Monitoring and Optimization for Smarter Power Grids Georgios B. - - PowerPoint PPT Presentation

Acknowledgement: Inst. on Renewable Energy and Environment RL-10-13

Monitoring and Optimization for Smarter Power Grids

• G. B. Giannakis, V. Kekatos, N. Gatsis, S.-J. Kim, H. Zhu, and B. Wollenberg, Monitoring and

Optimization for Power Grids: A Signal Processing Perspective,” IEEE SP Magazine, Sept. 2013.

Georgios B. Giannakis*, Vassilis Kekatos*, Nikolaos Gatsis§

1

*Digital Technology Center and Dept. of ECE, University of Minnesota

§Dept. of ECE, The University of Texas at San Antonio

Aging infrastructure

! Power outages cost: inadequate system understanding ! 2009 US Recovery Act includes $43 billion for energy ! 2008 PA Act 129 for energy efficiency ~ $1 billion in 4 years

Bell: telephone 1876 Edison: light bulbs Tesla: AC system 1888 Natural monopolies Single commodity Grew rapidly in the 2 world wars 1980-90s 1980-90s Deregulation started 1969: DARPAnet Deregulation started 2000s Enron, blackouts

2

The grid then, now, and ahead

“Most significant engineering achievement of 20th century” [NAE Report’10]

Picture source: FERC. “Final Report on the Aug. 14, 2003 Blackout in the US and Canada,” Apr. 2004.

• ! Several challenges ahead

" 99.97% reliable, but power outages still cost $150 billion/year " Customer engagement and environmental concerns 3

SMART GRID: Advanced infrastructure and information technologies

to enhance the current electrical power network

• Dept. of Energy, The smart grid: an introduction

controllable resilient efficient participation self-healing sustainable

4

situational awareness green

• ptimization, learning,

and signal processing toolbox

Enabling technology advances

sensing/metering distributed generation micro-grids

electric vehicles

renewables demand response communication networks power electronics

5

Outline

! Modeling ! Grid monitoring

" Power system state estimation " Observability and cyber attacks " Phasor measurement units (PMUs) " Learning and inference

! Optimal grid operation

" Economic operation " Demand response " Electric vehicles " Renewables

! Open issues

6

Power grid modeling

7

• A. R. Bergen and V. Vittal, Power System Analysis, Prentice Hall, 2000.

Preliminaries

! Power grids as electric circuits ! AC voltages, currents, and powers are sinusoids (in steady state) ! Phasor representation (at nominal frequency)

" Polar and rectangular coordinates:

V = V ejθ = Vr + jVi

8

IEEE 14-bus benchmark

! Buses (generators, loads) ! Tx lines and transformers ! From scalar to multivariate Ohm’s law ! Single-phase equivalent

V = ZI → v = Zi

9

Transmission lines

! Line series impedance ! Line series admittance ! Total charging susceptance bc,mn > 0

jbc,mn 2

ymn Vm Vn

equivalent model

zmn = rmn + jxmn (xmn > 0) ymn = 1 zmn = gmn + jbmn

gmn = rmn r2

mn + x2 mn

, bmn = − xmn r2

mn + x2 mn

< 0

π

10

Line currents

Imn = (jbc,mn/2 + ymn)Vm − ymnVn ymn Vn Vm jbc,mn 2 Imn Imn = ymn(Vm − Vn) + j bc,mn 2 (Vm − 0)

11

Bus injection currents

! Kirchoff’s current law (KCL)

Vm

Im = X

n∈Nm

Imn Im = @ X

n6=m

ymn + j X

n6=m

bc,mn/2 1 A Vm − X

n6=m

ymnVn

12

Multivariate Ohm’s law

" Symmetric and non-Hermitian

! Concatenating all injection currents ! Bus admittance matrix ! Bus impedance matrix

" Nonsparse " Not the matrix of line impedances

! Bottomline: Currents linearly expressed in terms of nodal voltages

i = Yv Z := Y−1

v = Zi

" Sparse efficient computations and storage " Fundamental for monitoring and optimization

I0

ms

13

[Y]mn = ⇢ P

k6=m ymk + j bc,mk 2

, m = n ymn , m 6= n

Complex power

! Power injection to bus m

Vm

Im

! (Re)active power generated or consumed at a bus ! Multivariate power model ! Power flow over line (m,n) ! Y in rectangular coordinates ! Polar versus rectangular coordinates for v

Y = G + jB s = diag(v)i∗ = diag(v)Y∗v∗ Smn = VmI∗

mn

Sm = VmI∗

m = Pm + jQm

14

Power flow equations

! Case 1: Polar coordinates ! Power depends only on phase differences ! Reference (slack or swing) bus θ1 = 0 ! Case 2: Rectangular coordinates ! Quadratic equations

Pm = P

n VmVn (Gmn cos(θm−θn) + Bmn sin(θm−θn))

Qm = P

n VmVn (Gmn sin(θm−θn) − Bmn cos(θm−θn))

Pm = Vr,m P

n (Vr,nGmn−Vi,nBmn) +Vi,m

P

n (Vi,nGmn+Vr,nBmn)

Qm = Vi,m P

n (Vr,nGmn−Vi,nBmn) +Vr,m

P

n (Vi,nGmn+Vr,nBmn)

15

Power flow problem

! No. of variables: ! No. of equations:

Goal: Given values of variables, find the rest unknown variables by solving the nonlinear power flow equations

2Nb 2Nb

! Typically, given values come from

" Generators (PV buses) " Loads (PQ buses) " Reference bus

(Pm, Vm) (Pm, Qm) (V1, θ1 = 0) {(Pm, Qm, Vm, θm)}Nb

m=1

! No. of buses:

Nb 2Nb 4Nb

16

DC flow model

(A1) Lossless lines (A2) Small angle differences (A3) Unit voltage magnitudes

rmn ⌧ xmn ! G ' 0 Pm ' X

n6=m

bmn(θm θn) Qm ' bmm X

n6=m

bmn(Vm Vn)

Pm = P

n VmVn (Gmn cos(θm−θn) + Bmn sin(θm−θn))

Qm = P

n VmVn (Gmn sin(θm−θn) − Bmn cos(θm−θn))

Vm ' 1 θmθn ' 0

17

DC bus admittance model

" Symmetric and positive semidefinite

! Power injections (and flows) linearly related to phase differences

Pm = − X

n6=m

bmn(θm − θn)

! Multivariate model

p = Bθ [B]mn = ⇢ P

n6=m x1 mn

, m = n x1

mn

, m 6= n B1Nb = 0Nb ⇒ pT 1Nb = 0

" Lossless lines " Weighted Laplacian of power grid graph 18

Power system state estimation (PSSE)

19

Motivation for PSSE

! Quantities of interest expressible as a function of bus voltages v

Goal: Given meter readings and grid parameters, find state vector v

! PSSE is of paramount importance for

" situational awareness " reliability analysis and planning " load forecasting " economic operations and billing

! Statistical problem formulation [Schweppe et al’70]

• F. C. Schweppe, J. Wildes, and D. Rom, Power system state estimation: Parts I, II, and III,

IEEE Trans. Power App. Syst., Jan. 1970.

20

SCADA-based PSSE

! Supervisory control and data acquisition (SCADA)

" Terminals forward readings to control center (~4 secs) " Phases cannot be used due to timing mismatches

! Available measurements (M)

z = h(v) + ✏ {Vm, Pm, Qm, Pmn, Qmn, Imn}

! Nonlinear (weighted) least-squares

ˆ v := arg min

v kz h(v)k2

! Possible constraints

" Zero-injection buses " Feasible ranges

Pm = Qm = 0 V min

m

≤ Vm ≤ V max

m

21

Popular solvers

! QR-based remedies for numerical stability ! Convergence to local minimum

(M1) Gauss-Newton iterations

ˆ v := arg min

v kz h(v)k2

(M2) Fast decoupled solver

! Active powers depend only on ; reactive only on ! Approximate

at flat voltage profile

{Vm} {θm}

• A. Monticelli and A. Garcia, “Fast decoupled state estimators,” IEEE Trans. Power Syst., May 1990.

" Approximate " Linear LS in closed form

v = 1 + j0

22

h(v) ' h(vk) + Jk(v vk), Jk : Jacobian at vk

vk+1 = vk + (JT

k Jk)−1JT k (z − h(vk))

(JT

k Jk)−1

Semidefinite relaxation

! Rectangular coordinates: measurements are quadratic in v

min

v M

X

m=1

(zm − hm(v))2 V = vvH

! Yet linear in

min

V M

X

m=1

(zm Tr (HmV))2 s.to V ⌫ 0 and rank(V) = 1

• H. Zhu and G. B. Giannakis, Estimating the state of AC power systems using semidefinite

programming,” in Proc. of North American Power Symposium, Aug. 2011.

! It can be cast as an SDP (PSSE convexified) ! SDR in SP and communications [Goemans et al ’95], [Luo et al’10] ! SDR for optimal power flow [Bai et al ’08] !

Dominant eigenvector approximation or randomization

ˆ V → ˆ v :

Pm + jQm = VmI∗

m = eT mv(Yv)Hem = Tr(YHemeT m

| {z }

Hm

vvH)

23

Numerical tests

! IEEE 30-bus benchmark grid !

Vm ∼ N(1, 0.01), θm ∼ U[−θ, θ]

! Closer to global optimum

at higher complexity

Average running time in secs.

24

Dynamic PSSE

! Well-motivated due to load and renewable variability ! Challenged by unknown system dynamics

" Random walk state model

[Monticelli00]

v(t + 1) = v(t) + w(t)

" With exogenous load

[Blood-Krogh-Ilic08]

v(t + 1) = F(t)v(t) + e(t) + w(t)

" Extended Kalman filter (EKF) [Monticelli00] " Unscented KF: more accurate but also complex [Valverde-Terzija’11] " Particle filters if affordable by real-time PSSE requirements

! Common prediction step; correction step options

25

Circuit breakers

1 2 15 16 17 18 19 20 23 22 21

bus/branch model bus section/switch model

! Protection and grid reconfiguration

Superbowl 2013

! Modeling

" closed: " open:

I18,19 = 0 V15 = V16

! Zero-impedance elements

26

Generalized state estimation

! Generalized state estimation (GSE) seeks state and grid topology ! Network topology processor collects circuit breaker (CB) statuses ! Topology errors easily detected, but hardly identified by PSSE ! Augmented state vector x: bus section voltages and CB flows

GSE:

min

x

kz Hxk2

2

s.to Cx = 0

! CB statuses effect constraints to ensure identifiability

• A. Monticelli, “Electric power system state estimation,” Proc. of the IEEE, Feb. 2000.

27

Identifying topology errors

" Least-absolute value (LAV) [Singh-Alvarado’95] " Largest normalized residual test (LNRT) [Clements-Costa’98] " Probabilistic modeling [Korres-Katsikas ’02] " Mixed-integer nonlinear program [Caro et al‘10]

! Validate the status of suspected/un-instrumented CBs

min

x

kz Hxk2

2 + λ

X

m∈S

kSmxk2 s.to Cx = 0

! Block-sparsity (Group-Lasso penalty) for suspected CBs

• V. Kekatos and G. B. Giannakis, Joint power system state estimation and breaker status

identification,” in Proc. IEEE PES North American Power Symposium, Sept. 2012.

28

! Account for CB status: C (S) for (un)-instrumented

IEEE 14-bus grid

65 bus sections 73 CBs 276 states 316 measurements Suspected CBs: 10-70 80% correct 20% erroneous

• A. Gomez-Exposito and A. de la Villa Jaen, Reduced substation models for generalized state

estimation,” IEEE Trans. Power Syst., Nov. 2001.

29

Numerical results

Generalized state estimator Novel method Execution time [secs] MSE

• No. of suspected CBs
• V. Kekatos and G. B. Giannakis, Joint Power System State Estimation and Breaker Status

Identification,” in Proc. IEEE PES North American Power Symposium, Sept. 2012.

CB status errors

30

Observability analysis

Given measurement set and grid parameters, assess state identifiability; if non-identifiable, find observable islands (maximally connected subgrids)

! Q: Why bother? A: To select (pseudo-)measurement sites

" generation schedules, load predictions, historical data

! Runs online to cope with meter failures, terminal delays, grid changes

Pm = − P

n6=m bmn(θm − θn)

Qm = −bmm − P

n6=m bmn(Vm − Vn)

DC model

! results carry over to

P−θ Q−V

• A. Monticelli, “Electric power system state estimation,” Proc. of the IEEE, Feb. 2000.

! Pairs of active-reactive meters

31

Numerical observability

! Identifiability (DC model):

H is full column rank

z = Hθ ⇔

! Voltage phase shift ambiguity anyway ! Branch-bus

incidence matrix ANl×Nb: [A]ln =    1 , l : n → m −1 , l : m → n , o.w.

• A. Monticelli and F. F. Wu, “Network observability,” IEEE Trans. Power App. Syst., May 1985.

! Observable: If ! For

the non-zeros of define unobservable lines

θ ∈ null(H), Aθ

! Systematic removal of unobservable branches reveals observable islands

null(H) ⊆ null(A) = {c · 1Nb, c ∈ R}

32

Topological observability

! Graph-theoretic approach

• K. A. Clements, P. R. Krumpholz, and G. W. Davis, “PSSE with measurement deficiency: An
• bservability/measurement placement algorithm,” IEEE Trans. Power App. Syst., July 1983.

! Builds a maximal spanning tree (o.w. forest)

" branches directly measured or incident to a metered bus " every branch corresponds to a different measurement

! Numerical observability implies topological observability ! Converse does not necessarily hold (x=y) [Monticelli’00]

33

Bad data

! Sources: time-skews, parameter uncertainty, uncalibrated meters,

reverse wiring

! Preprocessing (polarity and range tests)

z = Hx + ✏

" LSE ˆ

x =

• HT H

−1 HT z

" LSE residual r := z − Hˆ

x = P⊥

Hz = P⊥ H✏

" For ✏ ∼ N(0, I),

! DC (or linearized AC) model without bad data

H ∈ RM×N r ∼ N(0, P⊥

H), rank(P⊥ H) = M − N

• A. Abur and A. Gomez-Exposito, Power System State Estimation: Theory and Implementation,

Marcel Dekker, 2004.

34

Revealing outliers

Chi-squared test (detection)

• F. C. Schweppe et al, Power system state estimation: Part II, IEEE Trans. Power App. Syst., Jan. 1970.

! Because

then

r ∼ N(0, P⊥

H),

Largest normalized residual test-LNRT (identification)

! Remove bad datum and re-compute LSE (efficient RLS update) ! Equivalent to leave-one-out validation ! If

declare bad data

! If

the i-th measurement is bad

max

i

|ri| √Pii > t,

!

is the i-th diagonal entry of

P⊥

H

ri/ p Pii ∼ N(0, 1), Pii krk2

2 ⇠ χ2 M−N

krk2

2 > F −1 χ2

M−N (0.05),

35

Robust PSSE

• V. Kekatos and G. B. Giannakis, “Distributed Robust Power System State Estimation,” IEEE Trans. Power

Syst., May 2013.

! Least-median of squares (LMS) ! Least-absolute deviations (LAV)

ˆ xLAV := arg min

x kz Hxk1

ˆ xLMS := arg min

x medi(zi − hT i x)2

! Huber estimator ! Additive outlier model

z = Hx + ✏ + o min

x,o kz Hx ok2 2 + λ L

X

l=1

kolk2

! Systematic bad data cleansing

" decentralized algorithms " correlated noise, (block-)sparse " link to compressed sensing

ˆ xH := arg minx PM

i=1 h

• zi − hT

i x

• 36

Critical measurements

! For a critical measurement

" undefined NRT " no cross-validation (blindly trusted)

The i-th measurement is critical if once removed from the measurement set, the power system becomes unobservable

! Claim: The i-th column of

is zero; hence,

P⊥

H

ri = 0 (r = P⊥

H✏)

! Multiple corrupted readings

" communication link failures, cyber-attacks

|ri|/ p Pii

• O. Kosut, L. Jia, J. Thomas, and L. Tong, “Malicious data attacks on smart grid,” IEEE Trans. Smart

Grid, Dec. 2011.

! Bad data processing vulnerable to critical measurements

37

Cyber-attacks

Liu, Ning, and Reiter, “False data injection attacks against state estimation in electric power grids,” ACM Trans. Info. and System Security, May 2011.

! Grid is a continent-size cyber-physical system ! Challenged by increased sensing and networking ! No. of cyber-incidents on power SCADA: 3 (2009), 25 (2011) ! Types: GPS spoofing, generator controls, CB tripping ! Focus on PSSE (situational awareness, markets) ! Compromised meters at non-zero entries of a ! Stealth attacks can arbitrarily mislead PSSE by ! Deleting related rows of H deems the system unobservable

z = Hx + ✏ + a a = Hv

38

Non-stealth attacks

! Designs for attacker and defender [Kosut-Jia-Thomas-Tong’11]

" defender solves an l1-norm penalized GLRT " attacker trades identifiability for state divergence

! Nonlinear AC model attacks [Zhu-GG’12] ! Caveats

" attacker knows H " linearization around x

! Topology attacks [Kim-Tong’13]

39

! Summary on PSSE

Distributed PSSE

vertically organized markets tie lines deregulation long-distance power transfer privacy policies control areas with thousands of buses dense instrumentation

• A. Gomez-Exposito, A. Abur, A. de la Villa Jaen, and C. Gomez-Quiles, “A multi-level state estimation

paradigm for smart grids,” Proc. of the IEEE, June 2011.

40

Setups

! IEEE 14-bus grid with 4 control areas ! Area 2 states (buses): {3,4,7,8} ! Area 2 collects flow measurements

{(4,5), (4,9), (7,9)}...

! Option 1: Ignore tie-line meters

" statistically suboptimal " observability at risk (bus 11) " tie-line mismatches (trading)

! Option 2: Augment v2 to {3,4,7,8,5,9}

" consent with neighbors on shared states

• G. Korres, Distributed multi-area state estimation, IEEE Trans. Power Syst., Feb. 2011.

41

Distributed solvers

! Decentralized solvers ! Parallel and cascade (KF-type)

solvers [Schweppe et al70]

! Arbitrary interconnection graph

" Coordinator needed [Zhao-Abur05], [Korres et al ’11] " Block Jacobi [Conejo et al07]; consensus averaging [Xie et al11]

• A. Gomez-Exposito, A. de la Villa Jaen, C. Gomez-Quiles, P. Rousseaux, T. V. Cutsem, “A taxonomy of

multi-area state estimates,” Electric Power Systems Research, 2011.

! Caveats: Local observability, shared interconnection states,

slow convergence (if ensured), parameter tuning

42

ADMM solver

! Local linear(ized) model ! ADMM-based semidefinite programming (SDP) [Zhu-GG12]

" inter-area dependency graph affects decomposability

• V. Kekatos and G. B. Giannakis, “Distributed Robust Power System State Estimation,” IEEE Trans. on

Power Syst., May 2013.

43

! Regional PSSEs ! Coupled local

problems

! Framework features: minimum inter-area exchanges, linear

convergence guaranteed, disclosure policies

Decentralized LS-based PSSE

• L. Xie, C. Choi, and S. Kar, “Cooperative distributed state estimation: local observability relaxed,”
• Proc. IEEE PES General Meeting, July 2011.

44 44

S2. S1. Mean Square Error

MSE(decentralized-true) MSE(decen.[Xie etal]-centralized)

Decentralized bad data cleansing

45

S1. S3. S2.

f(x) := min

• 1

2kz Hx ok2

2 + λkok1

=

M

X

m=1

h(zm hT

mx) ! Pertinent bad data-aware

model

! Reveal outliers via

D-PSSE on a 4,200-bus grid

46

Phasor measurement units (PMUs)

47

SCADA PMU

measurements power, voltage, current magnitude voltage & current phasors, frequency (derivative)

• meas. model

non-linear linear reporting rate

• ne every 1-4 sec

30-60/sec wide-area sync poor precise (GPS signal) “Its like going from an X-ray to an MRI of the grid,” Terry Boston, PJM CEO

PMU versus SCADA

• A. G. Phadke and J. S. Thorp, Synchronized Phasor Measurements and their Applications,

Springer, 2008.

48

PMU architecture

! Correlation or (sliding) DFT ! Frequency offset estimation also of interest ! Other unknowns: DC and damping effects

IEEE Standard for Synchrophasor Measurements for Power Systems, IEEE Std. C37.118, Dec. 2011.

analog voltage/current transformers GPS

fs = Nf0

−Xm √ 2 sin φ

Xm √ 2 cos φ Xm cos(2πf0t + φ)

49

System architecture

! Phasor data concentrator (PDC)

" collects data streams from several PMUs " time-alignment, local cleansing, data compression

! Single PMU can measure voltage and several line currents

50

PMU uses

Power system state estimation

• A. G. Phadke and J. S. Thorp, Synchronized Phasor Measurements and their Applications,

Springer, 2008.

! Challenges

" different rates " (non)linear models (rect./polar) " phase alignment

! Solutions

" SCADA estimates as priors " PMU at reference bus

Monitoring, control, and protection (local and wide-area)

! Voltage stability ! Parameter estimation and dynamic line rating ! Oscillation and angular separation monitoring ! Visualization

51

52

Deployment of PMUs

Costs: acquisition, installation, networking Deployment: 2009: ~100; 2014: ~500

North American Synchrophasor Initiative, www.naspi.org

Q: Where should new PMUs be placed?

! Criteria: topological observability [Emami-Abur’10]

estimation accuracy [Li-Negi-Ilic’11]

Optimal experimental design

53

" PMU measurement model

zn = Hnx + ✏n x ∼ N(ˆ xSCADA, Σs)

# SCADA prior " Scalable projected gradient algorithm

• V. Kekatos, G. B. Giannakis, and B. Wollenberg, "Optimal Placement of Phasor Measurement

Units via Convex Relaxation," IEEE Trans. on Power Systems, vol. 27, pp. 1521-1530, Aug. 2012. # MMSE covariance

Σ(a) = N X

n=1

anHT

nΣ−1 n Hn + Σ−1 s

!−1

" E-optimal design (NP-hard)

min

a

• λmax(Σ(α)) : a ∈ {0, 1}N, aT 1 = k

# SDP relaxation

min

a

• λmax(Σ(α)) : a ∈ [0, 1]N, aT 1 = k

Additional learning and inference issues

54

! Blackouts cost $150 billion/year

Cascading failures

! Cascading: Lines exceed ratings and successively fail

55

! Linear DC model

" Internal-external states " Pre- and post-outage graph

p = Bθ θT = [θT

I θT E]

(N, E) → (N, E0)

internal system external system

Problem: Given pre- and post-outage internal states and basecase topology B, find the line outage set θI, θ0

I,

˜ E := E \ E0

! Exhaustive search [Tate-Overbye’09], [Emami-Abur’10]; GMRF [He-Zhang’11]

Linear DC model with outages

! Nodal power injections remain unchanged

p0 = p + n

! Post-/pre-outage

DC model

! Exploit ! Sparse representation over all transmission lines

• H. Zhu and G. B. Giannakis ”Sparse overcomplete representation for efficient identification of power

line outages,” IEEE Trans. Power Syst., Nov. 2012.

56

B0θ0 = Bθ + n B˜ θ = ˜ Bθ0 + n ˜ B = X

l∈ ˜ E

1 xl alaT

l

⇒

˜ B := B0 − B ˜ θ := θ0 − θ ˜ Bθ0 = X

l2E

alml + n, ml := (

aT

l θ0

xl

, l ∈ ˜ E , l / ∈ ˜ E

Grid Laplacian

57

p = Bθ + n

[B]mn = ⇢ P

n6=m x1 mn

, m = n x1

mn

, m 6= n

nodal powers nodal voltage phases bus admittance matrix

(m, n) ↔ l

(al := em − en)

·          ... . . . . . . · · · +1 · · · −1 · · · . . . ... . . . · · · −1 · · · +1 · · · . . . . . . ...         

+1 +1

• 1
• 1

n m n m

B = X

l∈E

x−1

l

alaT

l =

X

(m,n)∈E

x−1

mn

! Linear DC model ! From nodes to lines: weighted grid Laplacian

Lassoing line outages

! Given only , solve for

Running times (secs.)

branch-bus incidence matrix

! Sparse linear regression model with colored noise

" Orthogonal matching pursuit (greedy) " l1-norm penalized regression (coordinate descent)

IEEE 300-bus (detection probability)

OMP CD

58

A := ⇥ a1 · · · a|E| ⇤ (B, ˜ θI) ˜ θI = ⇥ B†⇤

I Am +

⇥ B†⇤

I n

B˜ θ = Am + n

Electromechanical modes

! Oscillations in voltage, frequency, and power flows (grid breakup) ! Inter-area oscillations in 0.1-1 Hz

• P. Kundur, Power System Stability and Control, McGraw-Hill, 1994.

! Probing signal u(t)

˙ x(t) = Axx(t) + Buu(t) + w(t)

! Modes described by the eigenvalues of Ax ! Linearized continuous-time differential equations ! Challenges in retrieving harmonics

" nonlinearity " time-varying power systems " closely-spaced frequencies 59

Mode estimation

(M1) Build Ax based on grid model (scalability issues)

• D. Trudnowski and J. Pierre, Signal processing methods for estimating small-signal dynamic properties from

measured responses, Springer, 2009.

! Batch modal analysis (Prony’s) ! Adaptive nodal analysis (LMS, RLS) ! Probing signal design for improved accuracy and minimal grid impact ! Measurement types

" ambient (regular) " ring-down (disturbance) " probing (engineered u(t))

(M2) Spectral estimates using measured x(t)

60

Load forecasting

! Spatial scales per substation, utility, and interconnection ! Challenges: deregulation, demand response, electric vehicles ! Based on historical load data and other (e.g., weather) predictors ! Data exhibit cycles (daily, weekly, seasonal) ! Outliers due to extreme weather, events, and strikes ! Time scales depend on application

" minute and hour (economic dispatch) " week (reliability assessment) " year (generation and transmission planning) 61

Popular load predictors

! Ordinary LS ! Kernel-based regression and SVMs ! Time series analysis (ARMA, ARIMA, ARIMAX) ! State-space models with KF and particle filtering ! Neural networks and artificial intelligence ! Low-rank models for load cleansing [Mateos-GG’12]

• M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric Power Systems: Forecasting,

Scheduling, and Management, Wiley Interscience, 2002.

62

Electricity price forecasting

MISO market

• M. Amjady and N. Hemmati, Energy price forecasting,“ IEEE Power Energy Mag., Apr. 2006.

! Important for asset owners,

system operators, government

! Challenges

" weather and load " hedging strategies " outages and security

! Approaches

" Time-series based predictors [Contreras et al’03], [Conejo et al’05] " Neural networks [Gonzalez et al’05], [Li et al’07], [Wu-Shahidehpour’10] " Nearest-neighbor approach [Lora-Exposito’07] " QP with outage combinations [Zhou-Tesfatsion-Liu’11] 63

• V. Kekatos, S. Veeramachaneni, M. Light, and G. B. Giannakis, ”Day-Ahead Electricity Market Forecasting

using Kernels,” Proc. of Innovative Smart Grid Technologies, Feb. 2013.

Spatiotemporal price forecasting

!

Learning over pricing network

! Separable kernel ! Gaussian kernel

ˆ f:= arg min

f∈H

X

t,n

(p(t, n) f(xt, n))2 + λkfk2

H

! inverse Laplacian of

balancing authority graphs

MISO; Jun-Aug 2012; 1732 nodes

kt : ks : k ((ti, ni), (tj, nj)) = kt

• xti, xtj
• ks (ni, nj)

64

• V. Kekatos, Y. Zhang, and G. B. Giannakis, “Electricity Market Forecasting via Low-Rank Multi-Kernel

Learning,” IEEE Journal of Selected Topics in Signal Processing (submitted).

Clustering the grid

! Modularization facilitates

" decentralized and parallel computing " minimum communication

! Static partitioning into reliability regions ! “Self-healing” (islanding under contingencies)

" active power balance (frequency stability) " reactive power balance (voltage stability)

! Zonal analysis via spectral clustering (reliability planning, market)

" bus adjacency (“small-world effect” [Watts-Strogatz’98]) " bus electrical adjacency (admittance matrix )

• J. Li, C.-C. Liu, and K. Schneider, “Controlled partitioning of a power network considering real and

reactive power balance,” IEEE Trans. Smart Grid, Dec. 2010.

65

Grid monitoring and learning

Observability Analysis PSSE Batch/adaptive Distributed Generalized Bad data Cyber-attacks PMUs Line

• utages

Forecasting Load, wind, prices Clustering L E A R N I N G M O N I T O R I N G E C O N O M I C O P E R A T I O N S

66

Economic Operation of Power Systems

• A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, 2nd ed., Wiley, 1996.
• A. Gomez-Exposito, A. J. Conejo, and C. Canizares, Electric Energy Systems: Analysis and Operation, CRC

Press, 2009.

67

! ED typically solved every 5-10 minutes

Generation cost

! Thermal generators ! Power output (MW) ! Generation cost ($/h or €/h)

PGi Ci(PGi)

~ ~ ~

Load

PD

PGNg PG1 PG2

Economic dispatch (ED): Find most economically generated power output to serve given load

68

Optimizing ED

! Generation cost typically convex and strictly increasing

" Piecewise linear or quadratic

Ci(PGi) min

{PGi} Ng

X

i=1

Ci(PGi) P min

Gi

≤ PGi ≤ P max

Gi

• subj. to

Ng

X

i=1

PGi = PD

! ED balances supply and demand most economically ! Convex optimization problem

Ci(PGi) PGi

69

Marginal price

! Lagrange multiplier for supply-demand balance ! Optimal generation output ! ED optimizes net cost: Generation cost minus revenue ! Price ($/MWh or €/MWh) ! Prices Lagrange multipliers

λ P ∗

Gi =

arg min

P min

Gi ≤PGi≤P max Gi

{C(PGi) − λ∗PGi} λ∗

Ng

X

i=1

PGi = PD

70

Market participants

Market Operator/ Independent System Operator (ISO) Dispatchable Generators Non-dispatchable Generators e.g., renewable energy sources

Ci(PGi)

Large Industrial Customers Load-Serving Entities Market clearing: Optimal schedules and prices Generation Demand (fixed or elastic) Serving commercial and residential end-users

71

• A. J. Conejo, M. Carrion, and J. M. Morales, Decision Making Under Uncertainty in Electricity Markets,

Springer, 2010, ch. 1.

DC power flow

! ED must account for transmission network constraints

" DC approximation

! Generator and load per bus

" At all buses: and

! Power flow from bus to bus ! Define the matrix ! Power flows at all lines

PGm m pG Nl × Nb pD

m n

Pmn = x−1

mn(θm − θn)

D = diag({x−1

mn}(m,n)∈E)

A = branch-bus incidence matrix PDm

72

xmn θm θn Pmn F = DA f = [Pmn] = Fθ

! Power balance per bus; flow limit per line

DC optimal power flow

min

pG, θ Nb

X

m=1

Cm(PGm)

• R. D. Christie, B. F. Wollenberg, and I. Wangensteen, “Transmission management in the deregulated

environment,” Proc. of the IEEE, Feb. 2000.

pmin

G

≤ pG ≤ pmax

G

• subj. to pG − pD = Bθ

! Convex optimization problem ! Locational marginal prices (LMPs)

" Multipliers for nodal balance

MISO market

73

|Fθ| ≤ f max

! System security must be ensured even under unplanned events

" Line flow and bus voltage limits

! Set of credible contingencies ! Contingency may include any of the following:

" Generation loss " Generation derating " Demand variation " Line outage gives rise to , ,

! Notation for generation limits and demand ! North American Electric Reliability Council (NERC) N-1 security rule

Contingencies

K k ∈ K P max,k

Gm

= P max

Gm − ∆P k Gm

P k

Dm = PDm + ∆P k Dm

f max,k (pmax,k

G

, pmin,k

G

, pk

D) ∈ Ck

P max,k

Gm

= P min,k

Gm

= 0 Bk

74

Fk

! Spinning reserve: Online generation capacity available to be

deployed under contingencies

! Scheduling of reserves

" Up-spinning ; down-spinning

! Reserve level available depends on generation schedule

Reserves

Rup

m

Rdown

m

PGm PGm P max

Gm

P min

Gm

Rup

m

Rdown

m

P k

Gm

Adjusted generation

75

! Decide reserve levels and adjustments for each contingency

Security-constrained DC OPF

pG − rdown ≤ pk

G ≤ pG + rup

rup ≤ pmax

G

− pG rdown ≤ pG − pmin

G ;

min

pG, θ, rup, rdown {pk

G, θk}k∈K

Nb

X

m=1

⇥ Cm(PGm) + Cup

m (Rup m ) + Cdown m

(Rdown

m

) ⇤

• subj. to

∀ k ∈ K ∀ k ∈ K ∀ k ∈ K pmin,k

G

≤ pk

G ≤ pmax,k G

(pmax,k

G

, pmin,k

G

, pk

D) ∈ Ck ∀ k ∈ K

pmin

G

≤ pG ≤ pmax

G

Network constraints in base case Network constraints at every contingency Reserve levels and adjustment constraints

• F. D. Galiana, F. Bouffard, J. M. Arroyo, and J. F. Restrepo, “Scheduling and pricing of coupled

energy and primary, secondary, and tertiary reserves,” Proc. of the IEEE, Nov. 2005.

pG − pD = Bθ pk

G − pk D = Bkθk

∀ k ∈ K

76

|Fθ| ≤ f max |Fkθk| ≤ f max,k

AC power flow

! Incorporate AC transmission network into ED

" AC model is exact " Ohmic losses - typically 5% of total load " Bus voltage constraints

! Real power injection: ! Reactive power injection: ! Complex power flowing over line ! Current injections

Smn = VmI∗

mn

(m, n) PGm − PDm QGm − QDm

Vm

Im = X

n∈Nm

Imn

i = Yv

77

! Quadratic equality constraints nonconvex problem ! Typical approaches rely on KKT conditions

AC optimal power flow

V min

m

≤ |Vm| ≤ V max

m

min

pG,qG,v Nb

X

m=1

Cm(PGm) |Re{Smn}| ≤ f max

mn ;

pmin

G

≤ pG ≤ pmax

G

; qmin

G

≤ qG ≤ qmax

G

|Smn| ≤ Smax

mn

• subj. to

pG − pD + j(qG − qD) = diag(v)(Yv)∗

78

! AC OPF with variables and additional constraints ! Nodal balance constraint linear in ! Works in many practical OPF instances and IEEE benchmarks ! Optimal in (three-phase) tree networks [Lam et al’12] [Dall’Anese et al’13]

SDP relaxation

! Canonical basis of

VmI∗

m = eH mv(Yv)Hem = tr[eH mvvHYHem] = tr[YHemeH mvvH]

{em}Nb

m=1

RNb V := vvH PGm − PDm + j(QGm − QDm) = tr[YHemeH

mV]

! Line flow and bus voltage constrains also linear in V

pG, qG, V rank[V] = 1

Nonconvex Drop

• J. Lavaei and S. Low, “Zero Duality Gap in Optimal Power Flow Problem,” IEEE TPS, Feb. 2012.
• X. Bai, H.Wei, K. Fujisawa, and Y.Wang, “Semidefinite programming for optimal power flow

problems,” Int. J. Elect. Power Energy Syst., 2008.

V ⌫ 0

79

! Scheduling horizon ! Unit commitment (UC) solved every day in many ISOs ! Binary variable: if gen. is ON at slot , 0 otherwise ! Practical constraints for thermal generators ! Ramp-up/down ! Minimum up/down time

" If a generator is turned on, it must remain on for the next slots " If a generator is turned off, it must remain off for slots

Multi-period scheduling

{1, . . . , T} ut

m = 1

m t P t

Gm − P t−1 Gm ≤ Um

P t−1

Gm − P t Gm ≤ Dm

T up

m

ut

m − ut−1 m

≤ uτ

m, τ = t + 1, . . . , min{t + T up m , T}

T down

m

80

! Start-up/shut-down costs

" Depend on previous on/off activity

Unit commitment

nodal balance constraints for every t line flow limits for every t ramp-up/down limits minimum up/down time constraints

• subj. to

ut

mP min Gm ≤ P t Gm ≤ ut mP max Gm ; ut m ∈ {0, 1}

St

m({uτ m}t τ=0)

min

{pt

Gm,θt m,ut m}m,t

T

X

t=1 Nb

X

m=1

⇥ Ct

m(P t Gm) + St m({uτ m}t τ=0)

⇤

81

Solution approaches

! UC is a mixed-integer program [Padhy’04]

" Further complication: Coupling of on/off status across time

! Classical approach [Takriti-Birge’00]

" Dualize nodal balance and line flow constraints " Lagrangian minimization can by solved by dynamic programming " Commitments from optimal multipliers " After commitments are set, solve DC OPF

! Benders decomposition [Shahidehpour et al’02] ! Duality gap vanishes as number of generators increases [Bertsekas et al’83]

82

Economic operations

! SCUC is solved for day-ahead market clearing in many ISOs

ED DC OPF AC OPF DC power flow AC power flow DC UC Multi-period

• n/off scheduling

SC DC OPF Security constraints SCUC AC UC SC AC OPF SCUC

83

Demand response

84

Demand response

! Changes in electricity consumption by end-users in response to

" Changes in electricity prices over time " Incentive payments at times of high wholesale prices

• r jeopardized system reliability

! Demand-side management, load control ! Incentive-based programs

" Direct load control/interruptible loads

!

Large customers enter into contracts with utility

!

Utility takes full control of their loads

" Demand-side bidding (DSB)

• M. Albadi and E. F. El-Saadany, “Demand Response in Electricity Markets: An Overview,” IEEE PES GM, 2007.
• K. Hamilton and N. Gulhar, “Taking Demand Response to the Next Level,” IEEE PES Mag., May 2010.

85

Demand-side bidding

! Large customers adjust consumption (ED with DSB) [Christie et al’00] ! Concave utility function : willingness-to-buy

" Piecewise linear or quadratic

P min

Gi

≤ PGi ≤ P max

Gi

Uj(PDj) min

{PGi},{PDj } Ng

X

i=1

Ci(PGi) −

Nd

X

j=1

Uj(PDj)

• subj. to

Ng

X

i=1

PGi =

Nd

X

j=1

PDj P min

Dj

≤ PDj ≤ P max

Dj

Maximize social welfare Supply-demand balance

PDj Uj(PDj)

86

! DSB elastic demand ! Slide from right to left on curve ! DSB tends to decrease marginal price

DSB can reduce marginal price

! Consider one generator and one load ! Lagrangian function ! Marginal price

L(PG, PD, λ) = C(PG) − U(PD) − λ(PG − PD) dC(PG) dPG dU(PD) dPD PG = PD PG = P inelastic

D

λinelastic dC/dPG λ∗ = dC(PG) dPG = dU(PD) dPD λ∗

87

Price-based programs

! Time of use (TOU)

" Prices vary throughout the day, e.g., peak and off-peak " TOU rates announced long time ahead not dynamic

! Critical peak pricing (CPP)/extreme day pricing (EDP)

" Higher prices on top of TOU for certain days " Called during emergencies/high wholesale prices " Effect may be announced a day in advance

! Real-time pricing (RTP)

" Typically hourly prices reflecting wholesale prices " Day-ahead or hour-ahead 88

Advanced metering infrastructure

! Real-time pricing enabled by the AMI ! Two-way communication network ! Utility smart meters at end-user premises ! Smart meter measures power consumption with frequency e.g., 15min ! “Prices to devices” Energy consumption scheduling Two-Way Communication Network (AMI) Electricity Distribution Network

Utility Company Main Grid

89

• L. Chen, N. Li, L. Jiang, and S. Low, “Optimal Demand Response: Problem Formulation and

Deterministic Case,” in Control and Optimization Methods for Smart Grids, Springer, 2012.

! Scheduling horizon ! Power consumption of appliance ! Concave utility function ! Elastic loads, e.g., lights ! Energy loads, e.g., PHEV charging

Adjustable power appliances

pt

a

{1, . . . , T} a Ua(pa) Ua(pa) =

T

X

t=1

U t

a(pt a)

pmin

a

≤ pt

a ≤ pmax a

Ua(pa) = Ua T X

t=1

pt

a

! Emin

a

≤

T

X

t=1

pt

a ≤ Emax a

pmin

a

≤ pt

a ≤ pmax a

90

! Thermostatically controlled loads, e.g., AC

" Room temperature ; ambient (external) " Preferred temperature

! Interruptible loads: If on, they consume nonzero power ! Noninterruptible and deferrable loads, e.g., dishwasher

" Fixed load profile, can be shifted " Integer constraints

Additional types of appliances

t pt

a

pt

a = 0

• r

pmin

a

≤ pt

a ≤ pmax a

ϑt = (1 − γ)ϑt−1 − γϑamb,t − ηpt

a

ϑmin ≤ ϑt ≤ ϑmax ϑt ϑamb,t

91

Energy consumption scheduling

! Cost of power for end-user: typically (piecewise) linear ! Tradeoff cost with utility ! Convexity depends on ! Solved by the smart meter with processor ! With interruptible appliances: Vanishing duality gap when horizon

length increases [Gatsis-GG’11]

! Price uncertainty: Prediction methods, robust/stochastic optimization

[Mohsenian-Rad,Leon-Garcia’10], [Conejo et al’10], [Kim-Poor’11], [Kim-GG’13]

min

{pa} T

X

t=1

ct X

a∈A

pt

a −

X

a∈A

Ua(pa)

• subj. to pa ∈ Pa, a ∈ A

Pa

92

Cooperative DR

! Set of users served by the same utility company ! Set of smart appliances per user ! Power consumption ! End-user utility function ! Cost of power procurement for utility company Two-Way Communication Network (AMI) Electricity Distribution Network

Utility Company Main Grid

{1, . . . , R} pt

ra

Ar r Ura(pra)

93

Ct R X

r=1

X

a∈Ar

pt

ra

!

! Motivation: Reduce peak demand respecting users’ preferences ! Convexity depends on ! Challenges

" Distributed scheduling over AMI " Privacy issues

Social welfare maximization

min

{pra} T

X

t=1

Ct R X

r=1

X

a∈Ar

pt

ra

! −

R

X

r=1

X

a∈Ar

Ura(pra)

• subj. to pra ∈ Pra, a ∈ Ar, r = 1 . . . , R

Pra

94

! Gradient projection, block coordinate descent, dual decomposition,

Vickrey-Clark-Groves mechanism

[Chen et al’12], [Mohsenian-Rad et al’10], [Papavasiliou et al’10], [Samadi et al’11], [Gatsis-GG’12]

! Dual decomposition: Introduce variable for total provided power

" Lagrange multiplier for supply-demand balance

! Upshot

" Separate subproblems for utility and smart meters " Privacy respect

Solution approaches

st

Supply-demand balance

λt

R

X

r=1

X

a∈Ar

pt

ra ≤ st

95

! Schedule update: At the utility company and smart meters ! Multiplier update: At utility company

Distributed algorithm

prices total hourly consumption

96

P

a∈Ar pt ra(`)

min

0≤st≤smax{Ct(st) − t(`)st}

min

pra∈Pra

( T X

t=1

t(`)pt

ra − Ura(pra)

) t(` + 1) = " t(`) + R X

r=1

X

a∈Ar

pt

ra(`) − st(`)

!#+

Lost AMI messages

! Messages in both ways may be lost

" Not transmitted, due to failure " Not received, due to noise " Cyber-attacks

! Use the latest message available ! Convergence established for different lost-message patterns

" Asynchronous subgradient method

! Benefit: Resilience to communication network outages

• N. Gatsis and G. B. Giannakis, “Residential load control: Distributed scheduling and convergence

with lost AMI messages,” IEEE Trans. Smart Grid, June 2012.

97

P

a∈Ar pt ra(`)

Scheduling with DR aggregators

! Goal: DR from small-scale end-users into day-ahead scheduling ! Challenges

" User preferences are private " Large-scale problem

! Approach: Decomposition algorithms

" Subproblems for market operator and aggregators

! Outcome: Scalable distributed solution

Market Operator Aggregator

A2 G1 BL3 A3 BL2

• Y. Zhang, N. Gatsis, and G. B. Giannakis, “Disaggregated bundle methods for distributed market

clearing in power networks,” in Proc. IEEE Global Conf. Signal and Information Process., Dec. 2013.

98

• N. Gatsis and G. B. Giannakis, “Decomposition algorithms for market clearing with large-scale

demand response,” IEEE Trans. Smart Grid, 2013.

Plug-in (Hybrid) Electric Vehicles

99

Plug-in electric vehicles

! PEVs feature batteries that can be plugged in

" at end-user premises " at charging stations

! Hybrid: Also consume fuel (PHEV) ! Benefits of high P(H)EV penetration

" environmental: reduce carbon emissions " economic: reduce dependency on oil 100

Opportunities and challenges

! PHEV battery charging is a controllable load ! Charging coordination required to avoid

" overloading of distribution networks [Clement-Nyns et al’10] " accentuating (or creating new) peaks

! Aggregator provides charging services to end-users [Wu et al’12]

" Can offer reserves [Sortomme et al’12], [Bessa et al’12]

! Vehicle-to-grid (V2G) [Kempton-Tomic’05]

" Unidirectional V2G: Modulation of charging rate system reserves " Bidirectional V2G: Discharge batteries to feed power in the grid

! Mitigation of renewable energy intermittency ! Spatiotemporal prediction of demand [Lojowska et al’12], [Bae-Kwasinski’12]

101

! Power provided at the substation at slot ! Load factor (LF) ! LF closer to 1 Total power consumption smoother

" Lower peak, higher valley

Load factor

t

0 ≤ LF ≤ 1 st = base load + PHEV charging + losses LF =

T

P

t=1

st max

t=1,...,T st · T

IEEE 14-node feeder

102

! Fleet of vehicles ! Charging rate at slot

" Vehicle plugged in at different slots

! Total power (ignoring losses) ! Charging coordination

Charging coordination

n = 1, . . . , N t rt

n

rt,min

n

≤ rt

n ≤ rt,max n

• subj. to

rmin

n

≤ rn ≤ rmax

n T

X

t=1

rt

n = En

103

st = Lt +

N

X

n=1

rt

n

min

{rn} T

X

t=1

C Lt +

N

X

n=1

rt

n

!

! Quadratic cost

" Under certain conditions, [Sortomme et al’11]

! Charging coordination problem similar to cooperative DR ! Pricing signal ; initialization ! Schedule update at smart charger

" Approximation of objective

! Price update at aggregator

Distributed charging

pt(`) C(st) = (st)2 max LF ⇔ min(st)2 rn(`) C(st) = (st)2

Aggregator

pt(`) rn(`)

• L. Gan, U. Topcu, and S. Low, “Optimal decentralized protocol for electric vehicle charging,”

in Proc. Control and Decision Conf., Dec. 2011.

104

pt(1) = Lt pt(` + 1) = Lt +

N

X

n=1

rt

n(`)

Renewables

105

Picture source: REN21, “Renewables 2011 global status report,” Paris, 2011, [Online]. http://www.ren21.net/Portals/0/documents/Resources/110929_GSR2011_FINAL.pdf

Renewable energy

! Milestones

" US DoE: 20% of demand covered by wind energy by 2030 " Denmark: 100% renewable energy by 2050

! Renewable energy production classification

" >1 MW: utility/wholesale scale " 10-100 kW: microgrid scale " <10 kW: residential scale 106

Microgrids

! Small-footprint power systems

" Distribution networks, campuses, isolated areas, military facilities " Distributed generation (renewable and conventional) " Distributed storage and controllable loads

LC = Local controller MGEM = Microgrid energy manager

! Why microgrids?

" Generation closer to demand " Bypass transmission congestion " Reduced bulk generation " Islanded mode in disasters 107

Challenging energy management

! Renewable energy sources (RES) are nondispatchable ! Optimization under uncertainty

" Forecast-based; chance constraints; robust/stochastic optimization

Uncertain

108

min

{PGm} Ng

X

m=1

Cm(PGm)

• subj. to

Ng

X

m=1

PGm = L min

{PGm} Ng

X

m=1

Cm(PGm)

• subj. to

Ng

X

m=1

PGm + W = L

Forecast-based methods

! RES power output forecast ! Model predictive control for multi-period scheduling [Ilic et al’11]

" Multi-period forecast; scheduling; dispatch first period only " Move the horizon and repeat

ˆ W

109

min

{PGm} Ng

X

m=1

Cm(PGm)

• subj. to

Ng

X

m=1

PGm + ˆ W = L

! Allow for insufficient generation at low risk level ! Chance constraints intractable due to spatiotemporal correlation ! Convex approximations: Gaussian modeling, scenario approximation

Risk-constrained ED

[Liu’10], [Varaiya-Wu-Bialek’11], [Zhang-Gatsis-GG’12], [Zhang-Gatsis-Kekatos-GG’13]

110

min

{P t

Gm}

T

X

t=1 Ng

X

m=1

Cm(P t

Gm)

• subj. to Prob

2 4

Ng

X

m=1

P t

Gm + Nw

X

i=1

W t

i − P t D ≥ 0 (t = 1, . . . , T)

3 5 ≥ 1 − α

Microgrid components

! Distributed generation units

" Capacity and ramp limits

! Elastic loads with concave utility ! Energy loads ! Distributed storage units

" State of charge " Capacity and charge rate limits

P t

Gm

111

Uj(P t

Dj) T

X

t=1

P t

Ek = Ek

Bt

n = Bt−1 n

+ P t

Bn

0 ≤ Bt

n ≤ Bmax n

P min

Bn ≤ P t Bn ≤ P max Bn

Midwest ISO, March 5, 2013

Renewable energy uncertainty

! Polyhedral uncertainty set for RES outputs ! Split horizon into disjoint sub-horizons

" Deterministic bounds from e.g., historical data

W t

i

T = {1, . . . , T}

Higher production in the evenings Lower production late night – early morning

112

W := ( w

• Wt

i ≤ W t i ≤ ¯

W t

i , W min h

≤ X

t∈Th Nw

X

i=1

W t

i ≤ W max h

,

H

[

h=1

Th = T ) H

P t

R

Lt P t

Gm

Transaction with the main grid

! Committed renewable energy ! Worst-case transaction cost

P t

R

[a]+ = max{a, 0} [a]− = max{−a, 0}

Surplus to sell Shortage to buy Export price Import price

P t

Ek

113

G({P t

R}T t=1) := max w∈W T

X

t=1

@αt " P t

R − Nw

X

i=1

W t

i

#+ − βt " P t

R − Nw

X

i=1

W t

i

#−1 A P t

Dj

P t

Bn

• Y. Zhang, N. Gatsis, and G. B. Giannakis, “Robust energy management with high-penetration

renewables,” IEEE Trans. Sustainable Energy, 2013.

Robust ED with benefits

! Computational challenge: Maximization over uncertainty set

t = 1, . . . , T

Supply- demand balance

! Unit commitment, transmission network, reserves

[Zhao-Zheng’12], [Bertsimas et al’13]

114

• subj. to generation, load, storage contraints

min

{pG,pB,pR, pD,pE} T

X

t=1 Ng

X

m=1

Cm(P t

Gm) − T

X

t=1 Nd

X

j=1

Uj(P t

Dj) + G({P t R}T t=1) Ng

X

m=1

P t

Gm − Ns

X

n=1

P t

Bn + P t R = Nd

X

j=1

P t

Dj + Ne

X

k=1

P t

Ek + Lt

Stochastic programming

! Two-stage approach

" First stage: Unit commitments in a day-ahead fashion " Second stage: Generation dispatch when uncertainty is revealed

! Uncertainty is RES output here ! Postulate plausible scenarios of RES output ! Scenario has probability ! First-stage decision variables: Commitments ! Second-stage decision variables: Generation dispatch ;

" Second-stage variables depend on first-stage decisions and scenario

u

115

s = 1, . . . , S s π(s) pG(u, s) θ(u, s) {W t

i (s)}i,t

! Consider a single period ! For fixed and , solve DC OPF ! Convex problem ! Multi-period extension includes ramp constraints

Second-stage problem

u w(k)

116

Q(u, s) = min

pG(u,s), θ(u,s) Nb

X

m=1

Cm(PGm(u, s))

• subj. to

pG(u, s) − pD + w(s) = Bθ(u, s) umP min

Gm ≤ PGm(u, s) ≤ umP max Gm

|Fθ(u, s)| ≤ f max

! Decide unit commitments

" Start-up/shut-down costs and expected cost of dispatch

! First stage includes second-stage decisions in ! First stage decides commitments ahead of time ! At the beginning of the horizon, becomes known through forecast ! Solve second-stage problem for generation dispatch ! Extensions: DSB, reserves, demand uncertainties

First-stage problem

minimum up/down time constraints

• subj. to

ut

m ∈ {0, 1}

w

[Bouffard-Galiana’08]

117

Q(u, s) min

u T

X

t=1 Nb

X

m=1

St

m({ut m}t τ=0) + S

X

s=1

π(s)Q(u, s)

! Two-stage stochastic programming for energy planning in microgrids

Stochastic optim. for microgrids

118

Time

Scheduling horizon T # First-stage decisions: Ahead of the horizon 1. Generation setpoints 2. Load setpoints P t

Gm

P t

Dj

# Second-stage decisions: Real-time 1. Load adjustments --- negative load adjustment penalized with price 2. Transaction with the main grid --- energy import/export at prices # Scenarios for RES output At

j

δt

j

wt(s) s = 1, . . . , S αt, βt

! Second stage: Convexity ensured if for all

The two stages

119

• subj. to

Ng

X

m=1

P t

Gm + P t R = Nd

X

j=1

(P t

Dj + At j) + Lt

P min

Dj

≤ P t

Dj + At j ≤ P max Dj

min

pG,pD T

X

t=1 Ng

X

m=1

Cm(P t

Gm) − T

X

t=1 Nd

X

j=1

Uj(P t

Dj) + S

X

s=1

π(s)

T

X

t=1

Qt(pt

G, pt D, wt(s))

• subj. to generation and load constraints

αt ≥ βt t

! First stage is convex as long as second stage is convex

• G. Martinez, N. Gatsis, and G. B. Giannakis, “Stochastic programming for energy planning in microgrids

with renewables,” in Proc. IEEE Workshop Comput. Advances in Multi-Sensor Adaptive Process., Dec. 2013.

Transaction cost Load adjustment cost

Qt(pt

G, pt D, wt) = min at,pt

R

Nd

X

j=1

δt

j[At j]− +αt

" P t

R − Nw

X

i=1

W t

i

#+ −βt " P t

R − Nw

X

i=1

W t

i

#−

Robust vs. stochastic optimization

! Solution approaches entail decomposition methods

Robust Stochastic Approach Worst-case (conservative) On the average Modeling requirements Uncertainty set, typically polyhedral Set of scenarios and corresponding probabilities Main computational challenge Maximization over uncertainty set Large number of scenarios required

120

Open issues

121

Big Data grid informatics

! Energy analytics: statistical learning ! Data deluge at different system levels ! Forecasting of loads, RES, prices, consumer patterns, PEV charging

Large-scale systems Microgrids Smart Buildings Measurement data SCADA, PMUs Voltage and power across distribution system Power meter readings, ambient condition sensors Needs and objectives Situational awareness, reliability, cyber-security, economic operation Self-sustainability,

• peration in

connected or islanded modes Energy conservation, anomalous consumption pattern detection

122

Renewables and PEVs

! Renewable energy enabled consumers

" Increased demand response capabilities

! Aggregator participation in markets

" Increased uncertainty

! Predicting consumer patterns and response ! Dynamic learning of consumer elasticity in PEV charging ! Interconnections: from transmission systems to microgrids ! Leverage SP expertise for resource management

Thank you!

123

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