dvaced Methods for Security ostraied Fiacial Trasmissio Rights (FTR) Stephe Elbert, Steve.Elbert@PNN.gov Kara Kalsi, Kurt Glaesema, Mark Rice, Maria Vlachopoulou, Nig Zhou Pacific Northwest Natioal aboratory, Richlad, W FER Staff Techical oferece o Icreasig Real-Time ad Day-head Market Efficiecy through Improved Software (Docket No. D0--003) Jue 5-7, 0 Washigto,D
Motivatio - halleges with FTR alculatios Fiacial Trasmissio Rights (FTR) improve power market operatio efficiecy by providig fiacial tool to hedge price risk associated with cogestio Mitigate icetives for iefficiet trasmissio ivestmet FTR auctio is formulated as a liear programmig optimizatio problem FTR calculatios are computatioally expesive because arge umber of security costraits (N- cotigecy aalysis) May FTR variables (obligatory ad optioal FTR bids) Multiple time periods (security costraits coupled & o. of costraits icrease expoetially with o. of categories) FTR computatio must be fiished i time to improve market efficiecy
Objectives Develop iovative mathematical reformulatio of the FTR problem ompare multiple solvers for FTR computatios Developed approaches will be able to Support N- Simultaeous Feasibility Test (SFT) e.g. D cotigecy aalysis Support both optioal ad obligatory FTR bids Support multi-period FTR calculatio (e.g. witer, summer ad aual) lgorithms desiged to solve FTR problem should be parallelizable to support large-scale implemetatio i a cloud eviromet 3
Problem Formulatio Power flow costraits is (sigular) admittace matrix θ i are the bus voltage agles is FTR locatio matrix Thermal costraits coverts voltage agles to lie flows i are trasmissio lie limits id-i costraits ombie dimesio is (costraits x bids) m m m m FTR FTR FTR b b b b m FH m FH FH FTR FTR FTR 0 0 0 b m m m m b b b FTR FTR FTR 3 3 3 3 33 3 3 3 3 to solve x b 4
Stadard FTR Solvers PEX (idustry stadard) Primal simplex; most basic P solver method Updates tableau cotaiig objective fuctio ad costrait iformatio at every iteratio osistetly slower o FTR tha dual simplex Dual simplex; fastest of the PEX methods Similar to primal simplex method, but uses dual formulatio of the P to improve covergece time of optimizatio ore computatio is a liear solve; scales as cube of size arrier; a iterior poit method (best for large sparse problems) primal-dual logarithmic barrier algorithm that geerates a sequece of strictly positive primal ad dual solutios Fewest iteratios but each is more computatioally itese 5
PNN FTR solver Parallel daptive No-liear Dyamical System (NDS) Trasform P ito coupled set of o-liear dyamical equatios Dyamical system coverges to stable states which are solutios of primal ad dual P problems respectively No-liear Dyamical System Primal maximize c T x subject to x b ad x 0 Dual miimize b T y subject to T y c ad y 0 dx = k dt c T y + k dy dt dy = k dt b + x + k dx dt k = K i i =,, M k = k Kerel is a pair of easily parallelized matrix-vector operatios: scale as square of problem size (costraits x variables) 6
Implemetatio otes Obligatory ad Optioal bids sorted ito separate blocks Obligatory bids may use dese matrix arithmetic Optioal bids use sparse matrix arithmetic (up to 50% sparse) matrix segmets commuicated oce at begiig ad T stored separately to maximize uit stride access Global sums for product vectors ad distributio of x ad y vectors are the oly commuicatio after iitializatio Symmetry of Obligatory bids reduces computatio by two 7
Validatio test cases ases ostraits ids a. WE 30 sigle period 5,36 5,790 a. WE 30 sigle period & may bids 5,36 00,000 3a. WE 30 multi-period 0,74 7,370 4a. WE 30 multi-period & may bids 0,74 300,000 b. WE 00 sigle period 9,094,455 b. WE 00 sigle period & may bids 9,094 00,000 3b. WE 00 multi-period 38,88 67,365 4b. WE 00 multi-period & may bids 38,88 300,000 WE 30 & 00 model power flow o trasmissio lies operatig at mi of 30 kv (,930 buses ad,68 braches) 00 kv (7,485 buses ad 9,547 braches) Multi-period problems have idepedet periods plus a couplig block FTR bids PF (witer) PF (summer) ids (witer) ids (summer) ids (aual,) 8
Results sigle period cases WE 30 WE 00 Primal simplex takes 0 mi, dual simplex ad serial NDS ( core) takes miutes Parallel NDS is six times faster tha dual simplex (PEX) t 4 hours, dual simplex ot yet coverged Parallel NDS 46X faster tha dual simplex 9
Results sigle period & may bids (00,000) WE 30 WE 00 NDS 56-core ad PEX comparable results (cross at 53 secods) NDS 00 times faster whe crossig PEX curve NDS scalig well 0
Results two period (summer/witer) cases WE 30 WE 00 Serial NDS is faster tha PEX NDS 8-core is 7 times faster The bigger the problem, The faster the relative performace PEX o loger practical time is divided by 0 ad ot coverged NDS 56-core is 85 times faster.7 billio o-zero matrix elem.
Results two periods & may bids (300,000) WE 30 WE 00 dditioal cores ad code improvemets solutio i uder 4 hours 5.3 billio o-zero matrix elem.
Real-world data PEX time per iteratio slows by 85x from begiig to ed due to backfill 3
Real World Data PEX time per iteratio slows by 69x from begiig to ed due to backfill 4
Summary Developed ovel o-liear dyamical system based FTR solver Easily parallelized to solve large liear programmig (P) problems for FTR applicatio withi few hours (cloud compatible) Parallel NDS more computatioally efficiet tha PEX for P omputatioal kerel of PEX is liear solver that scales as cube of problem size NDS kerel is matrix-vector multiplicatio that scales as square NDS avoids backfill (filig i zeros) of coupled blocks Maitais umerical stability through usig oly origial matrix Uses dese algorithm for obligatory bids, sparse (50%) for optioal bids Half the arithmetic for obligatory bids (two ier products differ oly i sig) Data loaded efficietly i parallel Further ehacemets Further improve parallelizatio (asychroous commuicatio, sparse ops) Refie adaptive time steppig ad explore ode time steppig for faster covergece 5
Future Develop quadratic programmig capability Improved FTR costraits Explore other applicatio eedig P ad/or QP capability Trasmissio plaig ocatioal Margial Pricig (MP) Optimal Power Flow (OPF) Explore usig method with discrete problems Mixed Iteger Programmig (MIP) Resource Schedulig ad ommitmet (RS) (aka Uit ommitmet) Stochastic RS 6
ckowledgemets For providig lstom data sets Xig Wag David Su Support This work was supported by the U.S. Departmet of Eergy s Office of Electricity Delivery ad Eergy Reliability, dvaced Grid alytics program at the Pacific Northwest Natioal aboratory. Pacific Northwest Natioal aboratory is operated for the U.S. Departmet of Eergy by attelle Memorial Istitute uder otract DE-05-76R0830. 7
Refereces Kalsi K, ST Elbert, M Vlachopoulou, N Zhou, ad Z Huag. 0. "dvaced omputatioal Methods for Security ostraied Fiacial Trasmissio Rights." to be preseted at the IEEE Power ad Eergy Society Geeral Meetig 0. -7 July 0, Sa Diego. Elbert ST, K Kalsi, M Vlachopoulou, MJ Rice, KR Glaesema, ad N Zhou. 0. "dvaced omputatioal Methods for Security ostraied Fiacial Trasmissio Rights: Structure ad Parallelism." to be preseted at the IF (Iteratioal Federatio of utomatic otrol) 8th PP&PS (Power Plat ad Power Systems otrol) Symposium, -5 Sept. 0, Toulouse, Frace. 8