to appear in SIAM Review
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLNeed for robustness!
min E[c(p)]
p,α
X
s.t.
αi = 1, α ≥ 0i∈GBδ = α, δn = 0
X
X
X
µi =
di
pi +
i∈G

i∈F

i∈D

f ij = βij (θi − θj ),Bθ = p + µ − d, θn = 0
X
sij2 ≥ βij2
σk2 (Bik+ − Bjk+ − δi + δj )2k∈F|f ij | + sij φ−1 (1 − ij ) ≤ fijmax
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustness: what do we want

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustness: what do we want1

We do not want to go crazyBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustness: what do we want1

We do not want to go crazy2

When data errors are big we want our solutions to degrade ina controlled manner

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustness: what do we want1

We do not want to go crazy2

When data errors are big we want our solutions to degrade ina controlled manner

3

When data errors are small we want our solutions to degradevery little

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustness: what do we want1

We do not want to go crazy2

When data errors are big we want our solutions to degrade ina controlled manner

3

When data errors are small we want our solutions to degradevery little from nominal behavior

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Sensitivity to data errors?

sij2 ≥ βij2X

σk2 (Bik+ − Bjk+ − δi + δj )2k∈F|f ij | + sij φ−1 (1 − ij ) ≤ fijmax(the f ij implicitly incorporate the µi )Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Sensitivity to data errors?

sij2 ≥ βij2X

σk2 (Bik+ − Bjk+ − δi + δj )2k∈F|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 > fijmax )?

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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:

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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(M) + O(V )Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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(M) + O(V )Here, the O() “hides” some constants dependent on e.g. reactances

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

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(M) + O(V )Here, the O() “hides” some constants dependent on e.g. reactances
In other words, solution quality degrades “gracefully”

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustness: small errors

Polyhedral data error model:|σ̃i2 − σi2 | ≤ γi ∀i,

X |σ̃ 2 − σ 2 |
i

i

i

γi

≤ Γ.

Ellipsoidal data error model:(σ̃ 2 − σ 2 )T A(σ̃ 2 − σ 2 ) ≤ b
Here A  0 and b > 0 are parameters.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustifing chance constraints
Nominal case:Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustifing chance constraints|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustifing chance constraints|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax

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

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustifing chance constraints|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax

Nominal case:→ a conic constraint
Robust case:maxE |E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax( E : data error model)
how to formulate?
sij2 ≥ βij2

X

(Bik+ − Bjk+ − δi + δj )2 σk2k∈FBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Robustifing chance constraints|E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax

Nominal case:→ a conic constraint
Robust case:maxE |E f ij | + var (f ij )φ−1 (1 − ij ) ≤ fijmax( E : data error model)
how to formulate?
sij2 ≥ βij2

X

(Bik+ − Bjk+ − δi + δj )2 σk2

k∈F

sij2 ≥

max
βij2
2{σk }∈E

X(Bik+ − Bjk+ − δi + δj )2 σk2k∈FBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLHow NOT to formulate: duality
Lemma: Let(U(γ, Γ)

=

σ

2

∈ RF
+

: |σi2 − σ̄i2 | ≤ γi ∀i ∈ F,

X |σ 2 − σ̄ 2 |ii∈FBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

i

γi

)
≤Γ .

Columbia University, LANLHow NOT to formulate: duality
Lemma: Let(U(γ, Γ)

=

σ

2

∈ RF
+

: |σi2 − σ̄i2 | ≤ γi ∀i ∈ F,

X |σ 2 − σ̄ 2 |
i

i∈F

i

γi

)
≤Γ .

Then
sij2 ≥

max{σk2 }∈U(γ,Γ)

βij2

X

(Bik+ − Bjk+ − δi + δj )2 σk2

(4)

k∈F

is equivalent to:

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLHow NOT to formulate: duality
Lemma: Let(U(γ, Γ)

=

σ

2

∈ RF
+

: |σi2 − σ̄i2 | ≤ γi ∀i ∈ F,

X |σ 2 − σ̄ 2 |
i

i

γi

i∈F

)
≤Γ .

Then
sij2 ≥

max{σk2 }∈U(γ,Γ)

βij2

X

(Bik+ − Bjk+ − δi + δj )2 σk2

(4)

k∈F

is equivalent to:"#1/2
X {i,j}
X
2
2
{i,j}
sij ≥
σ̄k (πik − πjk − δi + δj ) + Γa
+
bk
k∈F

k∈F

1 {i,j}{i,j}(πik − πjk − δi + δj )2 −a
− bk
≤ 0 ∀k ∈ F
γk
{i,j}

bk

≥ 0 ∀k ∈ F; a{i,j} ≥ 0.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLHow NOT to formulate: duality
Lemma: Let(U(γ, Γ)

=

σ

2

∈ RF
+

: |σi2 − σ̄i2 | ≤ γi ∀i ∈ F,

X |σ 2 − σ̄ 2 |
i

i

γi

i∈F

)
≤Γ .

Then
sij2 ≥

max{σk2 }∈U(γ,Γ)

βij2

X

(Bik+ − Bjk+ − δi + δj )2 σk2

(4)

k∈F

is equivalent to:"#1/2
X {i,j}
X
2
2
{i,j}
sij ≥
σ̄k (πik − πjk − δi + δj ) + Γa
+
bk
k∈F

k∈F

1 {i,j}{i,j}(πik − πjk − δi + δj )2 −a
− bk
≤ 0 ∀k ∈ F
γk
{i,j}

bk

≥ 0 ∀k ∈ F; a{i,j} ≥ 0.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

NOT CONVEX!
Columbia University, LANL

But the original constraint IS convex!

sij2 ≥

max
βij2
2{σk }∈E

X(Bik+ − Bjk+ − δi + δj )2 σk2

(5)

k∈F

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

But the original constraint IS convex!

sij2 ≥

max
βij2
2{σk }∈E

X(Bik+ − Bjk+ − δi + δj )2 σk2

(5)

k∈F

Algorithm.
1. Solve convex relaxation (initially: empty). Let δ ∗ be optimal.
2. (For each line (i, j)) computeP
max{σk2 }∈E βij2 k∈F (Bik+ − Bjk+ − δi∗ + δj∗ )2 σk2which is a convex problem in the above cases.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

But the original constraint IS convex!

sij2 ≥

max
βij2
2{σk }∈E

X(Bik+ − Bjk+ − δi + δj )2 σk2

(5)

k∈F

Algorithm.
1. Solve convex relaxation (initially: empty). Let δ ∗ be optimal.
2. (For each line (i, j)) computeP
max{σk2 }∈E βij2 k∈F (Bik+ − Bjk+ − δi∗ + δj∗ )2 σk2which is a convex problem in the above cases. Let {σ̂ 2 } be optimal.
3. Linearize (5) around δ ∗ and {σ̂ 2 } (and add cut)Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals. m > 1 intervals.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals. m > 1 intervals.

2

We will set the average operating point p̄i(end of) interval h.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch(h)for each generator i andColumbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals. m > 1 intervals.

2

We will set the average operating point p̄i for each generator i and(end of) interval h. → This allows us to model generator ramping.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch(h)Columbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals. m > 1 intervals.

2

We will set the average operating point p̄i for each generator i and(end of) interval h. → This allows us to model generator ramping.

3

Each interval h split into K ≥ 1 sub-intervals.

(h)Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals. m > 1 intervals.

2

We will set the average operating point p̄i for each generator i and(end of) interval h. → This allows us to model generator ramping.

3

Each interval h split into K ≥ 1 sub-intervals.

4

Actual expected output of generator i at subinterval k of interval h:

(h)

(h,k)

p̄i
5

=

K − k (h)k − 1 (h+1)
p̄ +p̄K −1 iK −1 i(h,k)αi= participation factor for generator i at subinterval k,interval h.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals. m > 1 intervals.

2

We will set the average operating point p̄i for each generator i and(end of) interval h. → This allows us to model generator ramping.

3

Each interval h split into K ≥ 1 sub-intervals.

4

Actual expected output of generator i at subinterval k of interval h:

(h)

(h,k)

p̄i
5

=

K − k (h)k − 1 (h+1)
p̄ +p̄K −1 iK −1 i(h,k)αi= participation factor for generator i at subinterval k,(h)interval h. Could be αi .

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals. m > 1 intervals.

2

We will set the average operating point p̄i for each generator i and(end of) interval h. → This allows us to model generator ramping.

3

Each interval h split into K ≥ 1 sub-intervals.

4

Actual expected output of generator i at subinterval k of interval h:

(h)

(h,k)

p̄i
5

=

K − k (h)k − 1 (h+1)
p̄ +p̄K −1 iK −1 i(h,k)αi= participation factor for generator i at subinterval k,(h)interval h. Could be αi . Or even αi .

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL

Part II. Multi-time-step chance-constrained OPF
1Covers multiple OPF planning intervals. m > 1 intervals.

2

We will set the average operating point p̄i for each generator i and(end of) interval h. → This allows us to model generator ramping.

3

Each interval h split into K ≥ 1 sub-intervals.

4

Actual expected output of generator i at subinterval k of interval h:

(h)

(h,k)

p̄i

=

K − k (h)k − 1 (h+1)
p̄ +p̄K −1 iK −1 i

(h,k)

5

αi
= participation factor for generator i at subinterval k,(h)interval h. Could be αi . Or even αi .

6

di

(h)

=

estimate for demand at bus i at interval h.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLReal-time power flow balance equation:
At (instantaneous) time t in subinterval k of interval h:Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLReal-time power flow balance equation:
At (instantaneous) time t in subinterval k of interval h:!
Bθ (h,k) (t) = p̄ (h,k) + µ(h,k) − d (h) + ω(t) −Xωi (t) α(h) ,iBent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLReal-time power flow balance equation:
At (instantaneous) time t in subinterval k of interval h:!
Bθ (h,k) (t) = p̄ (h,k) + µ(h,k) − d (h) + ω(t) −Xωi (t) α(h) ,iRandom quantities in bold.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLReal-time power flow balance equation:
At (instantaneous) time t in subinterval k of interval h:!
Bθ (h,k) (t) = p̄ (h,k) + µ(h,k) − d (h) + ω(t) −Xωi (t) α(h) ,iRandom quantities in bold.
(h,k)

θi

(t) = phase angle at bus i at time t.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLReal-time power flow balance equation:
At (instantaneous) time t in subinterval k of interval h:!
Bθ (h,k) (t) = p̄ (h,k) + µ(h,k) − d (h) + ω(t) −Xωi (t) α(h) ,iRandom quantities in bold.
(h,k)

θi

(t) = phase angle at bus i at time t.
(h,k)

Output at farm i = µi

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch+ ωi (t)Columbia University, LANLReal-time power flow balance equation:
At (instantaneous) time t in subinterval k of interval h:!
Bθ (h,k) (t) = p̄ (h,k) + µ(h,k) − d (h) + ω(t) −Xωi (t) α(h) ,iRandom quantities in bold.
(h,k)

θi

(t) = phase angle at bus i at time t.
(h,k)

Output at farm i = µi
+ ωi (t)(h)(E ωi (t) = 0, var (ωi (t)) = (σi )2 )Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLReal-time power flow balance equation:
At (instantaneous) time t in subinterval k of interval h:!
Bθ (h,k) (t) = p̄ (h,k) + µ(h,k) − d (h) + ω(t) −Xωi (t) α(h) ,iRandom quantities in bold.
(h,k)

θi

(t) = phase angle at bus i at time t.
(h,k)

Output at farm i = µi
+ ωi (t)(h)(E ωi (t) = 0, var (ωi (t)) = (σi )2 )Leads to conic formulation of multi-time-stepchance-constrained problem
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLReal-time power flow balance equation:
At (instantaneous) time t in subinterval k of interval h:!
Bθ (h,k) (t) = p̄ (h,k) + µ(h,k) − d (h) + ω(t) −Xωi (t) α(h) ,iRandom quantities in bold.
(h,k)

θi

(t) = phase angle at bus i at time t.
(h,k)

Output at farm i = µi
+ ωi (t)(h)(E ωi (t) = 0, var (ωi (t)) = (σi )2 )Leads to conic formulation of multi-time-stepchance-constrained problem
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLThe value of reduced uncertainty
An imaginary situation:
The OPF dispatching periods are five minutes long; theplanning horizon spans a total of three hours (36 intervals).

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLThe value of reduced uncertainty
An imaginary situation:
The OPF dispatching periods are five minutes long; theplanning horizon spans a total of three hours (36 intervals).
A weather disturbance (a set of storm cells) is expected toreach the geographical area under consideration at roughly thetwo-hour mark (i.e. at h = 24). This disturbance will either
affect the northern or the southern sectors of the grid.

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLThe value of reduced uncertainty
An imaginary situation:
The OPF dispatching periods are five minutes long; theplanning horizon spans a total of three hours (36 intervals).
A weather disturbance (a set of storm cells) is expected toreach the geographical area under consideration at roughly thetwo-hour mark (i.e. at h = 24). This disturbance will either
affect the northern or the southern sectors of the grid.
Which of the two cases takes place will be known by the 1.5
hour mark (i.e. at h = 18).

Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANLThe value of reduced uncertainty
An imaginary situation:
The OPF dispatching periods are five minutes long; theplanning horizon spans a total of three hours (36 intervals).
A weather disturbance (a set of storm cells) is expected toreach the geographical area under consideration at roughly thetwo-hour mark (i.e. at h = 24). This disturbance will either
affect the northern or the southern sectors of the grid.
Which of the two cases takes place will be known by the 1.5
hour mark (i.e. at h = 18).
We can take advantage of the possibility of the recourse in theformulation
Bent, Backhaus, Bienstock, Chertkov
Multi-time-step Chance Constrained Generation Re-dispatch

Columbia University, LANL