Seminar Course 392N ● Spring2012

Lecture 2 Intelligent Energy Systems: Monitoring Basics Dimitry Gorinevsky

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Traditional Grid • Worlds Largest Machine! – 3300 utilities – 15,000 generators, 14,000 TX substations – 211,000 mi of HV lines (>230kV)

• A variety of interacting information decision and control systems

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Smart Energy Grid Intelligent Energy Network Source IPS

energy subnet

Load IPS Intelligent Power Switch

Generation Transmission Distribution Load Conventional Electric Grid Conventional Internet ee392N - Spring 2012 Stanford University

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Outline 1. 2. 3. 4.

Monitoring Applications Statistical Process Control - SPC Multivariate SPC – MSPC Principal Component Analysis - PCA

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Internet Applications Tablet

Computer

Smart phone

Internet

Presentation Layer Backend

Business Logic

CRM and ad analytics Portfolio optimization Decision support Fraud detection

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Intelligent Energy Applications Tablet

Computer

Smart phone

Communications Internet Energy Application

Presentation Layer Application Logic Business Logic (Intelligent Functions) Database ee392N - Spring 2012 Stanford University

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Control Functions • Control function in a systems perspective – Closed loop

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Monitoring & Decision Support • Monitoring functions are open-loop - Data presentation to a user

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Power Generation Time Scales • Power generation and distribution • Energy supply side

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Anomalies & Sustainment Power Supply Scheduling

1000 Time (s)

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Power Demand Time Scales • Power consumption – DR, Homes, Buildings, Plants

• Demand side

Anomalies & Sustainment

Enterprise Demand Scheduling Building HVAC Home Thermostat Demand Response

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Time (s) 10

Monitoring Goals • Situational awareness – Anomaly detection – State estimation

• System health management – Fault isolation – Condition based maintenances

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Condition Based Maintenance • DOD CBM+ Initiative

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Outline 1. 2. 3. 4.

Monitoring Applications Statistical Process Control - SPC Multivariate SPC – MSPC Principal Component Analysis - PCA

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Anomaly Detection - SPC • SPC - Statistical Process Control – Introduced for monitoring of manufacturing processes – Warning for off-target quality

• SPC vs. EPC – Engineering Process Control = feedback control

• Main SPC method – Shewhart Chart (Control Chart)

• Other SPC methods – EWMA, CuSum, Western Electric Rules ee392N - Spring 2012 Stanford University

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Exceedance Monitoring • Currently used in most monitoring systems • Example: grid frequency deviation from 60Hz – Empirical exceedance threshold

Plant/System Data

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Monitoring Function: Exceedance

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Detected Anomalies

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SPC: Shewhart Control Chart W.Shewhart, Bell Labs, 1924 Statistical Process Control (SPC) UCL = µ + 3·σ LCL = µ - 3·σ Upper Control Limit

quality variable

• • • •

mean µ Lower Control Limit 3

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sample

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Walter Shewhart (1891-1967)

Exceedance / Out of control

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Shewhart Chart, cont’d • Quality variable assumed randomly changing around a steady state • Detection: y(t) > UCL = µ + 3·σ • For a normal distribution, false alarm probability is 0.27% LCL

z (t ) =

y (t ) − µ

σ

UCL

P(z > 3) = 1-Φ(3) = 0.1350·10-2 P(z < 3) = Φ(-3) = 0.1350·10-2 ee392N - Spring 2012 Stanford University

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SPC: Use Examples • SPC in manufacturing • Fault monitoring for PHM/CBM • Sensor integrity monitoring – Fault tolerance and redundancy management

Sensor Reference ee392N - Spring 2012 Stanford University

+ -

|v| < 3σ

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no

Fault

yes

Normal

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Outline 1. 2. 3. 4.

Monitoring Applications Statistical Process Control - SPC Multivariate SPC – MSPC Principal Component Analysis – PCA

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Multivariate SPC • Univariate process: y(t)  z =  2

y − µ0  2 ~ χ  σ  2

Chi-squared CDF

P (z 2 > c 2 ) = 1 − F ( c 2 ,1) = Φ ( −c) + 1 − Φ (c)

• Two univariate processes

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MSPC Explanation • MSPC=Multivariate Statistical Process Control • Scatter plot for correlated channels Time series data

Keep the data values, ignore the time stamp

y1(t) y2 y2(t) y1 ee392N - Spring 2012 Stanford University

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Multivariate SPC • Two correlated univariate processes y1(t), y2(t)  y1   µ1  y=  µ=   y2   µ2  cov(y) = P multivariate outlier: out of control

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Whitened Variables • Uncorrelated linear combinations

z = ( y − µ ) P −1 ( y − µ ) ~ χ 22

z(t) = L·[y(t)-µ] LTL= P-1

2

T

 cov(z) = I

• Declare fault (anomaly) if

( y − µ )T P −1 ( y − µ ) > c 2 P (z 2 > c 2 ) = 1 − F ( c 2 ;2) CDF for Chi-squared with 2 DOF ee392N - Spring 2012 Stanford University

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Multivariate Monitoring Plant / System y (t ) Data

Monitoring data processing: x = y − µˆ T 2 = x T Pˆ −1 x

Historical Data Set

Models: Performance, Noise µˆ , Pˆ

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Advisory Info: • Anomaly T 2 > c2

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Hotelling's T2 • Empirical parameter estimates 1 µˆ = N 1 ˆ P= N

N

∑ y (t ) ≈ E ( y ) t =1 N

T ˆ ˆ ( ( ) )( ( ) y t − µ y t − µ ≈ cov ( y − µ ) ) ∑ t =1

• Hotelling's T2 =

•

T2

N N +1

T2

two-sample statistics is

Harold Hotelling (1895-1973)

T ⋅ ( y ( N + 1) − µˆ ) Pˆ −1 ( y ( N + 1) − µˆ )

2 χ distribution differs from since Pˆ , µˆ are

considered as random variables, y(t) ~ N(µ,P)

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Multivariate SPC with

2 T

• The anomaly detection decision is T 2 > c2

• Threshold c is defined by the false positive/false negative tradeoff based on the distribution ( N − 1) p Fp , N − p T ~ ( N − p) 2

where F is the Fisher-Snedecor’s F-distribution p is the dimension of the data vector y N is the size of the training data set ee392N - Spring 2012 Stanford University

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Outline 1. 2. 3. 4.

Monitoring Applications Statistical Process Control - SPC Multivariate SPC – MSPC Principal Component Analysis – PCA

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PCA • PCA = Principal Component Analysis X = [ y (1) − µˆ

y ( 2) − µˆ  y ( N ) − µˆ ]

• What if empirical covariance P=XXT/N is not invertible? Cannot compute xTP-1x – This happens in most real cases

• SVD of the data and covariance matrix X = U⋅ S⋅ VT = ∑k uk skvkT XXT= U⋅ S2⋅ UT = ∑k uk sk2ukT VTV = I ee392N - Spring 2012 Stanford University

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Scores Loadings 29

PCA Structure • Singular vectors (principal components) U = [UR U0] UR - Range Space; nonzero singular values, sk > 0 U0 - Null Space; zero singular values, sk=0 • Singular values

nonzero

sk ‘zero’

k >>[U,S,W]=svd(X*X’) % 2ms for 100x100 ee392N - Spring 2012 Stanford University

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PCA,

2 T,

and Q statistics

• T2 statistics is used in Range Space of P xR = U RTU R x is Range Space projection of x • Range Space: covariance is invertible

T 2 = xRT PR−1 xR

• Must also monitor Null Space projection • Q statistics: Q = xTU0U0 x – a.k.a. SPE (squared prediction error)

Q = x − xR ee392N - Spring 2012 Stanford University

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PCA, T2, and Q Summary Q

T2

principal component #1

principal component #2

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PCA Prediction Model • Null space defines linear dependency between monitored variables U0x = v ≈ 0 m linear equations • Can be interpreted as a dependence between two subsets of variables  y m

x=  z k

y = bz + v

• SPE yields model prediction error:

y − bz ee392N - Spring 2012 Stanford University

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End of Lecture 2

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