Distributed State Es.ma.on Algorithms for Electric Power Systems

Transcription

1 Distributed State Es.ma.on Algorithms for Electric Power Systems Ariana Minot, Blue Waters Graduate Fellow Professor Na Li, Professor Yue M. Lu Harvard University, School of Engineering and Applied Sciences Blue Waters Symposium

2 Outline Mo.va.on Problem Statement Method Results Conclusions 2

3 Outline Mo.va.on Problem Statement Method Results Conclusions 3

4 Mo.va.on: why we need to improve the electric power grid. Source: htp:// Communica.on Networks Source: ROBERT GIROUX/ Time Fuel supply Source: htp:// Source: maporheweek.blogspot.com Support for cri.cal infrastructure: Communica.on networks and fuel and water supply systems Prevent blackouts Realize a robust efficient grid under increased renewable energy produc.on Source: htp:// 4

5 Scien.fic Mo.va.on: beter understand large, complex interconnected systems. Scien#fic*Compu#ng* Distributed*Algorithms* Scalable*with*respect*to*size*of*:* Network** Data* Asynchronous*SeMng* Simula#on*of*complex,*dynamic*systems* Challenge:*coordinate*control,* sensing,*and*op#miza#on*in*a* decentralized*way.* Es#ma#on*&*Detec#on* State*Es#ma#on* Bad*Data*Detec#on* Decision*Making*Under*Uncertainty* Line*Outage*Iden#fica#on* Op#miza#on*&*Control** Op#mal*Power*Flow* Economic*Dispatch* Demand*Response* Wide* *Area*Control* 5

6 Improving power grid opera.ons via High Performance Compu.ng Power Grid Control Room (California) Por.on of Midwest Power Grids Poland Europe # ver.ces N # edges # generators Source: htp:// Blue Waters Supercomputer (NCSA & UIUC) Efficient matrix inversion for real-.me applica.ons New sensing technology (PMUs) produces a lot of data 30 samples/sec - > 2 million samples/day For ~100 sensors, ~2*108 samples/day Source: htp://engineering.illinois.edu 6 Source: Franz Franchea, CMU

7 Outline Mo.va.on Problem Statement Method Results Conclusions 7

8 State Es.ma.on & Distributed Algorithms Monitor the electric grid in real-.me State Es.ma.on task: Given noisy system measurements, infer the state of the system. Reference : [Schweppe 1970] Measurements consist of subset of: Power flows along transmission lines (edges). Power injec.ons and voltage at ver.ces. State is the voltage (phase angle and magnitude) at each vertex in the network. Non- linear measurement model Distributed vs. centralized Advantages of distributed approaches: 1) avoid communica.on botleneck 2) reduc.on in computa.on and memory requirements per area. Node(1( Central(Coordinator( Node(2( Node(3( Hierarchical(Communica5on(Scheme( Fully(Distributed(Communica5on(Scheme( ( 8

9 Mathema.cal Problem Statement Goal: From noisy system measurements, determine state,, at every vertex in the power network. Weighted non- linear least squares op3miza3on z x =[ V ] min x f(x) (z h(x)) T W (z h(x)) Itera3ve solu3on using Newton s method x (k+1) = x (k) [r 2 f(x (k) )] 1 rf(x (k) ) Main challenge : Matrix inversion requires full, global knowledge of matrix entries. How to calculate inverse in a fully distributed fashion? Idea : use matrix spliang techniques (R. Varga) In general, matrix spliang A = M + N provides an itera.ve approach to solve the linear system, Ax = y x t+1 = M 1 Nx t + M 1 y The sequence {x t } converges to its limit x as t!1 if and only if the spectral radius of the matrix M 1 N is strictly less than 1. 9

10 Outline Mo.va.on Problem Statement Method Results Conclusions 10

11 Matrix- Spliang for Distributed State Es.ma.on Let A r 2 f(x (k) ) 0 nx and Ē ii A ij j=1,j6=i A" D" E" ["""]"="["""]"+"["""]" ="(D"+"E)"+"(E" "E)" " M" N" }" }" Proposition: Let M = D + Ē and N = E Ē. 1 Then, for 2, (M 1 N) < 1. 11

12 U.lizing Sparsity to Enable Distribu.on M is diagonal and easy to invert distributedly. Each vertex needs only its local informa.on. Power systems follow Kirchoff s current & voltage laws è N has a dis.nct sparsity patern related to the power network topology. Voltage measurement θ b and V b Power flow measurement P ab Power injec.on measurement P b b b b a c a c a c d d d This sparsity in N is what allows for distribu.on with limited communica.on. At most need to communicate 2- hop neighbor informa.on. 12

13 Distributed Newton Method for DSE Power injec.on measurement at vertex b b :{θ a, V a } a c Each vertex is assigned an MPI process. MPI Graph Communicator (MPI_Graph_create) used to mimic structure of power network. 13 Icon Source: htp://icons8.com/license/

14 Distributed Newton Method for DSE Power injec.on measurement at vertex b b :{θ b, V b } :{θ b, V b } a c Each vertex is assigned an MPI process. MPI Graph Communicator (MPI_Graph_create) used to mimic structure of power network. 14 Icon Source: htp://icons8.com/license/

15 Distributed Newton Method for DSE Power injec.on measurement at vertex b b :{θ c, V c } a c Each vertex is assigned an MPI process. MPI Graph Communicator (MPI_Graph_create) used to mimic structure of power network. 15 Icon Source: htp://icons8.com/license/

16 Distributed Newton Method for DSE Power injec.on measurement at vertex b b :{θ c, V c } :{θ a, V a } a c Each vertex is assigned an MPI process. MPI Graph Communicator (MPI_Graph_create) used to mimic structure of power network. 16 Icon Source: htp://icons8.com/license/

17 Outline Mo.va.on Problem Statement Method Results Conclusions 17

18 Results: Accuracy Performance I Objective Function Objective Function Vertex Network Newton Iterations 1354 Vertex Network Newton Iterations Objective Function Objective Function Newton Iterations 3374 Vertex Network Each uses 15 matrix- spliang itera.ons. Number of Newton itera.ons needed for convergence depends on power network size Vertex Network Newton Iterations

19 Results: Accuracy Performance II Objective Function Vertex Network Matrix Splitting Iterations: 100 Matrix Splitting Iterations: 200 Matrix Splitting Iterations: 400 Matrix Splitting Iterations: 600 # Matrix- Spli?ng Itera3ons Run3me (minutes) Newton Iterations Tradeoff between convergence and run.me for different number of matrix- spliang itera.ons. 19

20 Results: Accuracy Performance III Matrix Splitting Convergence: 3374 Vertex Network Vertex 254 Vertex 316 Vertex Absolute Error Absolute Error Zoom Inner Loop Iterations Matrix Splitting Iterations Convergence of matrix- spliang itera.ve scheme varies between different ver.ces in the power network. 20

21 Results: Timing Performance I Misc. Setup Matrix Splitting Communication Matrix Splitting Computation Outer Loop (Newton Update) Runtime (s) Network Size Using same number of itera.ons (15 matrix- spliang, 40 Newton), compare how computa.on and communica.on scale with power network size. 21

22 Results: Timing Performance II 1.6 Runtime (s) Misc. Setup Matrix Splitting Communication Matrix Splitting Computation Outer Loop (Newton Update) Process or Vertex Number For 14- vertex power network, study how.me spent on communica.on and computa.on varies among different ver.ces. 22

23 Outline Mo.va.on Problem Statement Method Results Conclusions 23

24 Conclusions Use of Blue Waters Allows tes.ng of algorithms on large- scale, realis.c systems. Each vertex in the power grid is assigned an MPI process. Acknowledgement: This research is part of the Blue Waters sustained- petascale compu.ng project, which is supported by the Na.onal Science Founda.on (awards OCI and ACI ) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana- Champaign and 24 its Na.onal Center for Supercompu.ng Applica.ons.

25 Conclusions Main result Developed a new fully distributed state es.ma.on algorithm using matrix- spliang techniques. Requires limited sharing of informa.on between neighboring ver.ces. Saves memory resources. Implementa.on available in C++ using MPI. Future Work Develop new local pre- processing and asynchronous communica.on schemes. Improve the robustness of state es.ma.on with respect to imperfect communica.on as well as reduce network traffic. Thank you to the Blue Waters Graduate Fellowship Program and NSF for suppor.ng this work, as well as to the NCSA staff and point of contact Craig Steffen, for their help in working with Blue Waters. 25

26 References F. C. Schweppe and J. Wildes "Power system sta.c- state es.ma.on, pt. I: exact model", IEEE Trans. Power Apparatus and Systems, vol. PAS- 89, pp A. Minot, N. Li, and Y. Lu, A Distributed Newton Method for Power System State Es.ma.on, SubmiTed to Trans. on Power Systems, 2015, htp://scholar.harvard.edu/files/aminot/files/distr_se_ tpsv1.2.pdf A. Minot and N. Li, A Fully Distributed State Es.ma.on Using Matrix Spliang Methods, American Control Conference, A. Abur and A. G. Exposito, Power System State Es.ma.on: Theory and Implementa.on. CRC Press, L. Xie, D.- H. Choi, S. Kar, and H. V. Poor, Fully Distributed State Es.ma.on for Wide- Area Monitoring Systems, IEEE Trans. Smart Grid, vol. 3, no. 3, pp , V. Kekatos and G. Giannakis, Distributed Robust Power System State Es.ma.on, IEEE Trans. Power Systems, vol. 28, no. 2, pp , May R. Varga, Matrix Itera.ve Analysis. Springer Series in Computa.onal Mathema.cs, E. Wei, A. E. Ozdaglar, and A. Jadbabaie, A Distributed Newton Method for Network U.lity Maximiza.on, Conference on Decision and Control, pp , D. P. Bertsekas and J. Tsitsiklis Parallel and Distributed Computa;on - Numerical Methods, 1989 :Pren.ce Hall 26

27 Backup Slides 27

28 Kirchoff s Voltage and Current Law Sum of all currents leaving and entering a node Σ I = 0 Total voltage around a closed loop Σ V = 0 28

29 Measurement Model Reference: [Wood & Wollenberg] B ab : susceptance of transmission line b/t ver.ces a and b G ab : conductance of transmission line b/t ver.ces a and b 29

30 Measurement Model 30