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Applications of Bilevel Mixed-Integer Programming to Power Systems Resilience Devendra Shelar Joint work with Saurabh Amin and Ian Hiskens January 19, 2018 Outline Motivation Modeling Network model Generalized disruption model

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Applications of Bilevel Mixed-Integer Programming to Power Systems Resilience

Devendra Shelar Joint work with Saurabh Amin and Ian Hiskens January 19, 2018

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Outline

  • Motivation
  • Modeling
  • Network model
  • Generalized disruption model
  • Multi-regime System Operator (defender) model
  • Grid-connected, cascade, islanding
  • Bilevel formulation
  • Benders decomposition
  • Resource dispatch
  • Controllable DGs, islanding capabilities
  • Trilevel formulation – solution approach

2

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SLIDE 3

Cyberphysical disruptions

Hurricane Maria (September 2017)

  • Customers facing

blackouts for months

3

Metcalf Substation (April 2013)

  • Sniper attack on 17

transformers

  • Telecommunication cables cut
  • 15 million $ worth of damage
  • 100 mn $ for security upgrades

Ukraine attack (Dec 2015- 2016)

  • First ever blackouts

caused by hackers

  • Controllers damaged for

months

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SLIDE 4

Attack scenarios

8

=> supply-demand imbalance (sudden / prolonged)

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SLIDE 5

Three regimes of SO operation

Distribution substation 𝑄

" #, 𝑅" #

π‘Š

" #

𝑄

" ', 𝑅" '

π‘Š

" '

Transmission network βˆ’Ξ”π‘Š TN level disturbance Attack-induced DN level supply-demand imbalance SO response DER disconnect -- cascade load disconnect Microgrid islanding

When TN and DN level disturbances clear, the system can return to its nominal regime

5

Grid-connected regime

  • Can absorb the impact of

disturbances Islanding mode regime

  • Larger disturbances may

force microgrid islanding Cascade regime

  • High severity voltage

excursions, then more DER disconnects (cascades), more load shedding

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SLIDE 6

Our approach

Most attacker-defender interactions can be modeled as

  • Supply-demand imbalance induced by attacker
  • Control (reactive and proactive) by the system operator
  • Abstraction: Bilevel (or multilevel) optimization problems
  • Flexible to allow for both continuous and discrete variables
  • Good solution approaches: Duality, KKT conditions, Benders cut, MILP
  • Provide practically useful insights to determine critical scenarios
  • Supplements simulation based approaches
  • For example, co-simulation of cyber and power simulators
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SLIDE 7

Our contributions

Bilevel problem Regime?

7

[1] Shelar D. and Amin. S - "Security assessment of electricity distribution networks under DER node compromises” [2] Shelar D., Amin. S and Hiskens I. – β€œTowards Resilience-Aware Resource Allocation and Dispatch in Electricity Distribution Networks” [3] Shelar D., Sun P., Amin. S and Zonouz S. - β€œCompromising Security of Economic Dispatch software”

Attacker model Regulation objectives Defender model Grid-Connected regime Cascade / Islanding regimes DER disruptions

  • Greedy Approach
  • IEEE TCNS 2016 [1]

DN vulnerability to simultaneous EV

  • vercharging [2]

Security of Economic Dispatch

  • KKT based reformulation
  • DSN 2017 [3]

Multiple regimes

  • Inner problem: mixed-integer vars
  • Benders decomposition
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SLIDE 8

Related Work (partial)

(T1) Interdiction and cascading failure analysis of power grids

  • R. Baldick, K. Wood, D. Bienstock: Network Interdiction, Cascades
  • A. Verma, D. Bienstock: N-k vulnerability problem
  • D. Papageorgiou, R. Alvarez, et al.: Power network defense
  • X. Wu, A. Conejo: Grid Defense Planning

(T2) Cyber-physical security of networked control systems

  • E. Bitar, K. Poolla, A Giani: Data integrity, Observability
  • H. Sandberg, K. Johansson: Secure control, networked control
  • B. Sinopoli, J. Hespanha: Secure estimation and diagnosis
  • T. Basar, C. Langbort: Network security games

8

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SLIDE 9

Network model

Power flow on tree networks - Baran and Wu model (1989):

  • 𝒣 = (π’ͺ, β„°)– tree network of nodes and edges
  • π‘žπ‘‘2,π‘Ÿπ‘‘2 - real and reactive nominal power demand at node 𝑗
  • π‘žπ‘•2, π‘Ÿπ‘•2 - real and reactive nominal power from uncontrollable generation at node 𝑗
  • π‘Š

2 - voltage magnitude at node 𝑗

  • z28 = r28 + 𝐀x28 - impedance on line (𝑗,π‘˜)
  • 𝑄

28, 𝑅28 - real and reactive power from node 𝑗 to node π‘˜

  • π‘ž2, π‘Ÿ2 - net real and reactive power consumed at node 𝑗

9

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Generalized disruption model

Attacker strategy: 𝑏 = πœ€, π‘žπ‘’A,π‘Ÿπ‘’A, Ξ”π‘Š

"

  • πœ€: attack vector, with πœ€2 = 1 if node 𝑗 is attacked and 0 otherwise
  • Satisfy βˆ‘ πœ€2

2

≀ 𝑁 (attacker’s resource budget)

  • π‘žπ‘’2

A, π‘Ÿπ‘’2 A - attacker’s active/reactive power disturbance at node 𝑗

(general model: captures various attack scenarios)

  • Ξ”π‘Š

": voltage drop at substation node

  • Due to physical disturbance or temporary fault in the TN

Attacker strategy:

  • Which nodes to compromise?
  • What set-points to choose?

10

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Defender model: Grid-connected regime

Defender response: 𝑒 = 𝛾

  • 𝛾2 ∈ 𝛾2,1 : load control parameter at node 𝑗
  • π‘žπ‘‘2 = 𝛾2 π‘žπ‘‘2, π‘Ÿπ‘‘2 = 𝛾2 π‘Ÿπ‘‘2

Defender response: How much load control should be exercised?

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π‘žπ‘‘2,π‘Ÿπ‘‘2 - nominal power demand at node 𝑗

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Defender model: Cascade regime

Defender response: 𝑒 = 𝛾, 𝑙𝑑, 𝑙𝑕

  • 𝑙𝑑2 = 0 if load is connected, 1 otherwise.
  • 𝑙𝑕2 = 0 if uncontrolled DG is connected, 1 otherwise.
  • Voltage constraints for connectivity:

𝑙𝑑2 = 0 ⟹ π‘Š

2 ∈ π‘Š '2,π‘Š '

L 2 𝑙𝑕2 = 0 ⟹ π‘Š

2 ∈ π‘Š M 2

, π‘Š

M

L

2

Defender response: Which loads and DGs to disconnect?

12

voltage bounds for load (resp. generation) connectivity

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SLIDE 13

Defender model: Islanding regime

Defender response: 𝑒 = 𝛾, 𝑙𝑑, 𝑙𝑕, π‘žπ‘ , π‘Ÿπ‘ , 𝑙𝑛

  • π‘žπ‘ , π‘Ÿπ‘  - dispatch of resources (DERs)
  • 𝑙𝑛28 = 1, if line 𝑗, π‘˜ ∈ πœ“ is open, 0 otherwise.
  • Microgrid formation affects power flows and voltages:

𝑙𝑛28 = 1 ⟹ Q 𝑄

28 = 𝑅28 = 0

π‘Š

8 = π‘ŠRST

𝑙𝑛28 = 0 ⟹ π‘žπ‘ 

8 = 0, π‘Ÿπ‘  8 = 0

Defender response: Which lines to disconnect?

13

πœ“ - set of lines which can be disconnected to form microgrids

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SLIDE 14

Power flow constraints before disruption

  • Net power consumed at a node
  • Linear Power flows (LPF)
  • Voltage drop equation

14

𝑄

28 = U 𝑄 8V V:8β†’V

+ π‘ž2 𝑅28 = U 𝑅8V

V:8β†’V

+ π‘Ÿ2 π‘Š

8 = π‘Š 2 βˆ’ (r28𝑄 28 + x28𝑅28)

π‘ž2 = π‘žπ‘‘2 βˆ’ π‘žπ‘•2 π‘Ÿ2 = π‘Ÿπ‘‘2 βˆ’ π‘Ÿπ‘•2 π‘Š

" = π‘Š " RST

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SLIDE 15

Power flow constraints after disruption

  • Net power consumed at a node
  • Linear Power flows (LPF)
  • Voltage drop equation

15

𝑄

28 = U 𝑄 8V V:8β†’V

+ π‘ž2 𝑅28 = U 𝑅8V

V:8β†’V

+ π‘Ÿ2 π‘Š

8 = π‘Š 2 βˆ’ (r28𝑄 28 + x28𝑅28)

π‘ž2 = π‘žπ‘‘2 βˆ’ π‘žπ‘•2 + πœ€2π‘žπ‘’A2

⋆

π‘Ÿ2 = π‘Ÿπ‘‘2 βˆ’ π‘Ÿπ‘•2 + πœ€2π‘Ÿπ‘’A2

⋆

π‘Š

" = π‘Š " RST βˆ’ Ξ”π‘Š "

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SLIDE 16

Power flow constraints after SO dispatch

  • Net power consumed at a node
  • Linear Power flows (LPF)
  • Voltage drop equation

16

𝑄

28 = U 𝑄 8V V:8β†’V

+ π‘ž2 𝑅28 = U 𝑅8V

V:8β†’V

+ π‘Ÿ2 π‘Š

8 = π‘Š 2 βˆ’ (r28𝑄 28 + x28𝑅28)

π‘ž2 = π‘žπ‘‘2 βˆ’ π‘žπ‘•2 + πœ€2π‘žπ‘’A2

⋆ βˆ’ π‘žπ‘  2

π‘Ÿ2 = π‘Ÿπ‘‘2 βˆ’ π‘Ÿπ‘•2 + πœ€2π‘Ÿπ‘’A2

⋆ βˆ’ π‘Ÿπ‘  2

π‘Š

" = π‘Š " #YZ βˆ’ Ξ”π‘Š "

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SLIDE 17

Losses

Cost of active power supply : Loss of voltage regulation : where 𝑒2 β‰₯ π‘Š

2 βˆ’ π‘ŠRST

Cost incurred due to load control : Loss in Grid-Connected regime :

17

𝑀^_ 𝑦 ≑ 𝑋

cd𝑄 "

𝑀ef 𝑦 ≑ 𝑋

ef U𝑒2 2∈g

, 𝑀h_ 𝑦 ≑ U 𝑋

h_,2(1 βˆ’ 𝛾2) 2∈g

𝑀id jkM2Zk 𝑦 = 𝑀^_ 𝑦 + 𝑀ef 𝑦 + 𝑀h_(𝑦)

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SLIDE 18

Attacker-Defender problem [AD] - Bilevel formulation

AD β„’ ∢= max

Aβˆˆπ’ min wβˆˆπ’  𝑀y_ z{|}T{ 𝑦 𝑏, 𝑒

  • Powerflows, DER capabilities, voltage bounds
  • Defender model (resources and capabilities)
  • Attacker model (resources and capabilities)

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System State 𝑦 = (π‘ž, π‘Ÿ, 𝑄, 𝑅, π‘Š)

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SLIDE 19

Attacker-Defender problem [AD] – Cascade regime

AD β„’ ∢= max

Aβˆˆπ’ min wβˆˆπ’  𝑀_~ z{|}T{ 𝑦 𝑏, 𝑒

  • Powerflows, DER capabilities, voltage bounds
  • Defender model (resources and capabilities)
  • Attacker model (resources and capabilities)

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Where 𝑀_~ z{|}T{ 𝑦 ≑ 𝑀y_ z{|}T{ 𝑦 + 𝑀~β€’ 𝑦

  • Cost of load shedding

𝑀~β€’ 𝑦 ≑ U 𝑋

~β€’,2 𝑙𝑑2 2∈π’ͺ

  • 𝑋

~β€’,2 : cost of unit load shedding

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SLIDE 20

Attacker-Defender problem [AD] – Islanding regime

AD β„’ ∢= max

Aβˆˆπ’ min wβˆˆπ’  𝑀€‒ z{|}T{ 𝑦 𝑏, 𝑒

  • Powerflows, DER capabilities, voltage bounds
  • Defender model (resources and capabilities)
  • Attacker model (resources and capabilities)

20

Where 𝑀€‒ z{|}T{ 𝑦 ≑ 𝑀y_ z{|}T{ 𝑦 + 𝑀€y 𝑦

  • Cost of microgrid islanding

𝑀€y 𝑦 ≑ U 𝑋

€y,28 𝑙𝑛28 (2,8)βˆˆβ€š

  • 𝑋

€y,28 : cost of a single microgrid island formation at node π‘˜

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SLIDE 21

Benders cut approach

2 5 6 7 8 12 11 9 1 3 4 10

21

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Computational results for Cascade regime

22

π‘žπ‘’A⋆ 𝑑𝑠 L

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SLIDE 23

Load shedding vs

β€žβ€¦ β€žβ€ 

23

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SLIDE 24

No response - (multi-round) cascade

Worst-case loss under no defender response An algorithm

  • Initial contingency
  • For r = 1,2,…
  • Compute new power flows
  • Determine a single loads or DG that maximally violates its voltage bounds
  • Disconnect that device accordingly

24

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SLIDE 25

Online vs Sequential vs Islanding

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Value of timely disconnections Value of timely Islanding

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SLIDE 26

Defender Response and Allocation: Diversification

  • Some DERs contribute

to 𝑀efmore than 𝑀^_, and vice versa

26

2 5 6 7 8 12 11 9 1 3 4 10

Left lateral (l) AC > V R Right lateral (r) V R > AC Attacked nodes

Special case of πœ“ = 0,1

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SLIDE 27

Defender Response and Allocation: Diversification

  • Diversification holds for

β€œheterogeneous allocation” with downstream DERs with more reactive power

27

2 5 6 7 8 12 11 9 1 3 4 10

Left lateral (l) AC > V R Right lateral (r) V R > AC Attacked nodes

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SLIDE 28
  • Post-contingency losses

are the same for uniform vs. heterogeneous resource allocations

  • Pre-contingency voltage

profile is better for heterogeneous resource allocation Heterogeneous resource allocation can support more loads than uniform one.

Defender Response and Allocation: Diversification

28

2 5 6 7 8 12 11 9 1 3 4 10

Left lateral (l) AC > V R Right lateral (r) V R > AC Attacked nodes

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SLIDE 29

Big picture: Where does it all fit?

min

jβˆˆβ„› 𝐷A‰‰Y' 𝑦Y 𝑠

+ max

Aβˆˆπ’ min wβˆˆπ’  𝑀 𝑦' 𝑠, 𝑏, 𝑒

  • Powerflows, DER capabilities, voltage bounds
  • Defender model (resources and capabilities)
  • Attacker model (resources and capabilities)

Resilience-Aware Optimal Power Flow (RAOPF)

29

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SLIDE 30

Voltage deviation model π‘Š#YZ βˆ’ π‘Š

" ' = βˆ’π‘ŠjkM 𝑄 " Y βˆ’ 𝑄 " '

Frequency deviation model 𝑔#YZ βˆ’ 𝑔' = βˆ’π‘”jkM 𝑅"

Y βˆ’ 𝑅" '

Pre-contingency resource allocation 𝑠 = (π‘žπ‘ Y, π‘Ÿπ‘ Y)

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Resiliency-Aware OPF - Trilevel formulation

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SLIDE 31

Final example: DN resiliency is indeed important

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DN 1 DN 2 DN 3 DN 4

60 MW 60 MW 60 MW 60 MW

𝐻Œ 𝐻‒

240 MW 120 MW DN 1 DN 2 DN 3 DN 4

60 MW 60 MW 60 MW 60 MW

𝐻Œ 𝐻‒

120 MW 120 MW 0 MW

𝑄

  • = 80 MW

𝑄

Ε’ = 80 MW

DN 1 DN 2 DN 3 DN 4

30 MW 30 MW 30 MW 30 MW

𝐻Œ 𝐻‒

40 MW 80 MW

  • Normal operating scenario
  • Lightning strikes - recloser opens temporarily
  • Voltage drops at the DN substations
  • Microgrid islanding reduces net load
  • Infeasible power flow in TN
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Summary

  • Resource allocation and dispatch in electricity DNs
  • under strategic cyber-physical failures
  • trilevel mixed-integer formulation
  • Multi-regime defender response
  • Application of Benders cut approach for solving bilevel MILPs
  • Structural results on worst-case attacks and tradeoffs for defender response

Future work

  • Design of decentralized defender response using message passing
  • Power restoration over multiple time periods

32

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SLIDE 33

Optimal attacker set-points

Typically,

  • Small line losses: in comparison to power flows
  • Small impedances: sufficiently small line resistances

Assume for simplicity:

  • No reverse power flows: power flows from substation

to downstream

33

What are optimal attacker set-points? Proposition: For a defender action 𝜚, and given attacker choice of πœ€, the optimal attacker disturbance is given by: π‘žπ‘’A⋆ = π‘žπ‘•2

Y, π‘Ÿπ‘’A⋆ = π‘Ÿπ‘•2 Y + 𝒕𝒉𝒋 (in case of attack on DERs)

π‘žπ‘’A⋆ = π‘žπ‘‘π‘“2

Y, π‘Ÿπ‘’A⋆ = π‘Ÿπ‘‘π‘“2 Y (in case of attack on EVs)

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SLIDE 34

Benders cut approach

Proposition (Bienstock 2009) Optimal value attack problem for a fixed attack cardinality is equivalent to a minimum cardinality attack problem for a fixed target loss value.

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Benders cut approach

Attacker Master problem

  • Initialize with no cuts

min Uπœ€2

2

  • s. t. cuts

πœ€2 ∈ {0,1}

Defender problem min

wβˆˆπ’  𝑀(𝑦)

s.t.

  • Powerflows, DER capabilities, voltage bounds
  • Defender model (resources and capabilities)

Optimal value attack problem for a fixed attack cardinality is equivalent to a minimum cardinality attack problem for a fixed target loss value. 𝑀‒AjMkβ€’ : minimum loss that the attacker wants to inflict upon the defender

35

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SLIDE 36

Benders cut

  • Let πœ€2β€’kj be fixed attacker strategy for current iteration
  • Let 𝜚ž (resp. 𝜚d) denote a defender response with fixed integer variables
  • Then the inner problem becomes a linear program (LP)

min 𝑑Ÿ𝑧 𝐷𝑧 = 𝑒 + π‘…πœ€2β€’kj 𝑑. 𝑒. 𝐡𝑧 β‰₯ 𝑐 𝑀𝑄 πœ€2β€’kj, 𝜚ž ≑

  • Let (πœ‡β‹†, 𝛽⋆) be the optimal dual variable solution to this LP

. Benders cut is given by : πœ‡β‹†ΕΈπ‘ + 𝛽⋆Ÿ 𝑒 + π‘…πœ€ β‰₯ 𝑀‒AjMkβ€’

  • This cut eliminates πœ€2β€’kj from feasible space of attacker strategies

36

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SLIDE 37

Controllable distributed generation model

pr qr qr

Reactive power Real power

Reactive power pr qr qr Real power sr

37

0 ≀ π‘žπ‘ 

2 ≀ π‘žπ‘ 2,

π‘žπ‘ 

2

  • + π‘Ÿπ‘ 

2

  • ≀ 𝑑𝑠

L2

  • π‘žπ‘ 2 - maximum active power capacity

𝑑𝑠 L2 - apparent power capability of inverter Polytopicmodel used for computational simplicity

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SLIDE 38

Uncontrolled cascade vs Sequential

38

Value of timely response

N = 37 nodes

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SLIDE 39

Microgrid island formation

2 5 6 7 8 12 11 9 1 3 4 10 2 5 6 7 8 12 11 9 1 3 4 10 2 5 6 7 8 12 11 9 1 3 4 10

  • πœ“ =

0,1 , 4,5 , 4,9

  • 3 out of 8 = 2 β€š possible configurations – 13 node network

39

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SLIDE 40

Linear power flows after dispatch

π‘ž2 = π‘žπ‘‘2 βˆ’ π‘žπ‘•2 βˆ’ π‘žπ‘ 

2 + πœ€2π‘žπ‘’A2 ⋆

π‘Ÿ2 = π‘Ÿπ‘‘2 βˆ’ π‘Ÿπ‘•2 βˆ’ π‘Ÿπ‘ 

2 + πœ€2π‘Ÿπ‘’A2 ⋆

Net power consumed at a node 𝑗 Power flow on line 𝑗 β†’ π‘˜ Voltage drop equations

40

π‘Š

" = π‘Š " Y βˆ’ Ξ”πœ‰

𝑄

28 = U 𝑄 8V V:8β†’V

+ π‘ž2 𝑅28 = U 𝑅8V

V:8β†’V

+ π‘Ÿ2 π‘Š

8 = π‘Š 2 βˆ’ (𝑠 28𝑄 28 + 𝑦28𝑅28)

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SLIDE 41

Islanding regime (cont’d)

Updated constraints

  • An (emergency) distributed generator is started at node π‘˜ in a microgrid island

π‘žπ‘ 

8 ≀ 𝑑𝑠

L8 𝑙𝑛28 π‘Ÿπ‘  ≀ 𝑑𝑠 L8 𝑙𝑛28 Where π‘žπ‘ 

8,π‘Ÿπ‘  8 is active and reactive power output; 𝑑𝑠

L8 is the apparent power capability of the emergency generator at node π‘˜

  • The net power flow into the node π‘˜ from the substation is 0, i.e.

𝑙𝑛28 = 1 ⟹ 𝑄28 = 𝑅28 = 0

  • The nodal voltage at node π‘˜ is the nominal voltage,

π‘Š

8 = Qπ‘Š 2 βˆ’ 𝑠 28𝑄 28 + 𝑦28𝑅28 ,

if 𝑙𝑛28 = 0 π‘ŠRST,

  • therwise.

41

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SLIDE 42

What’s next?

42

  • What is a good resiliency metric?
  • Allowable Ξ” π‘Š, π‘ž, π‘Ÿ without exceeding target 20% 𝑀¡¢
  • General case πœ“ > 1
  • Diversification?
  • Solution approach for RAOPF (trilevel)?
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SLIDE 43

Resiliency-aware Resource Allocation

Stage II - Adversarial node disruptions

  • a. Which nodes to compromise (πœ€)?
  • b. Set-point manipulation (π‘‘π‘žA)?

Stage I - Allocation of DERs over radial networks

  • a. Size and location
  • b. Active and reactive power setpoints (𝑦#)?

Stage III - Optimal dispatch / response (𝑦')

  • a. Maintain voltage
  • b. Exercise load control or not

Goals:

  • 1. Determine the best resource allocation
  • 2. Identify vulnerable / critical nodes
  • 3. Determine optimal dispatch post-contingency

43

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SLIDE 44

Microgrid formation (cont’d)

Updated constraints π‘ž2 = π‘žπ‘‘2 βˆ’ π‘žπ‘•2 βˆ’ π‘žπ‘ 

2 + πœ€2π‘žπ‘’A2 ⋆ βˆ’ π‘žπ‘“2

π‘Ÿ2 = π‘Ÿπ‘‘2 βˆ’ π‘Ÿπ‘•2 βˆ’ π‘Ÿπ‘ 

2 + πœ€2π‘Ÿπ‘’A2 ⋆ βˆ’ π‘Ÿπ‘“2

|𝑄28| ≀ π·π‘π‘ž28 1 βˆ’ 𝑙𝑛28 |𝑅28| ≀ π·π‘π‘ž28 1 βˆ’ 𝑙𝑛28 |πœ‰8 βˆ’ πœ‰#YZ| ≀ 1 βˆ’ 𝑙𝑛28 |πœ‰8 βˆ’ πœ‰2 βˆ’ 2 𝑠

28𝑄 28 + 𝑦28𝑅28

| ≀ 𝑙𝑛28

  • An emergency generator of microgrid is on only if it is in islanded state

π‘žπ‘“

8 ≀ 𝑑𝑓 8 𝑙𝑛28

π‘Ÿπ‘“

8 ≀ 𝑑𝑓 8 𝑙𝑛28

44