Operations Research Problems in Power Engineering: … - Bienstock Plenary Lecture.pdfRobust Network...

Operations Research Problems in PowerEngineering:

Robust Network control through DisjunctiveProgramming

Daniel Bienstock

Columbia University

ODS, 2017

Bienstock Columbia University

Operations Research Problems in Power Engineering:Robust Network control through Disjunctive Programming

Power engineering for non-power engineers

References

Andersson: Modelling and Analysis of Electric Power Systems

Bergen, Vittal: Power Systems Analysis

Glover, Sarma, Overbye: Power System Analysis and Design

Rebours, Kirschen: What is spinning reserve?




References

Andersson: Modelling and Analysis of Electric Power Systems

Bergen, Vittal: Power Systems Analysis

Glover, Sarma, Overbye: Power System Analysis and Design

Rebours, Kirschen: What is spinning reserve?




conductor

steammagnetic

field

statorrotor

sourceenergy

ω

current, voltage




conductor

steammagnetic

field

statorrotor

sourceenergy

ω

current, voltage
















conductor

steammagnetic

field

statorrotor

sourceenergy

ω

current, voltage

A generator produces current at a certain voltage.

Ohm’s law: power = current x voltage




conductor

steammagnetic

field

statorrotor

sourceenergy

ω

current, voltage

A generator produces current at a certain voltage.

Ohm’s law: power = current x voltage





AC Power Flows

Real-time:

k m

V(t)k

I(t) km

Voltage at bus k : vk(t) = Vmaxk cos(ωt + θVk )

Current injected at k into km: ikm(t) = Imaxkm cos(ωt + θIkm).

Power injected at k into km: pkm(t) = vk(t)ikm(t).

Averaged over period T :

pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax

km cos(θVk − θIkm).



AC Power Flows

Real-time:

k m

V(t)k

I(t) km





pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax




AC Power Flows

Real-time:

k m

V(t)k

I(t) km





pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax




AC Power Flows

Real-time:

k m

V(t)k

I(t) km





pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax




AC Power Flows

Real-time:

k m

V(t)k

I(t) km





pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax




k m

V(t)k

I(t) km

pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax

km cos(θVk − θIkm)

Vk.

=Vmaxk√

2e jθ

Vk , Ikm

.=

Imaxkm√

2e jθ

Imk

pkm = |Vk ||Ikm| cos(θVk − θIkm) = Re(Vk I∗km)

qkm.

= Im(VkmI∗km) and Skm

.= pkm + jqkm



k m

V(t)k

I(t) km

pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax


Vk.

=Vmaxk√

2e jθ

Vk , Ikm

.=

Imaxkm√

2e jθ

Imk


qkm.


.= pkm + jqkm



k m

V(t)k

I(t) km

pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax


Vk.

=Vmaxk√

2e jθ

Vk , Ikm

.=

Imaxkm√

2e jθ

Imk


qkm.


.= pkm + jqkm



k m

V(t)k

I(t) km

pkm.

= 1T

∫ T0 p(t) = 1

2Vmaxk Imax


Vk.

=Vmaxk√

2e jθ

Vk , Ikm

.=

Imaxkm√

2e jθ

Imk


qkm.


.= pkm + jqkm



Vk.

=Vmaxk√2e jθ

Vk , Ikm

.=

Imaxkm√2e jθ

Imk (voltage, current)

pkm = Re(Vk I∗km), qkm = Im(VkmI

∗km) (1)

Ikm = y{k,m}(Vk − Vm), y{k,m} = admittance of km. (2)

Network Equations

k



Vk.

=Vmaxk√2e jθ

Vk , Ikm

.=

Imaxkm√2e jθ



∗km) (1)

Ikm = y{k,m}(Vk − Vm),

y{k,m} = admittance of km. (2)

Network Equations

k



Vk.

=Vmaxk√2e jθ

Vk , Ikm

.=

Imaxkm√2e jθ



∗km) (1)


Network Equations

k



Vk.

=Vmaxk√2e jθ

Vk , Ikm

.=

Imaxkm√2e jθ



∗km) (1)


Network Equations

k



Vk.

=Vmaxk√2e jθ

Vk , Ikm

.=

Imaxkm√2e jθ



∗km) (3)


Network Equations

∑km∈δ(k)

pkm = Pk ,∑

km∈δ(k)

qkm = Qk ∀ k (5)

Generator: Pk , |Vk | given. Other buses: Pk , Qk given.



Vk.

=Vmaxk√2e jθ

Vk , Ikm

.=

Imaxkm√2e jθ



∗km) (3)


Network Equations

∑km∈δ(k)

pkm = Pk ,∑

km∈δ(k)

qkm = Qk ∀ k (5)




Vk.

=Vmaxk√2e jθ

Vk , Ikm

.=

Imaxkm√2e jθ



∗km) (3)


Network Equations

∑km∈δ(k)

pkm = Pk ,∑

km∈δ(k)

qkm = Qk ∀ k (5)




Optimal power flow (economic dispatch, tertiary control)

Used periodically to handle the next time window(e.g. 5 minutes, 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



DC = linearized Optimal Power Flow (OPF):

min c(p) (a quadratic)

s.t.

Bθ = p − d (6)

|yij(θi − θj)| ≤ uij for each line ij (7)

Pming ≤ pg ≤ Pmax

g for each bus g (8)

Notation:

p = vector of generations ∈ Rn, d = vector of loads ∈ Rn

B ∈ Rn×n, (bus susceptance matrix)

∀i , j : Bij =

−yij , ij ∈ E (set of lines)∑

k;{k,j}∈E ykj , i = j

0, otherwise



Managing changing demands







What happens when there is a generation/load mismatch

conductor

steammagnetic

field

statorrotor

sourceenergy

ω

current, voltage

Frequency response:mismatch ∆P ⇒ frequency change ∆ω ≈ −c ∆P




conductor

steammagnetic

field

statorrotor

sourceenergy

ω

current, voltage

Frequency response:

mismatch ∆P ⇒ frequency change ∆ω ≈ −c ∆P




conductor

steammagnetic

field

statorrotor

sourceenergy

ω

current, voltage

Frequency response:mismatch ∆P

⇒ frequency change ∆ω ≈ −c ∆P




conductor

steammagnetic

field

statorrotor

sourceenergy

ω

current, voltage

Frequency response:mismatch ∆P ⇒ frequency change ∆ω ≈ −c ∆P




1 Primary frequency control. Handles instantaneous (small)changes.

Agent: physics.

2 Secondary control. Handles changes that span more than afew seconds. Agent: algorithms, pre-set controls.

3 “Tertiary” control: OPF (Optimal power flow). Manageslonger lasting changes. Run every few minutes. Goal:economic generation that meets demands while maintainingfeasibility (stability). Agent: algorithmic computations,humans.

4 Once (?) a day: unit commitment problem. Chooses whichgenerators will operate in the next day or half-day. Agent:algorithms, humans.




1 Primary frequency control. Handles instantaneous (small)changes. Agent: physics.








2 Secondary control. Handles changes that span more than afew seconds.

Agent: algorithms, pre-set controls.















3 “Tertiary” control: OPF (Optimal power flow). Manageslonger lasting changes. Run every few minutes. Goal:economic generation that meets demands while maintainingfeasibility (stability).

Agent: algorithmic computations,humans.















4 Once (?) a day: unit commitment problem. Chooses whichgenerators will operate in the next day or half-day.

Agent:algorithms, humans.











s.t.

Bθ = p − d

|yij (θi − θj )| ≤ uij for each line ij


g for each bus g

How does the grid handle short-term fluctuations in demand (d)?Secondary frequency control:

Deployed a few seconds after ongoing change – “minute-by-minute”control

Generator output varies up or down proportionally to aggregatechange




s.t.

Bθ = p − d



g for each bus g






“Participation factors”

For each generator i , a parameter αi with∑i αi = 1

αi ≥ 0

αi > 0 only for selected generators

Assuming real-time generation/demand mismatch ∆, real-timeoutput of generator i :

pi = pi − αi∆

where pi = OPF computed output for generator i .

Note: the αi are precomputed e.g. uniform or based on economicconsiderations.





CIGRE -International Conference on Large High VoltageElectric Systems ’09

Large unexpected fluctuations in wind power can causeadditional 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 high



CIGRE -International Conference on Large High VoltageElectric Systems ’09

“Fluctuations” – 15-minute timespan

Due 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




s.t.

Bθ = p − d



g for each bus g




How does the grid handle short-term fluctuations in renewable output?Answer: Same mechanism, now used to handle aggregate wind powerchange ← Well, it doesn’t work




s.t.

Bθ = p − d



g for each bus g








s.t.

Bθ = p − d



g for each bus g




How does the grid handle short-term fluctuations in renewable output?Answer: Same mechanism, now used to handle aggregate wind powerchange

← Well, it doesn’t work




s.t.

Bθ = p − d



g for each bus g







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



What problem to focus on?

Control line temperature excursions through betterparticipation factors

How to use storage (batteries)? ←





How to use storage (batteries)?

←





How to use storage (batteries)? ←



A simple network setup

generator (source)controllable

uncontrollable

demand (load)

known

DC-OPF: Choose generator outputs so as to satisfy all demands,at minimum cost

Demands are known

Each arc has a capacity

Each generator has a (piecewise linear convex) cost function

This is an optimization problem with linear constraints



A simple network setup

generator (source)controllable

uncontrollable

demand (load)

known

DC-OPF: Choose generator outputs so as to satisfy all demands,at minimum cost

Demands are known

Each arc has a capacity

Each generator has a (piecewise linear convex) cost function

This is an optimization problem with linear constraints



A simple example:

100

100

Only one solution:

100

100200

100

200 100

But what if the red node suddenly injects power?



A simple example:

100

100

Only one solution:

100

100200

100

200 100




A simple example:

100

100

Only one solution:

100

100200

100

200 100




100

100

Only one solution:

100

100200

100

200 100

If red node suddenly injects power, offset using blue node:

100

100200

100

200 100

50

50

50



100

100

Only one solution:

100

100200

100

200 100

If red node suddenly injects power, offset using blue node:

100

100200

100

200 100

50

50

50



Better example: red node is cheap but unreliable source

100

100

50 + error

Base case solution:

100

100

150

150100

50 50

50

If the red node suddenly injects power, offset using blue node:

100

100

150

150100

50

50 + D

50 + D

D



Examples:

100

100

30

10100

160

160

− 15

− 10

75 0

X Y

75 + X Y

capacity = 15

Any combination with X + Y = 25 “works” so long as Y ≤ 15.

100

100

30

10100

160

160

X Y

+ 8

− 20

70 70 + X 1818 + Y

Any combination with X + Y = 12 “works” so long asY ≤ −3.



OPF + renewables + storage

A model for robust grid operation using storage (batteries) needs:

1 A control framework: how do we operate batteries

2 A model for data errors – renewable outputs and ’smart’loads

3 A model for battery dynamics

4 Multi-period model: T time periods of length ∆



























Battery model

Energy state bounds. Let E 0j = initial energy state of

battery at site i . Then

E tj = E 0

j + ∆t∑

h=1

δhj

is the energy state at the end of period t .Must have lower- and upper-bounds on E t

j .

Special bounds for ETj ?

Discharge rate (“speed”) bounds. We will want to upperbound the instantaneous rate of charge or discharge of anybattery.



Battery chemistry is complex. Traditional model:

1 0 < ηci ≤ 1, 0 < ηd

i ≤ 1: charging, discharging efficiencyfor battery i

2 If we inject electrical power Di ≥ 0 into battery i , energystate increases by ηc

i Di

3 If we extract electrical power Gi ≥ 0 from battery i ,energy state decreases by (ηd

i )−1Gi

4 Summary: if the electrical power injection into battery i isPi , energy state changes by

(−ηdi )−1 [Pi ]− + ηci [Pi ]

+

5 A nonconvex model: if Pi is a decision variable, must enforce

[Pi ]− [Pi ]

+ = 0




1 0 < ηci ≤ 1, 0 < ηd



i Di


i )−1Gi


(−ηdi )−1 [Pi ]− + ηci [Pi ]

+


[Pi ]− [Pi ]

+ = 0




1 0 < ηci ≤ 1, 0 < ηd



i Di


i )−1Gi


(−ηdi )−1 [Pi ]− + ηci [Pi ]

+


[Pi ]− [Pi ]

+ = 0




1 0 < ηci ≤ 1, 0 < ηd



i Di


i )−1Gi


(−ηdi )−1 [Pi ]− + ηci [Pi ]

+


[Pi ]− [Pi ]

+ = 0



Piecewise linear battery model:

e0

eK

starting state

energy inputx

C(x)

electrical

ending state

minE

maxE

y

C (y)(−1)



Piecewise linear battery model:

e0

eK

starting state

energy inputx

C(x)

electrical

ending state

minE

maxE

y

C (y)(−1)

This is a charging picture. Similar picture for discharging

Piecewise-constant charge/discharge efficiency

Piecewise-constant charge/discharge rate (speed): if batteryin piece k , output/input power at most Pk



Renewable forecast error model

w ti + w t

i = output of renewable at bus i at time t.w t

i = forecast, w ti = error (uncertain).

How to model the w ti ? Gaussian, iid?

Probably not independent.

Example: moving weather front




w ti + w t



How to model the w ti ?

Gaussian, iid?






w ti + w t









w ti + w t








A technical point

Suppose that x(t), t = 1, 2, . . . ,T is a stochastic process:

Even if we understand each x(t) well enough to provide tailprobabilities ...

Partial sums∑k

t=1 x(t) are a different story

Except e.g. in the gaussian case



A technical point

Suppose that x(t), t = 1, 2, . . . ,T is a stochastic process:

Even if we understand each x(t) well enough to provide tailprobabilities ...

Partial sums∑k

t=1 x(t) are a different story

Except e.g. in the gaussian case



Concentration models:real-time error set W is the set of all w such that

C 1w+ + C 2w− ≤ b, (9a)

wmink,t ≤ w t

k ≤ wmaxk,t for all t and k , (9b)

C1,C2 are nonnegative matrices

b, wmin,wmax: parameters of model

Caution! (10a) is not a linear model

Allows for spatial and temporal correlation




C 1w+ + C 2w− ≤ b, (9a)

wmink,t ≤ w t


C1,C2 are nonnegative matrices

b, wmin,wmax: parameters of model

Caution! (10a) is not a linear model

Allows for spatial and temporal correlation




C 1w+ + C 2w− ≤ b, (10a)

wmink,t ≤ w t


Special (symmetric) example: uncertainty budgets

|w ti | ≤ γti , all t and i∑

i

αti |w t

i | ≤ Γt all t

Here, the γti , αt

i and Γt are parameters.

Extra special case: αti = 1/γt

iMany variations, e.g. time-correlated models∑

t

αti |w t

i | ≤ Γi ,∑i

∑t

αti |w t

i | ≤ Γ




C 1w+ + C 2w− ≤ b, (10a)

wmink,t ≤ w t


Special (symmetric) example: uncertainty budgets

|w ti | ≤ γti , all t and i∑

i

αti |w t

i | ≤ Γt all t

Here, the γti , αt

i and Γt are parameters.Extra special case: αt

i = 1/γti

Many variations, e.g. time-correlated models∑t

αti |w t

i | ≤ Γi ,∑i

∑t

αti |w t

i | ≤ Γ



Control framework

w ti + w t

i = output of renewable at bus i at time t. w ti =

forecast, w ti = error (uncertain).

δtj = output of battery at bus j at time t.

Generic linear control:

δtj = −

∑i

λtji w ti

λt : decision variables

Complex? Requires many (accurate?) real-time observations



Control framework

w ti + w t




Generic linear control:

δtj = −

∑i

λtji w ti

λt : decision variables

Complex? Requires many (accurate?) real-time observations



Control framework

w ti + w t




Specialized linear control:

δtj = −λtj

∑i∈R(j)

wi

λtj : decision variables

R(j): set of buses that battery at j responds to



Optimization Problem

→ Linearized, or DC power flow model:

flow on km =θt

k − θtm

xkm, xkm = reactance

flow balance equations: Bθ = generation - demand vector



Optimization Problem

minPg ,λ

∑t

∑k

ctk(Pg ,tk )

s.t. the following system is feasible at all times t, for all w ∈ W:

Flow balance:

B θt = Pg ,t +

renewables︷︸︸︷w t + w t

batteries︷︸︸︷−

(∑i

w ti

)λt − Pd ,t

Battery dynamics constraints

Line limits:

| θtk − θt

m|xkm

≤ ukm ∀ km, t



How to build a good algorithm for this problem?

→ Theoretical fundaments: well, the problem is strongly-NP hard

Cutting-plane procedure, or a game

Start with a relaxation for the robust problem, e.g. the nominalproblem (no errors), and then

1 Solve relaxation, with solution (Pg∗, λ∗)

2 Play adversary: find a worst-case distribution w for(Pg∗, λ∗)

3 If (Pg∗, λ∗) revealed infeasible by w , separate and goto 1.







































Separating from battery speed constraint

e0

eK

starting state

energy inputx

C(x)

electrical

ending state

minE

maxE

y

C (y)(−1)

Initial electrical state for a given battery: yIf at time t, electrical state in (zk , zk+1), then output ≤ Pk

Violation at t, if

(1) zk < y +∑

h<t∑

i λhji wh

i < zk+1

(2)∑

i λtji w t

i > Pk




e0

eK

starting state

energy inputx

C(x)

electrical

ending state

minE

maxE

y

C (y)(−1)


If I am the attacker: given λ pick w ∈ W so that

(1) zk < y +∑

h<t∑

i λhji wh

i < zk+1

(2)∑

i λtji w t

i > Pk

Fact. This computation can be formulated as a linear program




e0

eK

starting state

energy inputx

C(x)

electrical

ending state

minE

maxE

y

C (y)(−1)


If I am the attacker: given λ pick w ∈ W so that

(1) zk < y +∑

h<t∑

i λhji wh

i < zk+1

(2)∑

i λtji w t

i > Pk

Fact. This computation can be formulated as a linear program



Violation at t, if

(1) zk < E0 +∑

h<t∑

i λhji wh

i < zk+1

(2)∑

i λtji w t

i > Pk

Moreover, given w ∈ W , a control λ satisfies the speedconstraint at t if:

1 Either E0 +∑

h<t∑

i λhji wh

i ≤ zk , or

2 E0 +∑

h<t∑

i λhji wh

i ≥ zk+1, or

3∑

i λtji w t

i ≤ Pk ,

a 3-way disjunction



Violation at t, if

(1) zk < E0 +∑

h<t∑

i λhji wh

i < zk+1

(2)∑

i λtji w t

i > Pk

Moreover, given w ∈ W , a control λ satisfies the speedconstraint at t if:

1 Either E0 +∑

h<t∑

i λhji wh

i ≤ zk , or

2 E0 +∑

h<t∑

i λhji wh

i ≥ zk+1, or

3∑

i λtji w t

i ≤ Pk ,

a 3-way disjunction



Six-period results on Polish Grid (2746wp)

penetration max error Cost Iterations Time (s)

18 % 9 % 7236431 18 348

18 % 14 % 7257286 28 515

18 % 17 % 7263172 25 498

base load ≈ 24.8 GW

32 wind farms, average predicted output ≈ 4.5 GW

50 batteries, total initial energy state ≈ 3.2 GW



Interesting results?

Solution is sparse! 50 batteries, but only ∼ 14 participatein the optimal control in any time period

Solution is non-uniform! Different batteries participate todifferent degree, even though batteries are all identical

Which batteries participate depends on the time period!Why?

This is work with: Gonzalo Munoz and Shuoguang Yang, Columbia University, Carsten Matke, NEXT ENERGY(Germany)To appear: CDC ’17

















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