A LASSO-Inspired Approach for Localizing Power System Infeasibility

A LASSO-Inspired Approach for Localizing Power System Infeasibility Shimiao Li, Amr itanshu Pandey, Aayu shya Agarwa l, Marko Je reminov, Larry P ileggi Dept. of Electr ical and Computer Eng ineering Carnegie Mellon University Pittsburgh, PA, USA Abstract — A method is propose d to i dentify and localize the cause of network collapse with augmented po wer flow analysis for a grid model with insufficient resources. Owing to he avy network loading, insufficient generation, component failure s and unavoidable d isturbances, power grid mod els can sometimes fail to converge to a fe asible soluti on for a stea dy-state power flow study. For applications such as system expa nsion planning, it i s desirable to lo cate the system buses that are contributing to network infeasibilities to facilitate correc tive actions . This pa per proposes a novel LASSO-inspired re gularization of the power flow matrix that enforc es sparsity to l ocalize and qu antify infeasibilities in the network . One of the examples demonstrat es how the proposed method is capable of localizing a source of blackout to a sin gle dominant bu s o n an 80k + bus eastern interconnection model . Index Terms — Equivalent circ uit formulation, infeasibility localization, LASSO, power system modeling , sparsity I. I NTRODUCTIO N To ensu re grid reliability and security , existing and planned power systems are evaluated on whether they can s urvive critical co ntingenc ies while serv ing curren t or forecasted loads . Often, due to severe contingenc ies, h eavy lo ading, and other limitations, th e simulatio n in dicates network collapse , which corr esponds to a grid that h as likely blacked out [1] . I n a tradition al p ower f low study , this collapsed grid state corresponds to no solution , and is characterized by divergence of the simulation [2]. Recently, methods [3],[4] have been developed that instead provide an infeasible po wer flow solution for such collapsed grid states. These infeasibility-b ased power-flow methods can converge for a collapsed grid state; however, th ey do n ot provide specific cause o f power outag e, nor do they identify localized lo cations that are disrupting system security and robustness. In most situatio ns, it would be d esirable to know the smallest possible set of d ominant nodes that are causing system collapse with so me quantifiable m etric. Accurate and efficien t loca lization of th is dominan t set of nodes ide ntifies the deficiency of power (real and reactive) and highlig ht s some critical lo cations for special attention in the planning process. For in stance, consider reactive power planning (RPP) p roblems [5]-[8] th at aim to find the optimal allocation of r eactive power support throug h capacitor b anks or FACTS d evices such as st atic VAR compensato rs (SVC). Such problem s correspond t o finding the sp arsest reactive power compensation vector that satisfies system p ower balance and oper ation limits in an optim ization-based power flow study. However, convergence o f such optimization-based studies becom es more difficult with increa sed system size and operating limits. Most state - of -the-art placemen t planning strategies [5] -[8] ar e only shown to han dle small cases wit h hundreds of buses or less and are known to suffer from lac k of robust convergen ce. Instead, a sparsity enfor cement method is preferred. The objective of th is method will be to provide a sparse set of nodes , that alon g with q uantified infeasibility power, can be added to each of the corresp onding set of nodes to make th e model feasible. This paper proposes a n ovel method to loca lize infeasibility within power grid models that have no solution . Unlike existing formulation s [3],[4] that distribute sy stem infeasibilities across all b uses in the system via minimization of the sum o f square o f their v alues (L2 -norm), the proposed approach localizes the infeasibility to a few system buses. The ap proach is inspired by LASSO [9],[10], a method that is us ed to enforce sparsity in featur e selection of a model by L1- regularization . In our approach, th e sparsity is enfor ced on the infeasibility solution vector to obtain infeas ibilities at on ly the dominant nodes in the system. This paper defines a new approach of enforcing sparsity such that the infeasibilities corresponding to geographically localized bu ses are correlated through a bus-wise sparsity enfo rcer . Mathematically, this pro blem is formulated as a no n - convex optimization problem and is imp lemented b ased on an equivalent circuit formulation (ECF) that has bee n demonstrated to ensure co nvergence for power flo w analysis and associated optimization application s (to a local optima) , such as that fo r identifying infeasibility . Since the power system is highly nonlinear, the major objective of our method can be summarized as follows: i. To reach a sufficiently sparse solution; i.e ., supply additional r eal or reactive power at minimum nu mber of possible lo cations for an infeasible n etwork model. ii. To f acilitate robu st convergen ce for larg e-scale systems. Section I I gives a background overview of the ECF [3],[11] ,[12] approach. Section III lists a series of power flow © 20 20 IEEE. Personal use of this material is permitted. Permission from IEEE mu st b e obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising o r p romotio nal purposes, creat ing new colle ctive works, for resal e or redistri bution to ser vers or lists, or re use of any copyri ghted compo nent of this w ork in other w orks. related f ormulations to illustrate the quantific ation of system infeasibility. Section IV presents the exploration of a spars ity enforcing me chanism, insight into problems with L1 - regularization based op timization problems , the p roposed infeasibility localization algorithm, and applicatio n to large scale systems with reliable c onvergen ce. Section V p resents some exper imental results on several larg e cases that include the U. S. Eastern Interconnection sized 80k + buses network , followed by o ur conclusions in section VI . II. B ACKGROUND A. Notation Table 1 s hows the symbols used in t his pa per. T ABLE 1. S YMBOLS AND D EFINITIONS Symbol Interpretation                   Real/imaginary voltage/current; Voltage magnit ude/angle; active/re active power  Power flow sol ution vector ,   󰇟     󰇠     Real or imagniar y infeasibility c urrent at bus #i   Infeasibility curre nt vector  󰇛  󰇜  0 KCL equations at a ll bus es  Sparse goal: to localize infeasi bility to k locations B. E quivalent Circuit Fo rmulation (ECF) A circuit-th eoretic formulation for power grid analysis was developed in [3],[11] ,[12]. Instead of describing components with ‘PQV’ par ameters , th e ECF framework models each component w ithin the p ower grid as an elect rical circuit element character ized by its I-V relationship . For computati onal analyticit y, the comple x relatio nships are split into real and imaginar y sub-circuits whose nodes corresponds to pow er system buses [3][11][12] . Table 2 shows a simple comparison between t he tra ditional PQV form ulation a nd ECF . T ABLE 2. C OMPARISON BETWEEN FORMULATIONS Property Comparison PQV formulati on ECF (I - V formu lation) Coordinate Polar Rectangle State variables             Network bala nce Zero power misma tch Zero current m ismatch Governing equatio ns Power balance eqn s KCL eqns Network constr aints Non-linear Linear Loads and generators Linear Non-Linear Under this ECF framewo rk, all branches, e.g. transmission lines, transformers, a nd shunts , ar e linea r comp onents, d ue to the intrinsic linearity of their I-V relationships. Other components such as generators and lo ads have nonlinear I- V models. The system balance is expressed b y a set o f nonlinear KCL equations:  󰇛  󰇜   (1) The se equations can b e iteratively linear ized and solved via Newton Raphson (NR) using bus voltage variables. Importantly, since ECF enables p ower systems of any size to be efficiently simulated as an equivalent circuit, numerous convergence techniques that were develo ped for circuit simulation (e.g . SPICE [1 5]) can be applied. III. P OWER FLOW RELAT ED PROBL EMS Next, we presen t a series o f power flow formulatio ns that introduc e ou r pro posed method fo r quantifying and lo calizing in feasibility. A. ( Problem Approa ch 1) Traditiona l power flow Traditional power flow outputs the bus voltag e solution by iteratively solving th e nonlinear n etwork balance eq u ations in (1). For a feasible network model, the traditional power flow converges to a fea sible solution; however , if no feasible solution exists for the n etwork, the methodology diverges resulting in no u seful solution. B. ( Problem Approa ch 2) Infea sibility-quantified power flow To av oid divergence with out providing informa tion , an approach to cap ture the infea sible-quantified power flow was developed. To effectively d istinguish betwe en hard- to -solve network case from infeasible netwo rk case , extensio ns have been p roposed and developed for both the ECF-based [3][11] and traditional [4] methods to quantify the potential infeasibility within th e g rid. The ECF-based appro ach [11] introduce s ‘ infe asibility current ’          at each bus #  . These values rep resent compensation term s that capture how much additional real/imaginary current flow is needed at each bus to make the network b alance cond itions hold . The infeasib ility- quantified power flo w study is formulated as a non -convex optimization p roblem:       󰇻  󰇻          󰇛  󰇜      (2) where   is infeasibility current vector that contains      for  . This formulation can clearly identify a fea sible case f rom an infeasible o ne: • Conver gence with zero infea sibility currents ev erywhere denotes system b alance. • Conver gence with nonzero      denotes an infeasible system with specific power flow deficiency at each bus  . • Diverg ence is totally attribu ted to insufficient convergence robustness of the algorith m. C. (Problem Approach 3) Proposed: I nfeasibility-localized power flow by L1 -regularization Mathematically, solving network balance equations with inclusion of   is an under-determined p roblem that h as inf inite solutions. Pro blem Approach 2 o utputs an optimal solution with minimal L2 norm, however, th is solu tion is not d esigned to b e spar se. Ther efore, the q uantit ies an d lo cations of the nonzero infeasibility currents is not necessarily an informative identifier of the dominant sources of the mo deled ou tage. Especially in larg e scale systems, we ca n hav e n umerous comparativ e infeasibility currents acro ss the grid. In reality, the most useful solution for expansion planning and corrective action is the not the one w ith the smallest L2 nor m a nd infeasible current values at m ultiple sources, but rather a solution that has non- zero infeasible currents in the least number of location s. To ad dress this, a classical m ethod for enforcin g sparsity can be applied using the L1 -norm in the objective function:       󰇻  󰇻     󰇻  󰇻      󰇛  󰇜      (3) However, this formulation neglects the correlatio n of real and imaginary counterparts during spar sity enf orcement , while in reality, the nonzero           at the same bus are often coupled terms emerging concurrently. Mo reover, the desired sparseness requ ires higher values assigned to regularization parameter  , m aking it difficu lt for th e algo rithm to converge. More detailed ex planations for th e co nvergence difficulty a re explored in section IV. IV. P ROPO SED M ETHOD To present our new approach, we first exploit some physical intuition of the system and develop a general methodology (within sub -section A) that enforces a sparse system solution using pr oblem approach 3. Also , in this sub-section, we d iscuss challenges of this approach from the convergence perspective. Next (within sub-sectio n B), based on the ob served me chanism, we propose our basic id ea of bus -wise sparsity enforcer, a novel regularization term to eliminate the aforemen tioned limitations. A. A n insight into sparsity inforcement by L1 reg ularization In the L1-r egularization-based sparse enfo rcement method ( problem approach 3 ) developed above, the inclusion of L1- norm leaves an undiff erentiable objection f unction. To tackle this, we introduce slack v ariable  an d convert the pr oblem in to the following constrained op timization form [ 13]:       󰇻  󰇻       (4)    󰇛  󰇜      (5)      (6)     (7) where the slack variable vector  represents the upper bound on the infeasibility currents ve ctor   . Each      corresponds to an upper- bound      such that             , as in (6 )-(7). We can write its Lagran gian function as:              󰇻  󰇻         󰇛  󰇜                     (8) T he perturbed KKT conditions of this proble m are:               (9)            ( 10 )             󰇛   󰇜 ( 11 )             󰇛 󰇜 ( 12 ) By further manipulation based on p roperties of L agrangian multiplier, the primal-du al pair     sho uld satisfy:            ( 13 ) This primal-dual relationship ca n b e clea rly illu strated by Figure 1, and inspires us to attach intuitiv e physical meanings : • Bus-wise Lag rangian multiplier      is a source of additional current flow into the network • Scalar  is a threshold such that any infeasibility quantities      below threshold are blocked out and o nly those above this threshold become the      ‘flow’ into the system. Figure 1. Relatio nship betwee n    and c: a blocki ng effect This rev eals a simple mechanism thro ugh which the threshold  enco urages a sparse solution by confining most   to nea r zero value. Whenever thresho ld  is add ed, the blocking effect redu ces the number of non- zero infeasibility sources in the network . As t he thr eshold  is increased, the number of non -zero in feasibility source s decrea se, and any remaining n on-zero infeasibility sources ad just their v alue to make the netwo rk feasible. Therefore, with a high enough threshold v alue, only a few sour ces turn out to be abo ve threshold and appea r as nonzero elements in   . In summary , o ur a pproach utilizes that: raising the va lue of c encourages more near - zero    elements by making the threshold hard- to -pass. However, there exists serious conv ergence p roblem with a single scalar  as tun ing par ameter for regularization. This challenge can be characterized by an u nwanted trade-off & inflexibility . Let u s illustrate this fur ther. With  repr esenting the upper bou nd of   , if there exists nonzero infeasibility (e.g.       ) at bus  , d ue to the minimization of  in the objective function , the u pper bound tends to b e very tight (i.e.,           ). Hence , if we uti lize a single lar ge scalar  value to achieve a su fficiently sparse solution, the tightness property of the algorithm results in convergence difficulties due to the s teep and highly n on -linear regions of the complem entarity slackness condition s giv en by ( 11 )-( 12 ). Figure 2. High c value causes st eep convergen ce region on the complementary slackness cur ve: (a)upper bound curve               , (b)lower bound curve               . When      , we have            . (             ) converge o n a difficult reg ion of the u pper curve This problem can be illustrated in Figure 2. As the number of inf easible bu ses increases, n umerous buses enco unter difficult steep region s of this kind , making it difficult for the algorithm to converge. Thus, the selection of the value of th e  parameter is a      trade-of f betwe en sufficient sparsity an d ro bust co nvergence, both of wh ich are essential for meeting our eventual goal . With  being a single scalar value, ther e is little freedom fo r us to manipulate its v alue and achieve the desired performance. B. P roposed metho d: Bus-wise sparsity enfo rcer To address th e af orementioned challenges, we p ropose a new method th at d efines thr eshold   for each bus  . T his   parameter is a bus-wise sparsity en forcer such that, according to the thr esholding effect, raising   encourages a zero      value at bus  in the solution. The infeasibility localization problem can now be reformulated a s problem 4. (Problem Approach 4) I nfeasibility-localized power flow by bus -wise sparsity enforcer       󰇻  󰇻       󰇛          󰇜      󰇛  󰇜      ( 14 ) In this approach, we convert a single scalar  to a vector of bus -wise sparsity enforcers. Then , to determine the values of   , we use the following ass umptions that are based on the grid physics: • Unev en distribution of in feasibility sources: system infeasibility is caused by and can be c haracterized by failures on isolated locations, rather th an outages of equal seriousness at each b us. • Ther e is a high probability that the dominant source s (locations) of failure in the system are reflected by the nodes with h ighest magnitude of      in the simulatio n. Based on these assumptio ns, we can simply ma ke flexible adjustments to   at each bus, accor ding t o the qualitativ e classification of bus-wise in feasibilities, as shown in Algorithm 1. For simp lification an d efficiency, we simply classify all buses into ‘ major ’ and ‘ minor ’ categor ies, according to their infeasibility current m agnitude and the sparsity goal. For bus es in the ‘major’ group, i.e. with high infeasibility qu antities (        󰇜 , we a ssume th at th ey are very likely th e dominan t sources of failu re and assign a low value   . This encourage s nonzero infeasibility curr ent on those locations. Fo r buses in the minor group, we assign a higher threshold   such that we can force their inf easibility values to zero or near zero values. Algorithm 1: B us -wise Sparse Enforce r Assignment Input : sparse goal k, threshold (      ), infeasibili ty current   Output : updated b us-wise sparsity e nforcer           1. Calculate infe asibility curre nt magni tude at all buses   2. Classify bus c ategory: ‘major’ buses index:    󰇛    󰇜 ‘minor’ buses index:   contains th e remaining bus es 3. Assign             󰇛   󰇜     Our in feasibility lo calization m ethod is summarized in Algorithm 2, where k defines the n umber of locations wh ere non -zero value o f infeasibility sour ces might b e allowed. From another intuitive viewpoint, this method unevenly penalizes infeasibility valu es at differ ent buses . By assigning high   values to certain buses, we de liberately attach high penalty to infeasibility currents in those buses, thereby forcing the infeasibility cu rrents to make the n etwork f easible from o ther sets of b uses with lo w   attached to them. More im portantly, this (      ) config uration remov es the need for high valu es of parameter   , as spar sity is dependent on the ratio of    and   , not the abso lute value of th e threshold. Simple p rinciples for selecting (      ) are: •   is ch osen to be suff iciently larger than      such that ‘minor’ infeasibility sources result in zero or near zero values. This en ables sufficiently sparse solution with small enoug h (      ) values, thereby avoiding convergence difficulties. •   is chosen to be sufficien tly lower than   such that the threshold is ‘easy - to - p ass’ for both          making nonzero     ,     coexist at in feasible buses. Th is is a necessary condition for practical application s. Due to the nature of th e p ower flow equations, grid devices provide both real and imaginary curren ts. Therefore, for any corrective actions, it is preferab le to achiev e sparse solutions th at h ave infeasibilities localized to the fewe st number of buses, rather than fewe st n umber of nonze ro       . Additionally, if t he k-sparse g oal is n ot practical,      in the ‘minor’ group leaves r oom for in feasible sources on more than k locations, and the final solu tion can be (    )-spar se. Algorithm 2: Infe asibility L ocalization with k-sparse g oal Input : testcase, i nitial guess           , sparse goal k , thres hold (      ) Output : a sparse   vector 1. Initialize       2. Bus-wise sparse enforcer ass ignment 3. Infeasibility-localize d power flow by bus-wise sparsity enforcer ( Problem4 ) C. Extension to large- scale systems For a practical large-scale power system, we do not have accurate k nowledge in advance about the sever ity of the system collapse, and therefore, it is hard to define a reasonable guess of th e k -sparse goal. Importantly , since infinite possible combination s o f {      } can m ake the system netwo rk balance equations correspond to a feasible network , the ‘major’ locations in a den se solution are likely to be the dominant sources with a hig h probability; however, we m ust note that this is not always tru e. With these co nsiderations we extend our method to large - scale networks by iterativ ely ad justing sparse enforcers and Algorithm 3: Infe asibility L ocalization for l arge-scale systems Input: testcase, shri nkage rate r Output : a sparse   vector 1. I nitialize           by infeasibi lity-quantified po wer flow ( Problem2 ) 2. Initialize 󰇛     󰇜 3. Initialize sparsi ty goal:       3. while not sparse enough, do Bus-wise spa rse enforcer ass ignment Infeasibilit y Localization w ith k-sparse goal Check curre nt solution spar isty  Update spars ity goal:      ( Optional ) ad just 󰇛      󰇜 if needed ( Optional ) ad just shrinka ge rate  i f needed gradually reaching sp arser solution from d enser ones. Fo r robust convergen ce, we start from a dense solution from ( problem approach2 ) r obust regular power flow [3][14] with quantified infeasibility in all locations an d gradually update the k-sparse goal by some shr inkage rate. This is equivalent to splitting the original p roblem into a series of subp roblems, where each subproblem uses a solution fro m the previous one as its initial guess, and easily reach es its optimal solution within a few iteratio ns. Ou r method is shown in Algorith m 3. V. E XPERI MENTS This paper conducts experiments to p rove the efficac y and scalabilit y of our proposed method. To create an infeasible scenari o (past the nose curve) on these cases, we incr ease the ir loading factor  . An d paramet ers of our proposed met hod are set to default val ues    10     0.1    0 .75 . We first test standard CASE14 which is infeas ible at   4.5 . Table 3 prese nts infeas ibility, quantifi ed first by infeasibl e power flow [3] ( Probl em2 ) and spa rsity enforcement using L1 - regulari zation ( Problem3 ) and our proposed method ( Problem4 ). Comparis on shows t hat our met hod reac hes 1- sparse solut ion and localizes infeas ibility to bus#14, where as the standard infeasibil ity approach [3] localizes infeasibili ty to almost all buses making the approach imp ractical for expansi on plannin g or appl ying c orrectiv e action. T ABLE 3. M ETHOD COMPARISON RESULTS ON CASE 14 Next, we test 5 large syst em cases. Table 4 shows our method efficiently localizes sy ste m infeasibilit y to sparse distrib utions . T ABLE 4. R ESULTS VI. C ON CLUSION This paper presents a novel app roac h t o localize the s ource o f infeasi bility in a grid network model . This is mathemat icall y equivale nt to finding a sparser soluti on to an underdetermi ned nonlinear system. With L1-regulariz ation sufferi ng from limited sol ution sparsit y due to the unwa nted t rade-o ff and lack of flexibilit y in the parameter adjustment , we propose a new method based on the physics-bas ed models and mechanis ms correspon ding to bus -s parsit y enf orcement o f the L1 -norm. The prima ry contri butio ns of our a pproac h are: • Definition of bus -wise spars e enforcer   to replace the scal ar para meter in L1-norm . • Creation o f a new r egulari zation term with uneve n blockin g effect (penalizat ion) on each bus . • Manipulation of spars ity by adj usting enforcers, based on t he o bserved s parse mec hanism. A CKNOWLEDGMENT This work was supporte d in part by the Defense Advanced Research Projects Agency (DARPA) under awar d no. FA8750- 17 -1-0 059 for RADICS program, and the National Scienc e Foundati on (NSF) under co ntract no. ECCS-18 00812. R EFERENCES [1] Pou rbeik, P ouyan, Prabha S. Kundur, and Carson W. Taylor. "T he anatomy of a power grid blackout-Root causes and dynamics of recent major black outs." IEEE Power and E nergy M agazine 4.5 (2006): 22-29. [2] Ajj arapu, Venkataramana, and Colin Christy. "The continuation power flow: a tool for steady state voltage stability analysis." IEEE transactions on Power Sys tems 7.1 (1992): 4 16-423. [3] M . Jereminov, D. M. B romberg, A. Pandey, M. R. Wagner, L. Pileggi, “Evaluating Feasib ility within Power Flow,” in IEEE Transactions on Power Systems, 2018 . (submitte d) [4] T. J. O verbye, “A power flo w measure for unsolvable cases”, I EEE Transactions on P ower Systems, vol. 9, no. 3, pp. 1359-1356, 1994. [5] M andala, Manasarani, and C. P. Gupta. 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Englewoo d Cliffs: Prentice Hall, 1995. Bus ID Infeasibility cur rent magnitude solution  Non-sparse [3] L1 -regul arization Proposed meth od 1 0 0 0 2 0.00858402 0 0 3 0.0561223 0 0 4 0.05097014 0 0 5 0.04278203 0 0 6 0.08877886 0.16111856 0 7 0.07740694 0 0 8 0.09593462 0.33915759 0 9 0.08860328 0 0 10 0.09134275 0 0 11 0.08889756 0 0 12 0.09065051 0.1244972 0 13 0.09368859 0.27381069 0 14 0.10908567 0.1824952 0.80006182 Case Name  k-sparse Dominant infe asible buses name MMWG80K 1.07 1 ‘155753’ ACTIVSg25K 1.8 42 Not listed here CASE9241pegas e 1.15 1 ‘2159’ CASE6515rte 1.15 2 ‘3576’,’ 4356’ CASE6468rte 1.29 1 ‘3718’