Applications of Bilevel Mixed-Integer Programming to Power Systems - PowerPoint PPT Presentation

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 • 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

Cyberphysical disruptions Metcalf Substation (April 2013) Ukraine attack (Dec 2015- Hurricane Maria • Sniper attack on 17 2016) (September 2017) transformers • First ever blackouts • Customers facing • Telecommunication cables cut caused by hackers blackouts for • 15 million $ worth of damage • Controllers damaged for months • 100 mn $ for security upgrades months 3

Attack scenarios => supply-demand imbalance (sudden / prolonged) 8

Three regimes of SO operation DER disconnect -- cascade load disconnect TN level Grid-connected regime Distribution Transmission disturbance Can absorb the impact of • Attack-induced substation network DN level disturbances supply-demand # ' −Δ𝑊 𝑊 𝑊 " " imbalance Islanding mode regime SO response Larger disturbances may • # , 𝑅 " # ' , 𝑅 " ' 𝑄 𝑄 " " force microgrid islanding Microgrid islanding Cascade regime High severity voltage • excursions, then more DER disconnects (cascades), more load shedding When TN and DN level disturbances clear, the system can return to its nominal regime 5

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 •

Our contributions Defender model Attacker model Regulation objectives Bilevel problem Regime? Cascade / Islanding regimes Grid-Connected regime Multiple regimes DN vulnerability to Security of Economic Dispatch DER disruptions Inner problem: mixed-integer vars • simultaneous EV KKT based reformulation Greedy Approach • • Benders decomposition • overcharging [2] DSN 2017 [3] IEEE TCNS 2016 [1] • • [1] Shelar D. and Amin. S - "Security assessment of electricity distribution networks under DER node compromises” [2] Shelar D., Amin. S and Hiskens I. – “Towards Resilience-Aware Resource Allocation and Dispatch in Electricity Distribution Networks” [3] Shelar D., Sun P., Amin. S and Zonouz S. - “Compromising Security of Economic Dispatch software” 7

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

Network model Power flow on tree networks - Baran and Wu model (1989): • 𝒣 = (𝒪, ℰ) – tree network of nodes and edges • 𝑞𝑑 2 ,𝑟𝑑 2 - real and reactive nominal power demand at node 𝑗 • 𝑞𝑕 2 , 𝑟𝑕 2 - real and reactive nominal power from uncontrollable generation at node 𝑗 • 𝑊 2 - voltage magnitude at node 𝑗 • z 28 = r 28 + 𝐤x 28 - impedance on line ( 𝑗,𝑘 ) • 𝑄 28 , 𝑅 28 - real and reactive power from node 𝑗 to node 𝑘 • 𝑞 2 , 𝑟 2 - net real and reactive power consumed at node 𝑗 9

Generalized disruption model Attacker strategy: 𝑏 = 𝜀, 𝑞𝑒 A ,𝑟𝑒 A , Δ𝑊 " • 𝜀: attack vector, with 𝜀 2 = 1 if node 𝑗 is attacked and 0 otherwise • Satisfy ∑ 𝜀 2 ≤ 𝑁 (attacker’s resource budget) 2 A - attacker’s active/reactive power disturbance at node 𝑗 A , 𝑟𝑒 2 • 𝑞𝑒 2 (general model: captures various attack scenarios) • Δ𝑊 " : 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

Defender model: Grid-connected regime Defender response: 𝑒 = 𝛾 • 𝛾 2 ∈ 𝛾 2 ,1 : load control parameter at node 𝑗 𝑞𝑑 2 ,𝑟𝑑 2 - nominal power • 𝑞𝑑 2 = 𝛾 2 𝑞𝑑 2 , 𝑟𝑑 2 = 𝛾 2 𝑟𝑑 2 demand at node 𝑗 Defender response: How much load control should be exercised? 11

Defender model: Cascade regime Defender response: 𝑒 = 𝛾, 𝑙𝑑, 𝑙𝑕 • 𝑙𝑑 2 = 0 if load is connected, 1 otherwise. • 𝑙𝑕 2 = 0 if uncontrolled DG is connected, 1 otherwise. • Voltage constraints for connectivity: L 2 voltage bounds for load (resp. 𝑙𝑑 2 = 0 ⟹ 𝑊 2 ∈ 𝑊 '2 ,𝑊 ' generation) connectivity L 𝑙𝑕 2 = 0 ⟹ 𝑊 2 ∈ 𝑊 , 𝑊 2 M M 2 Defender response: Which loads and DGs to disconnect? 12

Defender model: Islanding regime Defender response: 𝑒 = 𝛾, 𝑙𝑑, 𝑙𝑕, 𝑞𝑠, 𝑟𝑠, 𝑙𝑛 • 𝑞𝑠, 𝑟𝑠 - dispatch of resources (DERs) 𝜓 - set of lines which can • 𝑙𝑛 28 = 1, if line 𝑗, 𝑘 ∈ 𝜓 is open, 0 otherwise. be disconnected to form microgrids • Microgrid formation affects power flows and voltages: 𝑄 28 = 𝑅 28 = 0 𝑙𝑛 28 = 1 ⟹ Q 8 = 𝑊 RST 𝑊 𝑙𝑛 28 = 0 ⟹ 𝑞𝑠 8 = 0, 𝑟𝑠 8 = 0 Defender response: 13 Which lines to disconnect?

Power flow constraints before disruption 𝑞 2 = 𝑞𝑑 2 − 𝑞𝑕 2 • Net power consumed at a node 𝑟 2 = 𝑟𝑑 2 − 𝑟𝑕 2 • Linear Power flows (LPF) 𝑄 28 = U 𝑄 + 𝑞 2 8V V:8→V 𝑅 28 = U 𝑅 8V + 𝑟 2 V:8→V 𝑊 8 = 𝑊 2 − (r 28 𝑄 28 + x 28 𝑅 28 ) • Voltage drop equation RST 𝑊 " = 𝑊 " 14

Power flow constraints after disruption ⋆ 𝑞 2 = 𝑞𝑑 2 − 𝑞𝑕 2 + 𝜀 2 𝑞𝑒 A2 • Net power consumed at a node ⋆ 𝑟 2 = 𝑟𝑑 2 − 𝑟𝑕 2 + 𝜀 2 𝑟𝑒 A2 • Linear Power flows (LPF) 𝑄 28 = U 𝑄 + 𝑞 2 8V V:8→V 𝑅 28 = U 𝑅 8V + 𝑟 2 V:8→V 𝑊 8 = 𝑊 2 − (r 28 𝑄 28 + x 28 𝑅 28 ) • Voltage drop equation RST − Δ𝑊 𝑊 " = 𝑊 " " 15

Power flow constraints after SO dispatch ⋆ − 𝑞𝑠 𝑞 2 = 𝑞𝑑 2 − 𝑞𝑕 2 + 𝜀 2 𝑞𝑒 A2 • Net power consumed at a node 2 ⋆ − 𝑟𝑠 𝑟 2 = 𝑟𝑑 2 − 𝑟𝑕 2 + 𝜀 2 𝑟𝑒 A2 2 • Linear Power flows (LPF) 𝑄 28 = U 𝑄 + 𝑞 2 8V V:8→V 𝑅 28 = U 𝑅 8V + 𝑟 2 V:8→V 𝑊 8 = 𝑊 2 − (r 28 𝑄 28 + x 28 𝑅 28 ) • Voltage drop equation #YZ − Δ𝑊 𝑊 " = 𝑊 " " 16

Losses 𝑀 ^_ 𝑦 ≡ 𝑋 cd 𝑄 Cost of active power supply : " Loss of voltage regulation : 𝑀 ef 𝑦 ≡ 𝑋 ef U𝑢 2 , 2 − 𝑊 RST where 𝑢 2 ≥ 𝑊 2∈g Cost incurred due to load control : 𝑀 h_ 𝑦 ≡ U 𝑋 h_,2 (1 − 𝛾 2 ) 2∈g 𝑀 id jkM2Zk 𝑦 = 𝑀 ^_ 𝑦 + 𝑀 ef 𝑦 + 𝑀 h_ (𝑦) Loss in Grid-Connected regime : 17

Attacker-Defender problem [AD] - Bilevel formulation w∈𝒠 𝑀 y_ z{|}T{ 𝑦 𝑏, 𝑒 AD ℒ ∶= max A∈𝒝 min • Powerflows, DER capabilities, voltage bounds • Defender model (resources and capabilities) • Attacker model (resources and capabilities) System State 𝑦 = (𝑞, 𝑟, 𝑄, 𝑅, 𝑊) 18

Attacker-Defender problem [AD] – Cascade regime w∈𝒠 𝑀 _~ z{|}T{ 𝑦 𝑏, 𝑒 AD ℒ ∶= max A∈𝒝 min • Powerflows, DER capabilities, voltage bounds • Defender model (resources and capabilities) • Attacker model (resources and capabilities) Where 𝑀 _~ z{|}T{ 𝑦 ≡ 𝑀 y_ z{|}T{ 𝑦 + 𝑀 ~• 𝑦 • Cost of load shedding 𝑀 ~• 𝑦 ≡ U 𝑋 ~•,2 𝑙𝑑 2 2∈𝒪 ~•,2 : cost of unit load shedding • 𝑋 19

Attacker-Defender problem [AD] – Islanding regime w∈𝒠 𝑀 €• z{|}T{ 𝑦 𝑏, 𝑒 AD ℒ ∶= max A∈𝒝 min • Powerflows, DER capabilities, voltage bounds • Defender model (resources and capabilities) • Attacker model (resources and capabilities) Where 𝑀 €• z{|}T{ 𝑦 ≡ 𝑀 y_ z{|}T{ 𝑦 + 𝑀 €y 𝑦 • Cost of microgrid islanding 𝑀 €y 𝑦 ≡ U 𝑋 €y,28 𝑙𝑛 28 (2,8)∈‚ €y,28 : cost of a single microgrid island formation at node 𝑘 • 𝑋 20

Benders cut approach 0 1 2 3 4 5 9 6 10 7 11 8 12 21

Computational results for Cascade regime 𝑞𝑒 A ⋆ 𝑡𝑠 L 22

„… Load shedding vs „† 23

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