Generalized state estimation ! Network topology processor collects circuit breaker (CB) statuses ! Topology errors easily detected, but hardly identified by PSSE ! Generalized state estimation (GSE) seeks state and grid topology k z � Hx k 2 min 2 GSE: x s.to Cx = 0 ! Augmented state vector x : bus section voltages and CB flows ! CB statuses effect constraints to ensure identifiability A. Monticelli, “Electric power system state estimation,” Proc. of the IEEE , Feb. 2000. 27
Identifying topology errors ! Validate the status of suspected/un-instrumented CBs " 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] ! Account for CB status: C (S) for (un)-instrumented ! Block-sparsity (Group-Lasso penalty) for suspected CBs X k z � Hx k 2 k S m x k 2 min 2 + λ x m ∈ S s.to Cx = 0 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
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 CB status errors Novel method MSE Execution time [secs] 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. 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 DC model P m = − P n 6 = m b mn ( θ m − θ n ) ! results carry over to P − θ Q − V ! Pairs of active-reactive meters Q m = − b mm − P n 6 = m b mn ( V m − V n ) A. Monticelli, “Electric power system state estimation,” Proc. of the IEEE , Feb. 2000. 31
Numerical observability ! Identifiability (DC model): H is full column rank z = H θ ⇔ ! Voltage phase shift ambiguity anyway 1 , l : n → m ! Branch-bus A N l × N b : [ A ] ln = − 1 , l : m → n incidence matrix 0 , o.w. ! Observable: If null( H ) ⊆ null( A ) = { c · 1 N b , c ∈ R } ! For the non-zeros of define unobservable lines θ ∈ null( H ) , A θ ! Systematic removal of unobservable branches reveals observable islands A. Monticelli and F. F. Wu, “Network observability,” IEEE Trans. Power App. Syst., May 1985. 32
Topological observability ! Graph-theoretic approach ! 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] K. A. Clements, P. R. Krumpholz, and G. W. Davis, “PSSE with measurement deficiency: An 33 observability/measurement placement algorithm,” IEEE Trans. Power App. Syst., July 1983.
Bad data ! Sources: time-skews, parameter uncertainty, uncalibrated meters, reverse wiring ! Preprocessing (polarity and range tests) ! DC (or linearized AC) model without bad data H ∈ R M × N z = Hx + ✏ � − 1 H T z H T H " LSE ˆ � x = x = P ⊥ H z = P ⊥ " LSE residual r := z − H ˆ H ✏ r ∼ N ( 0 , P ⊥ H ) , rank( P ⊥ " For ✏ ∼ N ( 0 , I ) , 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) r ∼ N ( 0 , P ⊥ ! Because then k r k 2 2 ⇠ χ 2 H ) , M − N k r k 2 2 > F − 1 ! If declare bad data M − N (0 . 05) , χ 2 Largest normalized residual test-LNRT (identification) p P ii ∼ N (0 , 1) , P ii P ⊥ ! r i / is the i -th diagonal entry of H | r i | ! If the i -th measurement is bad > t, max √ P ii i ! Remove bad datum and re-compute LSE (efficient RLS update) ! Equivalent to leave-one-out validation F. C. Schweppe et al, � Power system state estimation: Part II, � IEEE Trans. Power App. Syst. , Jan. 1970 . 35
Robust PSSE ! Least-absolute deviations (LAV) x k z � Hx k 1 x LAV := arg min ˆ x med i ( z i − h T i x ) 2 ! Least-median of squares (LMS) x LMS := arg min ˆ P M ! Huber estimator � � z i − h T x H := arg min x ˆ i =1 h i x ! Additive outlier model z = Hx + ✏ + o L X ! Systematic bad data cleansing x , o k z � Hx � o k 2 k o l k 2 min 2 + λ l =1 " decentralized algorithms " correlated noise, (block-)sparse " link to compressed sensing V. Kekatos and G. B. Giannakis, “Distributed Robust Power System State Estimation,” IEEE Trans. Power Syst. , May 2013. 36
Critical measurements The i-th measurement is critical if once removed from the measurement set, the power system becomes unobservable P ⊥ r i = 0 ( r = P ⊥ ! Claim: The i -th column of is zero; hence, H ✏ ) H ! For a critical measurement p " undefined NRT | r i | / P ii " no cross-validation (blindly trusted) ! Bad data processing vulnerable to critical measurements ! Multiple corrupted readings " communication link failures, cyber-attacks O. Kosut, L. Jia, J. Thomas, and L. Tong, “Malicious data attacks on smart grid,” IEEE Trans. Smart Grid , Dec. 2011. 37
Cyber-attacks ! Grid is a continent-size cyber-physical system ! No. of cyber-incidents on power SCADA: 3 (2009), 25 (2011) ! Challenged by increased sensing and networking ! Types: GPS spoofing, generator controls, CB tripping ! Focus on PSSE (situational awareness, markets) z = Hx + ✏ + a ! Compromised meters at non-zero entries of a ! Stealth attacks can arbitrarily mislead PSSE by a = Hv ! Deleting related rows of H deems the system unobservable Liu, Ning, and Reiter, “False data injection attacks against state estimation in electric power grids,” ACM Trans. Info. and System Security , May 2011. 38
Non-stealth attacks ! Caveats " attacker knows H " linearization around x ! Designs for attacker and defender [Kosut-Jia-Thomas-Tong’11] " defender solves an l 1 -norm penalized GLRT " attacker trades identifiability for state divergence ! Topology attacks [Kim-Tong’13] ! Nonlinear AC model attacks [Zhu-GG’12] ! Summary on PSSE 39
Distributed PSSE vertically organized markets control areas with thousands of buses deregulation tie lines long-distance power transfer dense instrumentation privacy policies 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 v 2 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 ! Parallel and cascade (KF-type) solvers [Schweppe et al � 70] ! Arbitrary interconnection graph " Coordinator needed [Zhao-Abur � 05], [Korres et al ’11] ! Decentralized solvers " Block Jacobi [Conejo et al � 07]; consensus averaging [Xie et al � 11] ! Caveats: Local observability, shared interconnection states, slow convergence (if ensured), parameter tuning 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. 42
ADMM solver ! Local linear(ized) model ! Regional PSSEs ! Coupled local problems ! Framework features: minimum inter-area exchanges, linear convergence guaranteed, disclosure policies ! ADMM-based semidefinite programming (SDP) [Zhu-GG � 12] " 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
Decentralized LS-based PSSE S 1. S2. MSE(decen.[Xie etal]-centralized) Mean Square Error MSE(decentralized-true) L. Xie, C. Choi, and S. Kar, “Cooperative distributed state estimation: local observability relaxed,” Proc. IEEE PES General Meeting , July 2011 . 44 44
Decentralized bad data cleansing ! Pertinent bad data-aware model 1 2 k z � Hx � o k 2 f ( x ) := min 2 + λ k o k 1 ! Reveal outliers via o M X h ( z m � h T = m x ) m =1 S1 . S2. S3. 45
D-PSSE on a 4,200-bus grid 46
Phasor measurement units (PMUs) 47
PMU versus SCADA SCADA PMU measurements power, voltage, voltage & current phasors , current magnitude frequency (derivative) meas. model non-linear linear reporting rate one every 1-4 sec 30-60/sec wide-area sync poor precise (GPS signal) “It � s like going from an X-ray to an MRI of the grid,” Terry Boston, PJM CEO A. G. Phadke and J. S. Thorp, Synchronized Phasor Measurements and their Applications , Springer, 2008. 48
PMU architecture analog X m voltage/current 2 cos φ f s = Nf 0 √ transformers − X m X m cos(2 π f 0 t + φ ) 2 sin φ √ GPS ! 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. 49
System architecture ! Single PMU can measure voltage and several line currents ! Phasor data concentrator (PDC) " collects data streams from several PMUs " time-alignment, local cleansing, data compression 50
PMU uses Power system state estimation ! Challenges ! Solutions " different rates " SCADA estimates as priors " (non)linear models (rect./polar) " PMU at reference bus " phase alignment Monitoring, control, and protection (local and wide-area) ! Voltage stability ! Parameter estimation and dynamic line rating ! Oscillation and angular separation monitoring ! Visualization A. G. Phadke and J. S. Thorp, Synchronized Phasor Measurements and their Applications , Springer, 2008. 51
Deployment of PMUs Costs : acquisition, installation, networking Deployment: 2009: ~100; 2014: ~500 Q: Where should new PMUs be placed ? ! Criteria: topological observability [Emami-Abur’10] estimation accuracy [Li-Negi-Ilic’11] North American Synchrophasor Initiative, www.naspi.org 52
Optimal experimental design " PMU measurement model z n = H n x + ✏ n # SCADA prior x ∼ N (ˆ x SCADA , Σ s ) ! − 1 N X # MMSE covariance a n H T n Σ − 1 n H n + Σ − 1 Σ ( a ) = s n =1 " E-optimal design (NP-hard) λ max ( Σ ( α )) : a ∈ { 0 , 1 } N , a T 1 = k � min a # SDP relaxation λ max ( Σ ( α )) : a ∈ [0 , 1] N , a T 1 = k � min a " 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. 53
Additional learning and inference issues 54
Cascading failures ! Blackouts cost $150 billion/year ! Cascading: Lines exceed ratings and successively fail ! Linear DC model p = B θ θ T = [ θ T I θ T E ] " Internal-external states external internal system system ( N , E ) → ( N , E 0 ) " Pre- and post-outage graph Problem: Given pre- and post-outage internal states and θ I , θ 0 I , ˜ basecase topology B , find the line outage set E := E \ E 0 ! Exhaustive search [Tate-Overbye’09], [Emami-Abur’10] ; GMRF [He-Zhang’11] 55
Linear DC model with outages p 0 = p + n ! Nodal power injections remain unchanged B θ 0 + n B 0 θ 0 = B θ + n B ˜ θ = ˜ ! Post-/pre-outage ⇒ DC model ˜ B := B 0 − B ˜ θ := θ 0 − θ 1 X ˜ a l a T ! Exploit B = l x l l ∈ ˜ E ! Sparse representation over all transmission lines ( a T l θ 0 , l ∈ ˜ E B θ 0 = X ˜ a l m l + n , m l := x l ∈ ˜ , l / E 0 l 2 E H. Zhu and G. B. Giannakis ”Sparse overcomplete representation for efficient identification of power line outages,” IEEE Trans. Power Syst. , Nov. 2012. 56
Grid Laplacian ! Linear DC model p = B θ + n nodal voltage phases nodal powers n 6 = m x � 1 ⇢ P , m = n bus admittance [ B ] mn = mn matrix � x � 1 , m 6 = n mn ! From nodes to lines: weighted grid Laplacian ( m, n ) ↔ l m n . . ... . . . . +1 m -1 + 1 − 1 · · · · · · · · · X x − 1 X x − 1 a l a T B = . . ... l = . . mn l · . . +1 n -1 l ∈ E ( m,n ) ∈ E − 1 + 1 · · · · · · · · · . . ... . . . . ( a l := e m − e n ) 57
Lassoing line outages branch-bus ⇥ ⇤ B ˜ A := a 1 · · · a |E| θ = Am + n incidence matrix ( B , ˜ ! Given only , solve for ˜ B † ⇤ B † ⇤ θ I ) ⇥ ⇥ θ I = I Am + I n ! Sparse linear regression model with colored noise " Orthogonal matching pursuit (greedy) " l 1 -norm penalized regression (coordinate descent) Running times (secs.) IEEE 300-bus (detection probability) OMP CD 58
Electromechanical modes ! Oscillations in voltage, frequency, and power flows (grid breakup) ! Inter-area oscillations in 0.1-1 Hz ! Challenges in retrieving harmonics " nonlinearity " time-varying power systems " closely-spaced frequencies ! Linearized continuous-time differential equations x ( t ) = A x x ( t ) + B u u ( t ) + w ( t ) ˙ ! Probing signal u (t) ! Modes described by the eigenvalues of A x P. Kundur, Power System Stability and Control , McGraw-Hill, 1994. 59
Mode estimation (M1) Build A x based on grid model (scalability issues) (M2) Spectral estimates using measured x (t) ! Measurement types " ambient (regular) " ring-down (disturbance) " probing (engineered u (t) ) ! Batch modal analysis (Prony’s) ! Adaptive nodal analysis (LMS, RLS) ! Probing signal design for improved accuracy and minimal grid impact D. Trudnowski and J. Pierre, Signal processing methods for estimating small-signal dynamic properties from measured responses , Springer, 2009. 60
Load forecasting ! 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) ! Spatial scales per substation, utility, and interconnection ! Challenges: deregulation, demand response, electric vehicles 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 ! Important for asset owners, system operators, government ! Challenges " weather and load " hedging strategies " outages and security ! Approaches MISO market " 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] M. Amjady and N. Hemmati, � Energy price forecasting,“ I EEE Power Energy Mag., Apr. 2006 . V. Kekatos, S. Veeramachaneni, M. Light, and G. B. Giannakis, ”Day-Ahead Electricity Market Forecasting 63 using Kernels,” Proc. of Innovative Smart Grid Technologies , Feb. 2013.
Spatiotemporal price forecasting ( p ( t, n ) � f ( x t , n )) 2 + λ k f k 2 ˆ X Learning over pricing network f := arg min ! H f ∈ H t,n ! Separable kernel � � k (( t i , n i ) , ( t j , n j )) = k t k s ( n i , n j ) x t i , x t j ! Gaussian kernel k t : ! inverse Laplacian of k s : balancing authority graphs MISO; Jun-Aug 2012; 1732 nodes 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). 64
Clustering the grid ! Static partitioning into reliability regions ! Modularization facilitates " decentralized and parallel computing " minimum communication ! “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 65 reactive power balance,” IEEE Trans. Smart Grid , Dec. 2010.
Grid monitoring and learning Batch/adaptive Distributed Load, wind, Generalized prices Observability Forecasting Clustering PSSE Analysis L E A R N I N G PMUs Bad data Cyber-attacks E C O N O M I C Line O P E R A T I O N S outages M O N I T O R I N G 66
Economic Operation of Power Systems A. Gomez-Exposito, A. J. Conejo, and C. Canizares, Electric Energy Systems: Analysis and Operation , CRC Press, 2009. A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control , 2nd ed., Wiley, 1996. 67
Generation cost ~ P G 1 ~ P G 2 ! Thermal generators Load ! Power output (MW) P G i ! Generation cost ($/h or € /h) P D C i ( P G i ) P G Ng ~ Economic dispatch (ED): Find most economically generated power output to serve given load ! ED typically solved every 5-10 minutes 68
Optimizing ED ! Generation cost typically convex and strictly increasing C i ( P G i ) " Piecewise linear or quadratic C i ( P G i ) P G i N g X min C i ( P G i ) { P Gi } i =1 N g X subj. to P G i = P D i =1 P min ≤ P G i ≤ P max G i G i ! ED balances supply and demand most economically ! Convex optimization problem 69
Marginal price ! Lagrange multiplier for supply-demand balance λ N g X P G i = P D i =1 ! Optimal generation output P ∗ { C ( P G i ) − λ ∗ P G i } G i = arg min P min Gi ≤ P Gi ≤ P max Gi ! ED optimizes net cost: Generation cost minus revenue ! Price ($/MWh or € /MWh) λ ∗ ! Prices Lagrange multipliers 70
Market participants Large Industrial Dispatchable Customers Generators Market Operator/ C i ( P G i ) Independent System Operator (ISO) Non-dispatchable Load-Serving Generators Entities Serving commercial and e.g., renewable residential end-users energy sources Market clearing: Demand Optimal schedules (fixed or elastic) Generation and prices A. J. Conejo, M. Carrion, and J. M. Morales, Decision Making Under Uncertainty in Electricity Markets , 71 Springer, 2010, ch. 1.
DC power flow ! ED must account for transmission network constraints " DC approximation θ m θ n P mn ! Generator and load per bus P G m P D m m " At all buses: and p G p D ! Power flow from bus to bus x mn m n P mn = x − 1 mn ( θ m − θ n ) ! Define the matrix N l × N b A = branch-bus incidence matrix F = DA D = diag( { x − 1 mn } ( m,n ) ∈ E ) ! Power flows at all lines f = [ P mn ] = F θ 72
DC optimal power flow ! Power balance per bus; flow limit per line ! Convex optimization problem N b ! Locational marginal prices (LMPs) X min C m ( P G m ) p G , θ " Multipliers for nodal balance m =1 subj. to p G − p D = B θ | F θ | ≤ f max p min ≤ p G ≤ p max G G MISO market R. D. Christie, B. F. Wollenberg, and I. Wangensteen, “Transmission management in the deregulated 73 environment,” Proc. of the IEEE , Feb. 2000.
Contingencies ! System security must be ensured even under unplanned events " Line flow and bus voltage limits ! Set of credible contingencies K ! Contingency may include any of the following: k ∈ K P max ,k = P min ,k " Generation loss = 0 G m G m P max ,k G m − ∆ P k " Generation derating = P max G m G m " Demand variation P k D m = P D m + ∆ P k D m " Line outage gives rise to , , B k F k f max ,k ( p max ,k , p min ,k , p k D ) ∈ C k ! Notation for generation limits and demand G G ! North American Electric Reliability Council (NERC) N -1 security rule 74
Reserves ! Spinning reserve: Online generation capacity available to be deployed under contingencies ! Scheduling of reserves " Up-spinning ; down-spinning R down R up m m ! Reserve level available depends on generation schedule P G m R down R up m m P min P max P G m G m G m P k Adjusted generation G m 75
Security-constrained DC OPF ! Decide reserve levels and adjustments for each contingency N b X C m ( P G m ) + C up m ( R up m ) + C down ( R down ⇥ ⇤ min ) m m p G , θ , r up , r down m =1 { p k G , θ k } k ∈ K subj. to p k G − p k D = B k θ k ∀ k ∈ K p G − p D = B θ | F θ | ≤ f max | F k θ k | ≤ f max ,k ∀ k ∈ K p min ≤ p G ≤ p max p min ,k G ≤ p max ,k ≤ p k ∀ k ∈ K G G G G D ) ∈ C k ∀ k ∈ K Network constraints ( p max ,k , p min ,k , p k G G in base case r down ≤ p G − p min r up ≤ p max − p G G ; G Network constraints p G − r down ≤ p k G ≤ p G + r up at every contingency ∀ k ∈ K Reserve levels and adjustment constraints F. D. Galiana, F. Bouffard, J. M. Arroyo, and J. F. Restrepo, “Scheduling and pricing of coupled 76 energy and primary, secondary, and tertiary reserves,” Proc. of the IEEE , Nov. 2005.
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: P G m − P D m V m ! Reactive power injection: Q G m − Q D m ! Complex power flowing over line ( m, n ) S mn = V m I ∗ mn X I m = I mn ! Current injections i = Yv n ∈ N m 77
AC optimal power flow N b X min C m ( P G m ) p G , q G , v m =1 subj. to p G − p D + j ( q G − q D ) = diag( v )( Yv ) ∗ | Re {S mn }| ≤ f max |S mn | ≤ S max mn ; mn V min ≤ |V m | ≤ V max m m p min ≤ p G ≤ p max ; q min ≤ q G ≤ q max G G G G ! Quadratic equality constraints nonconvex problem ! Typical approaches rely on KKT conditions 78
SDP relaxation ! Canonical basis of { e m } N b R N b m =1 m = e H m v ( Yv ) H e m = tr[ e H m vv H Y H e m ] = tr[ Y H e m e H m vv H ] V m I ∗ ! Nodal balance constraint linear in V := vv H P G m − P D m + j ( Q G m − Q D m ) = tr[ Y H e m e H m V ] ! Line flow and bus voltage constrains also linear in V ! AC OPF with variables and additional constraints p G , q G , V rank[ V ] = 1 Nonconvex Drop V ⌫ 0 ! Works in many practical OPF instances and IEEE benchmarks ! Optimal in (three-phase) tree networks [Lam et al’12] [Dall’Anese et al’13] X. Bai, H.Wei, K. Fujisawa, and Y.Wang, “Semidefinite programming for optimal power flow problems,” Int. J. Elect. Power Energy Syst. , 2008. 79 J. Lavaei and S. Low, “Zero Duality Gap in Optimal Power Flow Problem,” IEEE TPS , Feb. 2012.
Multi-period scheduling ! Scheduling horizon { 1 , . . . , T } ! Unit commitment (UC) solved every day in many ISOs ! Binary variable: if gen. is ON at slot , 0 otherwise u t m = 1 t m ! Practical constraints for thermal generators ! Ramp-up/down P t G m − P t − 1 P t − 1 G m − P t G m ≤ U m G m ≤ D m ! Minimum up/down time " If a generator is turned on, it must remain on for the next slots T up m u t m − u t − 1 m , τ = t + 1 , . . . , min { t + T up m , T } ≤ u τ m " If a generator is turned off, it must remain off for slots T down m 80
Unit commitment ! Start-up/shut-down costs S t m } t m ( { u τ τ =0 ) " Depend on previous on/off activity T N b X X C t m ( P t G m ) + S t m ( { u τ m } t ⇥ ⇤ min τ =0 ) { p t Gm , θ t m ,u t m } m,t t =1 m =1 u t m P min G m ≤ P t G m ≤ u t m P max G m ; u t m ∈ { 0 , 1 } subj. to nodal balance constraints for every t line flow limits for every t ramp-up/down limits minimum up/down time constraints 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 DC power flow AC power flow ED DC OPF AC OPF Security Multi-period constraints on/off scheduling SC DC OPF DC UC SC AC OPF AC UC SCUC SCUC ! SCUC is solved for day-ahead market clearing in many ISOs 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 or 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. 85 K. Hamilton and N. Gulhar, “Taking Demand Response to the Next Level,” IEEE PES Mag., May 2010.
Demand-side bidding ! Large customers adjust consumption (ED with DSB) [Christie et al’00] ! Concave utility function : willingness-to-buy U j ( P D j ) U j ( P D j ) " Piecewise linear or quadratic P D j N g N d X X min C i ( P G i ) − U j ( P D j ) Maximize social welfare { P Gi } , { P Dj } i =1 j =1 N g N d X X subj. to P G i = P D j Supply-demand balance i =1 j =1 P min ≤ P G i ≤ P max G i G i P min ≤ P D j ≤ P max D j D j 86
DSB can reduce marginal price ! Consider one generator and one load ! Lagrangian function L ( P G , P D , λ ) = C ( P G ) − U ( P D ) − λ ( P G − P D ) ! Marginal price λ ∗ = d C ( P G ) = d U ( P D ) d U ( P D ) d C ( P G ) d P G d P D d P D d P G λ inelastic λ ∗ ! DSB elastic demand ! Slide from right to left on curve d C/ d P G ! DSB tends to decrease marginal price P G = P inelastic P G = P D D 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) Main Utility Grid Company Electricity Distribution Network 89
Adjustable power appliances ! Scheduling horizon { 1 , . . . , T } ! Power consumption of appliance p t a a ! Concave utility function U a ( p a ) ! Elastic loads, e.g., lights T X U t a ( p t U a ( p a ) = a ) p min a ≤ p max ≤ p t a a t =1 ! Energy loads, e.g., PHEV charging T ! T X p t X E min p t a ≤ E max U a ( p a ) = U a ≤ a a a t =1 t =1 p min a ≤ p max ≤ p t a a L. Chen, N. Li, L. Jiang, and S. Low, “Optimal Demand Response: Problem Formulation and 90 Deterministic Case,” in Control and Optimization Methods for Smart Grids, Springer, 2012.
Additional types of appliances ! Thermostatically controlled loads, e.g., AC ϑ t = (1 − γ ) ϑ t − 1 − γϑ amb ,t − η p t a ϑ t " Room temperature ; ambient (external) ϑ amb ,t ϑ min ≤ ϑ t ≤ ϑ max " Preferred temperature ! Interruptible loads: If on, they consume nonzero power p min a ≤ p max p t ≤ p t a = 0 or a a ! Noninterruptible and deferrable loads, e.g., dishwasher p t " Fixed load profile, can be shifted a " Integer constraints t 91
Energy consumption scheduling ! Cost of power for end-user: typically (piecewise) linear ! Tradeoff cost with utility T X c t X X p t min U a ( p a ) a − { p a } a ∈ A a ∈ A t =1 subj. to p a ∈ P a , a ∈ A ! Convexity depends on P a ! 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] 92
Cooperative DR ! Set of users served by the same utility company { 1 , . . . , R } ! Set of smart appliances per user A r r ! Power consumption p t ra ! End-user utility function U ra ( p ra ) R ! ! Cost of power procurement for utility company X X C t p t ra r =1 a ∈ A r Two-Way Communication Network (AMI) Main Utility Grid Company Electricity Distribution Network 93
Social welfare maximization R ! T R X X X X X C t p t min U ra ( p ra ) − ra { p ra } t =1 r =1 a ∈ A r r =1 a ∈ A r subj. to p ra ∈ P ra , a ∈ A r , r = 1 . . . , R ! Motivation: Reduce peak demand respecting users’ preferences ! Convexity depends on P ra ! Challenges " Distributed scheduling over AMI " Privacy issues 94
Solution approaches ! 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 s t R X X p t ra ≤ s t Supply-demand balance r =1 a ∈ A r " Lagrange multiplier for supply-demand balance λ t ! Upshot " Separate subproblems for utility and smart meters " Privacy respect 95
Distributed algorithm ! Schedule update: At the utility company and smart meters 0 ≤ s t ≤ s max { C t ( s t ) − � t ( ` ) s t } min prices ( T ) X � t ( ` ) p t min ra − U ra ( p ra ) p ra ∈ P ra t =1 a ∈ A r p t P ra ( ` ) total hourly consumption ! Multiplier update: At utility company !# + " R X X � t ( ` + 1) = � t ( ` ) + � p t ra ( ` ) − s t ( ` ) r =1 a ∈ A r 96
Lost AMI messages ! Messages in both ways may be lost " Not transmitted, due to failure " Not received, due to noise " Cyber-attacks a ∈ A r p t P ra ( ` ) ! 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 97 with lost AMI messages,” IEEE Trans. Smart Grid , June 2012.
Scheduling with DR aggregators ! Goal: DR from small-scale end-users into day-ahead scheduling ! Challenges G 1 " User preferences are private " Large-scale problem ! Approach: Decomposition algorithms A 2 A 3 BL 3 BL 2 " Subproblems for market operator and aggregators Market Aggregator Operator ! Outcome : Scalable distributed solution N. Gatsis and G. B. Giannakis, “Decomposition algorithms for market clearing with large-scale demand response,” IEEE Trans. Smart Grid , 2013. Y. Zhang, N. Gatsis, and G. B. Giannakis, “Disaggregated bundle methods for distributed market 98 clearing in power networks,” in Proc. IEEE Global Conf. Signal and Information Process. , Dec. 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
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