Mode Estimation using Synchrophasors Arezoo Rajabi and Rakesh B. - - PowerPoint PPT Presentation

A Resilient Algorithm for Power System Mode Estimation using Synchrophasors

Arezoo Rajabi and Rakesh B. Bobba

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Outline

• Introduction
• Background and Problem
• Prony Algorithm
• Standard ADMM
• False Data Injection
• Related Work
• Our Proposed Method
• Evaluation
• Analytical Intuition
• Conclusion

1 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Power System Large synchronous distributed system of interconnected electrical components used for generation, transmission and distribution of electric power

• Generators
• Transmission (and distribution) lines
• Transformers
• Substations

2

* Image Source: http://www2.econ.iastate.edu

Basic structure of Power System* 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Introduction

Stability In Power Systems

• The ability of operating an AC power network with:
• All generators in synchronism and
• Retaining synchronism even after a large disturbance
• Faults can lead to instability in power systems
• Instability problems in power systems can lead to brownouts or

in extreme cases blackouts

3 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Introduction

Northeast Blackout – August 2003

• Impacted 50 million people
• Estimated loss: $4-$10 billion
• At least 2 deaths in New York

city attributed to the blackout

4 Northeast Blackout Map*

*Image Source: http://naturalhistory.si.edu/exhibits

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Introduction

• In the presence of a fault, two or more coherent groups of

generators may start swinging against each other leading to frequency oscillations

• It is important to detect unstable oscillations and take

corrective action

5

Stable Power Oscillations Unstable Power Oscillations

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Inter-Area Oscillation Modes

Introduction

Oscillation Mode Detection Approaches

Model-Based Methods Measurements- Methods Time Efficiency

×

Scalability

×

On-line

×

Accuracy

×

Topology Independency

×

6 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Introduction

Prony Algorithm [Hauer 1990]

• Prony algorithm is a popular measurement-based method
• Consider a power system with 𝑛 synchronous generators
• Assume that each synchronous generator is modeled by a

second-order swing equation

• [𝑧𝑗 𝑢0 , … , 𝑧𝑗(𝑢𝑜)] is a set of measurements provided by 𝑗𝑢ℎ

Phasor Measurement Units at time 𝑢 𝑧𝑗 𝑢 = ෍

𝑙=1 2𝑛

𝑠

𝑗,𝑙𝑓𝜏𝑙+𝑘Ω𝑙 + 𝑠′𝑗,𝑙𝑓𝜏𝑙−𝑘Ω𝑙

7 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Background and Problem

Prony Algorithm

• Goal: To estimate damping factors(𝜏𝑙) and , frequencies (Ω𝑙)
• f oscillation modes
• Finds coefficient vector Ԧ

𝑏 :

• Obtains the roots 𝑎1, … , 𝑎𝑜 of discrete-time characteristic

polynomial equation

𝑎𝑜 + 𝑏𝑜𝑎𝑜−1 + 𝑏𝑜−1𝑎𝑜−2 + ⋯ + 𝑏1 = 0 𝜏𝑗 ± Ω𝑗 = log 𝑎𝑗 𝑈

8

𝑧𝑗(𝑢0 + 𝑜𝑈) 𝑧𝑗(𝑢0 + (𝑜 + 1)𝑈) ⋮ 𝑧𝑗(𝑢0 + (𝑜 + 𝑚)𝑈)

Ԧ 𝐷

= 𝑧𝑗 𝑢0 + 𝑜 − 1 𝑈 𝑧𝑗(𝑢0 + 𝑜𝑈) ⋮ 𝑧𝑗(𝑢0 + (𝑜 + 𝑚 − 1)𝑈) 𝑧𝑗(𝑢0 + (𝑜 − 1)𝑈) 𝑧𝑗(𝑢0 + (𝑜 − 2)𝑈) ⋮ 𝑧𝑗(𝑢0 + (𝑜 + 𝑚 − 2)𝑈) ⋯ ⋯ ⋮ ⋯ 𝑧𝑗(𝑢0) 𝑧𝑗(𝑢0 + 𝑈) ⋮ 𝑧𝑗(𝑢0 + 𝑚𝑈)

𝐼

ถ 𝑏1 𝑏2 ⋮ 𝑏𝑜

𝑏

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Background and Problem

Power Grid: A Large Distributed Network

• Power systems are usually divided into multiple areas of

control

9

North American Interconnections* *Image source: [Andersson (2005)]

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Background and Problem

Power Grid: A Large Distributed Network

• Power systems are usually divided into multiple areas of

control

• Using Alternating Direction Method of Multipliers (ADMM) to

implement Prony Algorithm in a distributed fashion [Wei 2013]:

• Local objective function of 𝑗𝑢ℎ area: (𝑔

𝑗 𝑏 =

𝐼𝑗𝑏 − 𝐷𝑗 )

• Goal: to find a solution for:

min

𝑏

෍

𝑗=1 𝑂

𝐼𝑗𝑏𝑗 − 𝐷𝑗 𝑡. 𝑢 𝑏𝑗 − 𝑨 = 0

10 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Background and Problem

Local Phasor Data Concentrator (PDC):

• Gathers measurements to create Henkel

matrix 𝐼𝑗 and vector 𝐷𝑗

• Updates the local optimal estimate value

(𝑏𝑗

(𝑙+1))

• Shares its local optimal estimate value with

central PDC and obtains the global optimal estimate value (𝑨𝑙+1) from Central PDC

Central PDC:

• Gathers local optimal estimates from local

PDCs

• Computes the global optimal estimate vale

(𝑨𝑙+1) and shares it with local PDCs

Standard ADMM (S-ADMM)[Nabavi 2015]

11

PDC 1 y11(t) ... y1n1(t)

PMUs

PDC 5

y51(t) .. y5n5(t) PMUs

PDC 4

y41(t) ... y4n4(t) PMUs

PDC 3

y31(t) ... y3n3(t) PMUs

PDC 2

y21(t) ... y2n2(t) PMUs

central PDC

z a1

z a

2

z a3 z a4 z a

5

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Background and Problem

Disadvantage: S-ADMM is not robust against false data injection Compromised areas can send corrupted data to mislead other areas or disrupt convergence

Impact of False Data Injection on Convergence

12

Without Attack With Attack

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Background and Problem

• Disrupting the mode estimation by preventing convergence :
• Random Value Attack
• Driving the estimate away from the real modes (potentially to

desired modes)

• Desired Value Attack
• Remaining Undetected
• Periodic Attack

Potential Adversary Goals

13 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Background and Problem

Related Work

• Round-Robin ADMM[Liao 2016]
• Central PDC updates the global optimal estimate value by using a local
• ptimal estimate value from only one area in each iteration (𝑨𝑙+1 =

𝑏𝑗

𝑙+1)

• Central PDC removes the local optimal estimate which causes the

most change in global optimal

• D-ADMM[Nabavi 2015]
• Fully distributed version of S-ADMM
• Areas send their local optima estimate values to each other
• Each area uses its objective function to detect compromised area
• CON:
• They need two runs: one for compromised area detection and one for

mode estimation

• Not robust against periodic attack

14 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Related Work

Our Contributions

• Unlike previous methods that localize the false data, our

approach aims to tolerate the false data

• Our approach needs only one run to estimate oscillation

modes

• We considered different attack scenarios to evaluate our

methods

15 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Our Proposed Method

• Central PDC will identify outlier and remove it

from 𝑨 𝑙+1 calculation

• Direction of

𝑤𝑗

(𝑙+1) = 𝑏𝑗 (𝑙+1) − 𝑨𝑙 points to the

location of optimal value from view of area i

• Dissimilarity matrix (𝑁𝑒𝑗𝑡(𝑗, 𝑘)) keeps the angle

between 𝑤𝑗

𝑙+1and 𝑤𝑘 𝑙+1

𝑁𝑒𝑗𝑡 = 𝜄5 𝜄1 𝜄1 + 𝜄2 + 𝜄3 𝜄1 + 𝜄2 𝜄5 𝜄4 + 𝜄2 + 𝜄3 𝜄4 𝜄4 + 𝜄3 𝜄1 𝜄4 + 𝜄2 + 𝜄3 𝜄2 + 𝜄3 𝜄2 𝜄1 + 𝜄2 + 𝜄3 𝜄4 𝜄2 + 𝜄3 𝜄3 𝜄1 + 𝜄2 𝜄4 + 𝜄3 𝜄2 𝜄3

Fault Tolerance Approach

16

θ4 θ3 θ2 θ1 θ5 v2

(k+1)

v1

(k+1)

v4

(k+1)

v5(k+1) v3

(k+1)

𝑨𝑙 𝑏𝑗

𝑙+1

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

• To resist against periodic attacks, central PDC has

a local memory with size W to track attacker.

Our Proposed Method

Fault Tolerance Approach’s Impact

• n Convergence

17

With Attack and With Using Attack Tolerance Approach No Attack and With Using Attack Tolerance Approach Without Attack and No Tolerance Approach With Attack and Without Tolerance Approach

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Our Proposed Method

Evaluation

• IEEE 68-bus power system divided

into 5 areas

• Generated measurements using

Power System Toolbox (PST)

• Generators in this model

are 6𝑢ℎ order

• Many of modes have small residues
• Inter-area oscillation modes have small

frequency

• Therefore, we consider about 40 modes

18 IEEE 68-bus*

*Image Source: [Nabavi 2015]

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Evaluation

Evaluation (Cont.)

19

Periodic Desired Value Attack Different Attack Scenarios

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Evaluation

Evaluation (Cont.)

20

Periodic Random Value Attack Different Attack Scenarios

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Evaluation

Evaluation (Cont.)

21

Window Size = 5

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Evaluation

Evaluation (Cont.)

22

Window Size=10

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Evaluation

Evaluation (Cont.)

23

Window Size=15

Evaluation

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Analytical Intuition Theorem 1. Let f1(x),···, fn(x) be convex functions with the same optimal value (x∗ , fi′ (x∗) = 0), and whose derivative exists at every point, then

f1

′ x

f2

′ x

= ⋯ =

fn

′ x

fn

′ x 24

Theorem 2. Let f1(x)andf2(x)be convex functions whose derivative exists in every point, and ∥ x1

∗ − x2 ∗ ∥< ε ≈ 0,

∥ x − x1

∗ ∥≫ ε, ∥ x − x2 ∗ ∥≫ ε then cosθ ≈ 1 where

xi

∗ is the optimal value at which fi(x) has its minimum

value and θ is the angle between f1

′ x and f2′(x)

Theorem 3. Let for all objective functions ∀x ∈ Si = {x| ∥ xi

∗ − x ∥< ε → 0}, fi′ (x∗) ≈ 0 and 1 ≤ i, j ≤ n,∥ xi ∗ − xj ∗ ∥

<

ε

• 2. Then, false data by an intelligent attacker will be

identified and dis-carded unless its fbad

′

(x) points to ∪ Si

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Evaluation

Conclusions

• We proposed a promising byzantine fault tolerant mode

estimation method based on S-ADMM

• Our proposed method does not localize the attacker but can

tolerate byzantine attackers

• Our proposed method works well under different attack

scenarios

25 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Conclusion

Future Directions We plan to:

• Evaluate this approach further both empirically and

analytically

• Provide a formal analysis of our approach and characterize its

limitations

• Apply machine learning algorithms to partition areas into

non-faulty and faulty areas

26 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Conclusion

References

[Hauer 1990] Hauer, J. F., Demeure, C. J., & Scharf, L. L. (1990). Initial results in Prony analysis of power system response signals. IEEE Transactions on power, 5(1), 80-89. [Andersson 2005] Andersson , G., Donalek, et.al.. (2005). Causes of the 2003 major grid blackouts in North America and Europe, and recommended means to improve system dynamic

• performance. IEEE transactions on Power Systems, 20(4), 1922-1928

[Wei 2013] Wei, E., & Ozdaglar, A. (2013, December). On the o (1= k) convergence of asynchronous distributed alternating direction method of multipliers. In Global Conference on Signal and Information Processing (GlobalSIP), 2013.IEEE (pp. 551- 554). IEEE. [Nabavi 2015] Nabavi, S., & Chakrabortty, A. (2015, December). An intrusion-resilient distributed

• ptimization algorithm for modal estimation in power systems. In 2015 54th IEEE

Conference on Decision and Control (CDC) (pp. 39-44). IEEE. [Liao 2016] Liao, M., & Chakrabortty, A.(2016). A round-robin ADMM algorithm for identifying data-manipulators in power system estimation. In Proc. Amer. Control Conf.

27 2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

28

rajabia@oregonstate.edu rakesh.bobba@oregonstate.edu

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016

Iteration k:

1. Local PDCs updating local optima 2. Central PDC compute the global optima: 𝑨(𝑙+1) = ෍

𝑗=1 𝑂

𝑏𝑗

(𝑙+1)

3. Local PDC update dual parameter 𝑥𝑗

(𝑙+1) = 𝑥𝑗 (𝑙) + 𝜍(𝑏𝑗 𝑙 − 𝑨 𝑙+1 )

S-ADMM (Cont.)

29

𝑏𝑗

(𝑙+1) = 𝐼𝑗 ′𝐼𝑗 + 𝜍𝐽 −1

𝐼𝑗

′𝐷𝑗 − 𝑥𝑗 𝑙 + 𝜍𝑨 𝑙

PDC 1 y11(t) ... y1n1(t)

PMUs

PDC 5

y51(t) .. y5n5(t) PMUs

PDC 4

y41(t) ... y4n4(t) PMUs

PDC 3

y31(t) ... y3n3(t) PMUs

PDC 2

y21(t) ... y2n2(t) PMUs

central PDC

z a1

z a2 z a3 z a4 z a

5

2nd Industrial Control System Security (ICSS) Workshop, December 6th 2016