Exact and Efficient Algorithm to Discover Extreme Stochastic Events in Wind Generation over Transmission Power Grids
EExact and Efficient Algorithm to Discover Extreme Stochastic Eventsin Wind Generation over Transmission Power Grids
Michael Chertkov, Mikhail Stepanov, Feng Pan, and Ross Baldick
Abstract — In this manuscript we continue the thread of[M. Chertkov, F. Pan, M. Stepanov, Predicting Failures inPower Grids: The Case of Static Overloads, IEEE Smart Grid2011] and suggest a new algorithm discovering most probableextreme stochastic events in static power grids associatedwith intermittent generation of wind turbines. The algorithmbecomes EXACT and EFFICIENT (polynomial) in the case ofthe proportional (or other low parametric) control of standardgeneration, and log-concave probability distribution of therenewable generation, assumed known from the wind forecast.We illustrate the algorithm’s ability to discover problematicextreme events on the example of the IEEE RTS-96 model oftransmission with additions of , and of renewablegeneration. We observe that the probability of failure may growbut it may also decrease with increase in renewable penetration,if the latter is sufficiently diversified and distributed. I. I
NTRODUCTION
In progress to becoming smarter, the power grid of todayis undergoing multiple transformations [1], [2]. One of theenvisioned changes consists in replacing a significant portionof the traditional fossil thermal plants by renewable gener-ation [3], in particular by solar and wind farms. This task,motivated by ecological and political reasons, will not be asimple substitution. The renewable sources of energy, willalso be much less predictable and thus much more difficultto control. Both wind and solar fluctuate temporally andspatially. Even when forecasted, the renewable generationcan be described only in probabilistic terms, suggesting thatthe existing toolbox of power engineering, which is largelydeterministic, needs to be upgraded with computationallymore challenging probabilistic tools. For any configurationof aggregated loads, which stays roughly constant for tensof seconds or even minutes, one ought to consider anensemble of possible configurations of renewable generationundergoing significant and unpredictable changes during the
The work of MC and FP at LANL was carried out under the auspicesof the National Nuclear Security Administration of the U.S. Departmentof Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. The work of MC, MS and FP was funded in partby DTRA/DOD under the grant BRCALL06-Per3-D-2-0022 on “NetworkAdaptability from WMD Disruption and Cascading Failures”. The work ofMS is also partially supported by NSF grant DMS-0807592. RB is funded,in part, by the Department of Energy under Contract 09EE0001385.M. Chertkov is with Theory Division & Center for Nonlinear Studies atLANL, Los Alamos, NM 87545 and also with New Mexico Consortium,Los Alamos, NM 87544 [email protected]
M. Stepanov is with Department of Mathematics, University of Arizona,Tucson, AZ 85721 [email protected]
F. Pan is with Decision Division at LANL, Los Alamos, NM 87545 [email protected]
R. Baldick is with Department of Electrical and Computer Engineer-ing Engineering, The University of Texas at Austin, Austin, TX 78712 [email protected] same time. This uncertain (but probabilistically predictable)configuration of renewable generation will be complementedby standard adjustable generation. The adjustment is requiredto stabilize the system and complete matching between re-quested load and total generation. Many configurations fromthis ensemble of matchings will be feasible, i.e. not violatingtransmission constraints. However, there will always be somenumber of probabilistically rare but strong fluctuations of therenewable sources for which an instantaneous adjustmentof generation to loads will be problematic, as resulting inoverload of some number of lines. Discovering extremestochastic events of this rare but damaging type becomes animportant practical problem which requires fast and largelynon-existent algorithmic solutions.This manuscript is devoted to resolving (at least some partof) the aforementioned challenge. We assume that configu-ration of loads over the network is fixed, while generation issplit into two parts: (a) the bigger, consisting of conventionalgeneration, controlled and adjustable as a group, for examplevia common re-scaling (roughly representing, for example,so-called “droop” speed control with the proportion of eachgenerator response to the frequency variation fixed, or rep-resenting “regulation” response when deployed regulation isalso in proportion to the frequency variation); and (b) renew-able generation, say coming from wind farms, uncertain andnot controlled, described in terms of a probability distributionfunction, assumed known from a forecast. We are interestedto discover the most probable configuration(s) of the wind(more generally any fluctuating renewable source) whichis troublesome, in terms of possibly violating transmissionand/or controlled generation constraints, when no additionalcontrol efforts, such as curtailment of wind generation, loadshedding, line switching, etc, are in place. With diversityof wind production across multiple wind sites, the extremestochastic events will not be frequent, thus making theproblem of finding the rare configuration causing the troubleas challenging as finding a needle in a haystack. To discoverthe extreme stochastic events we exploit and develop furtherthe approach, originally introduced in theoretical physics, seee.g. [4], [5], [6], [7] then used to analyze performance oferror-correction codes [8], and recently applied to predictstatic failures in power grids associated with fluctuationsin loads [9]. We call this rare but most probable troubledinstance of the wind the instanton. In a significant technicalimprovement in comparison with [9] this manuscript sug-gests a direct way, which is Exact and Efficient (polynomialin the size of the grid), to explore the structure of the problemfor finding the instanton(s). This algorithmic improvement is a r X i v : . [ c s . S Y ] S e p chieved via mapping the instanton problem to minimizationof a convex function, characterizing forecasted distribution ofthe wind, over the exterior of a tractable polytope, describingfeasible solutions of the power flow equations.The material in the manuscript is organized as follows.We give some technical background, formulate the problemand briefly describe history of the instanton methodologyin Section II. Section III formulates the main theoreticalresult of the manuscript: exact and efficient algorithm forfinding most probable wind extreme stochastic event forgiven configuration of the grid. The performance of thealgorithm is illustrated in Section IV on example of theIEEE RTS96 system with 10% ,
20% and 30% of renewablepenetration. The results are summarized and discussed inSection V, where we also discuss future research challenges.II. Q
UASI -S TATIC P OWER F LOWS , W
IND M ODELINGAND F ORMULATING THE I NSTANTON P ROBLEM
Modern power grid is always in motion. Moreover manyof the changes which take place on multiple time scalesare inherently uncertain, being probabilistic in nature. Withsufficient penetration of the renewable sources, wind turbinesbring a particularly important source of fluctuations into themodern power grid. Production of wind can be forecastedonly to a degree. Even with a perfect forecast, and dueto a turbulent nature of the wind, one ought to describeoutput of the wind turbine integrated directly to the powergrid (without curtailing) in probabilistic, rather than deter-ministic, terms. Relevant times scales, where fluctuations ofwind generation dominate other sources of uncertainty (inparticular these associated with fluctuations of demands),are in the range from tens of seconds to tens of minutes.At these temporal scales transient (sub-second) phenomenaare already settled and quasi-static description of the powerflows is appropriate [10], [11]. Taking a standard DirectCurrent (DC) power flow approximation (well justified fortransmission network, where resistivity of power lines issignificantly smaller that respective inductance, voltage vari-ations are small and change in phase between neighboringbusses is small too) one arrives at the following relationsbetween the vector of injected/consumed real powers,
PPP , andthe vector of phases, ϕϕϕ , over the power grid defined as agraph, G = ( G , G ) ( G and G represent the set of nodesand the set of edges respectively), written in the form ofconditions COND f low = (cid:32) BBB ϕϕϕ = PPP , & ϕ = (cid:33) , (1) BBB = , { i , j } / ∈ G − / x i j , { i , j } ∈ G ∑ { i , k }∈ G k x − ik , i = j , (2) PPP = − d i , i ∈ G d ρ i , i ∈ G r α p i , i ∈ G g , i / ∈ G d ∪ G r ∪ G g . (3) In the remainder of the paragraph we detail notations, andunderlying notions, assumed in Eq. (3). BBB in Eq. (1) isthe matrix of the graph Laplacian constructed from lineinductive reactances, xxx , and the second condition in Eq. (1)sets the phase at an arbitrarily chosen (zero) node to zero.Different components of the vector,
PPP , associated with nodesof the graph, are split in Eq. (3) into four groups: set ofdemand/consumption nodes, G d ; set of renewable nodes, G r ;set of standard generation nodes, G g ; and the remaining setof other (junction) nodes. Any component of the demandis assumed negative and fixed (on the time scale of interestmeasured in seconds-to-minutes). The renewable generationfluctuates according to the forecasted probability distribution P ( ρρρ ) ∼ exp ( − S ( ρρρ )) , (4)where S ( ρρρ ) is a known convex function of its multi-dimensional argument, achieving minimum at the most prob-able, equilibrium, configuration of the wind generation, ρρρ .This distribution function represents statistics of the windfluctuations, collected over the time interval when loadsdo not change or change very little. (The assumption of S convexity is reasonable for a sufficiently wide, and thusmost probable, vicinity of the maximum output, in particularfor popular modeling of the wind statistics via the Weibullfunction [12].) Standard (controllable) generation adjusts tomatch the imbalance between total consumption and totalrenewable generation (assuming that the loads stay constant).The adjustment is relatively fast (instantaneous within thequasi-static description) and we will simplify by assumingthat the mismatch is shared between the controllable gener-ators in some pre-defined, automatic fashion. In particular,we will consider the case where the adjustment for eachgenerator is proportional to its nominal generation, so that α entering Eq. (3), is α = ∑ i ∈ G d d i − ∑ i ∈ G r ρ i ∑ i ∈ G g p i , (5)and it is the only degree of freedom on the standard gener-ation side which reacts absorbing changes in the renewablegeneration, ρρρ = ( ρ i | i ∈ G r ) . We assume that α is set tounity at the equilibrium configuration, ρρρ . Then, ( p i | i ∈ G g ) constitutes output of the controllable generation preset in thebeginning of the time interval of interest. Normally, thisredispatch of the controllable generation is the output ofthe Optimum Power Flow (OPF) analysis, accounting fordiversity in the generation cost at different sites, and executedperiodically on the scale ranging from minutes to tens ofminutes. The model of Eq. (5) schematically represents eitherdroop control or regulation response, although details of bothof these control modes differ from our model in details.One expects that a feasible Power Flow (PF) solution,representing a perturbed OPF, satisfies, in addition to thebasic power flow relations (1), the following set of trans-mission (thermal) conditions representing line constraints onthe amount of power which can flow safely trough the lineswithout overheating or damage) COND edge = (cid:32) ∀{ i , j } ∈ G : | ϕ i − ϕ j | ≤ x i j u i j (cid:33) , (6)where u i j is the line { i , j } rating. One also assumes that allthe generators included in the proportional control are withintheir capacity bounds, which translates into the followingcumulative constraints on the proportional control coefficient COND power = (cid:32) α ≤ α ≤ α (cid:33) , (7)where α and α are defined in accordance with the maxi-mum low and minimum high constraints, respectively, overindividual controllable generators. In fact, typical controlmodes would allow for some generators reaching their limits;however, we simplify that issue here.Even though the most probable configuration of thewind generation corresponds to the power flow solution of COND f low , which is safely within the feasibility region of
COND edge ∪ COND power , other less probable configurationsof wind can violate one or more of the feasibility constraints.Naturally, we are first of all interested to discover the mostprobable instance of the wind, the instanton, which liesoutside of the feasibility region: ρρρ inst = argmin ρρρ S ( ρρρ ) (cid:12)(cid:12)(cid:12) ρρρ / ∈ D int , (8) D int ≡ Projection ( COND ( ϕϕϕ , ρρρ , α )) ρρρ , (9) COND ≡ COND f low ∪ COND edge ∪ COND power , (10)where Projection ( COND ( ϕϕϕ , ρρρ , α )) ρρρ is the projection of thepolytope COND to the ρρρ -space. Therefore, by construction D int is also a polytope. Alternatively we can rewrite Eq. (8)as ρρρ inst = argmin ρρρ S ( ρρρ ) (cid:12)(cid:12)(cid:12) ρρρ ∈ D ext , (11)where D ext is a non-convex set, defined as an exterior of theconvex set (polytope) D int , i.e. D ext = R | G u | + \ D int . Eq. (11)states succinctly the instanton problem addressed in thismanuscript.In more general formulations, not yet utilizing specialstructure of the optimization domain D ext specific to ourproblem, the instanton problem (11) can be solved withinthe machinery of non-convex optimization methods. Forexample, and as described in details in [13], [14], [9], onemay search for the minimum of S ( ρρρ ) with the help ofa general-purpose optimization technique, specifically thedownhill simplex (or “amoeba”) method [15], [16]. Sincethe optimization domain is not concave, we expect to findmany (candidate) instanton solutions. Each initialization ofthe instanton-amoeba could lead to a new instanton. Theinitialization selects a simplex, built on N d + N d is the dimensionality of the ρρρ space,which is equal in our case to the number of renewablegenerators) from the error-surface separating D int and D ext .Then the instanton-amoeba method evolves the simplex, via a sequential set of shifts, contractions and extensions, towardsits eventual collapse to a local minimum of S ( ρρρ ) . Differ-ent random initiations will sample the space of instantons,thus generating the so-called instanton spectrum describingthe frequency of a given instanton occurrence and alsosuggesting an estimation for the ordered list (with respectto their probability of occurrence and frequency) of topinstantons. Repeated infinite number of times, the samplingwould output the most probable instanton. However giventhat the number of initiations will be finite in reality, themost probable one (of the finite number of instantons found)gives a heuristic estimate from below for the probability ofthe most probable instanton. To ensure sampling quality oneneeds to continue random initiations till the most probable in-stantons would appear multiple number of times. (Typically,this require hundreds of initiations for the network measuredin hundreds of nodes, of the type discussed below in SectionIV.)However, the specific structure of our problem (11) allowsa much faster and moreover exact resolution, than the oneprovided by the general purpose but computationally heavyinstanton-amoeba method of [13], [14], [9]. As shown in thenext Section, very specific features of D ext , associated withthe linear nature of the DC power flow and also with thesingle-parametric and linear control of fluctuations on thestandard generation side, allow to solve Eq. (11) efficiently.The new approach, describe below, offers a significantalgorithmic improvement in comparison with the generalapproach of the instanton-amoeba type.III. E XACT AND E FFICIENT A LGORITHM TO D ISCOVERTHE I NSTANTON ( S )Given Eqs. (1) and Eq. (5), one can express ϕϕϕ and α via ρ , thus arriving at the following explicit (tractable polytope)expression for D int : D int = COND edge ∪ COND power , (12) COND edge (13) = (cid:32) ∀{ i , j } ∈ G : | ( ˜ BBBPPP ) i − ( ˜ BBBPPP ) j | ≤ x i j u i j (cid:33) , COND power (14) = (cid:32) ∑ i ∈ G d d i − α ∑ i ∈ G c p i ≥ ∑ i ∈ G u ρ i ≥ ∑ i ∈ G d d i − α ∑ i ∈ G c p i (cid:33) , where matrix ˜ BBB is the quasi-inverse of
BBB accounting forthe ϕ = BBB .With this explicit formulation of D int the instanton prob-lem (11) becomes tractable, as it reduces to a minimum overa tractable set of convex problems:min a = , ··· , K { M a } , M a = min ρρρ S ( ρρρ ) (cid:12)(cid:12)(cid:12)(cid:12) c a ∪ ( D int \ c a ) , (15)where K = | COND edge ∪ COND power | ; c a stands for any ofthe inequality constraints in COND edge ∪ COND power ; and c a is the saturated version of c a by replacing inequalityith equality. When the feasibility set of a sub-problem a in Eq. (15) is empty we formally set M a to infinity.Eq. (15) is computationally tractable because it splits into K convex optimizations. The proof of the transition fromEq. (11) to Eq. (15) is straightforward and it simply relayson testing the faces of D int sequentially. (See, e.g., detaileddiscussion of similar problem in [17].) For any of the internalminimizations in Eq. (15), which is feasible, having one facetof D int saturated is guaranteed (by construction) and unlessthere is a degeneracy in S ( ρρρ ) there will be only one saturatedfacet per any minimization problem M a .Two comments are in order. First, our scheme is general,and can account for other types of the generation controlbesides adjusting generation linearly in response to thevariation in ρρρ . However, one should also expect that thedifficulty of the generalized problem will grow exponentiallywith the number of the control degrees of freedom. Thisexponential explosion is associated with the fact that pro-jection of the respective generalization of COND ( ϕϕϕ , ρρρ , ααα ) ,where ααα is now multi-parametric, will generate generalized D int , which is much more complex as described via anexponentially large (in the dimensionality of ααα ) number ofcontraints/inequalities. This is in spite of the fact that theoriginal polytope, COND ( ϕϕϕ , ρρρ , ααα ) , is tractable. (Indeed, it iswell known that projection of a tractable polytope along asubspace results in a polytope characterized, in the worstcase, in terms of the set of constraints which is exponentiallylarge in the size of the subspace. See [18], [19] for examplesand related algorithmic discussions.) Second, given that theinstanton problem is split into K tractable optimizations inEq. (15), one also finds not only the instanton itself (absoluteminimum) but also the ranked list of other extreme stochasticevents associated with other faces of D int . We will call theseother instantons, second-, third- etc according to their rankingin the derived hierarchy. This ranking is useful to discover thelist of extreme stochastic events. Note, however, that strictlyspeaking this ranking does not necessarily corresponds to theactual ranked list of all possible extreme stochastic events.There are two reasons for that. First, any point from a smallcontinuous vicinity of the (top ranked) instanton, sitting atthe same face of D int as the instanton, will have a lower S ( ρρρ ) weight than other instantons. (This is under assumption thatthe situation is not degenerate and the instanton correspondsto an interior point of the face.) Second, one may find a(discrete) configuration, which will be second in rankingwithin a minimization ranked k in Eq. (15), but will stillhave a lower S ( ρρρ ) weight than the top result of anotherminimization ranked k , even when k < k .IV. N UMERICAL E XAMPLE
We test the algorithm on the standard IEEE RTS-96 model[20] extended with renewables. The results are shown inFigs. 1,2,3,4) and commented upon below. To imitate effectof renewables, we took the base configuration of the model(standard generators, loads and transformer nodes are keptaccording to the data from [20]). Then, we add 3 , GeneratorLoadRenewable
Fig. 1. Graph of the IEEE RTS-96 model with nine new renewablegenerators (
Bus ID
Instanton1Instanton2Instanton3 (a) Configuration of renewable generation for the topthree instantons.
Instanton1Instanton2Instanton3 (b) Top instantons are shown. Vertexes marked inred/orange/blue colors are of the most stressed renew-able generator from the instanton output. Edges markedred/orange/blue colors are the saturated ones for therespective instantons.Fig. 2. Instanton(s) for base configuration + 3 renewable generators: 10%of renewable penetration in average production. The instantons are orderedaccording to their cost values, S ( ρρρ ) . .001.002.003.00 72 73 74 75 76 77 78 79 Bus ID
Instanton1Instanton2Instanton3 (a) Configuration of renewable generation for the topthree instantons.
Instanton1Instanton2Instanton3 (b) Top instantons are shown. Vertexes marked inred/orange/blue colors are of the most stressed renew-able generator from the instanton output. (Bus S ( ρρρ ) . of them to some number of other randomly selected nodes,where number of connections per new node is distributedaccording to the degree distribution of generators in the baseconfiguration. To facilitate comparison, we also create thethree configurations sequentially, such that the bigger oneis built on the top of the smaller one. The new graph isshown in Fig. (1). The capacities of added lines are chosenequal (and roughly correspond to the median capacity of theexisting lines). Inductances of the new lines are distributeduniformly in a median range of inductances of the base case.We choose α = α =
2. We pick the simplest possiblemodel for statistics of renewable generation, assuming that ρρρ is site-uncorrelated, positive (component by component)Gaussian, thus represented by S ( White ) ( ρρρ ) ≡ (cid:26) ∑ i ∈ G u ( ρ i / ρ i − ) , ∀ i ∈ G r : ρ i > + ∞ , otherwise , (16)where one chooses ρρρ = ( ρ i | i ∈ G u in a way that the total oftypical renewable generation, ∑ i ∈ G u ρ i , corresponds to 10%,20% and 30% (for the three test cases respectively) of thestandard generation of the RTS-96 base case. Selecting ρρρ wealso make sure that the resulting configurations are all in the Bus ID
Instanton1Instanton2Instanton3 (a) Configuration of renewable generation for the topthree instantons.
Instanton1Instanton2Instanton3 (b) Top instantons are shown. Vertexes marked inred/orange/blue colors are of the most stressed renew-able generator from the instanton output. (Bus S ( ρρρ ) . regime where no transmission or generation constrains areviolated, i.e. ρρρ ∈ D int . Note, that the choice of the positiveGaussian and site-uncorrelated distribution is made here forillustrative purposes only. Actual correlations of wind willbe more elaborate and interesting. However, one expects thatthe realistic, S ( ρρρ ) , representing actual wind forecast, will stillbe a convex function of ρρρ , thus making application of ouralgorithm to the more realistic situation as straightforwardas for the synthetic positive Gaussian case discussed here.The results of our numerical tests, which main goal was toillustrate utility of the algorithm as of an exact and fast tool,are shown in Figs. (2,3,4) for the three levels of the renewablepenetration respectively. For each example we present twofigures showing in, (a) configuration of renewable generation ρρρ for the top three leading instantons in comparison with thebase case, and in (b) structure of the top three instantons(renewable sites with the largest values of ρ / ρ marked) andrespective saturated edges. A complete instanton analysis ofan instance (graph+distribution) is very fast, it takes fewseconds on a laptop.Here is the summary and brief discussion of the simulationresults: In the cases correspondent to 10% and 30% of renew-able penetration the top instantons are well localizedon a site/node, in a sense that production at this singlesite is significantly larger than the typical value whiledeviation in production at the other renewable sites aremuch weaker. On the contrary, the top instanton is of ade-localized type in the intermediate case of 20% ofrenewable penetration. Note that the cost of the topinstanton in the intermediate 20% case , S ( ρρρ ) = .
04, issignificantly lower than in the 10% case, S ( ρρρ ) = . S ( ρρρ ) = . • We also monitor the value of α at the instantons, andobserve that it decreases with increase of the level ofrenewables: 0 .
75, 0 .
63 and 0 .
54 for the top instantonsin the 10%, 20% and 30% cases respectively. This ex-presses the fact that the addition of renewables translatesinto reduction of the standard generation. • There is always one saturated edge per instanton. Somehow remarkably, this saturated edge is alwaysone of the “old” edges (present in the base structure),and it is also positioned in majority of cases relativelyfar from the renewable sites which over-generate atthe instanton. This observation emphasizes non-locality(and thus intrinsic difficulty) of the problem. • The standard generation constraints are not violated atthe instantons, i.e. for each of the instantons found, α = < α < α = α is unity at the base case andit (naturally) decreases with the level of renewablegeneration increase.V. C ONCLUSIONS AND P ATH F ORWARD
Summarizing, the main results of this work are: • We posed the problem of discovering extreme stochasticevents in power grids, the instanton, associated withtransmission overflows caused by fluctuations of renew-able generation. • We showed that the aforementioned problem is com-putationally tractable within the DC power flow settingand with a low parametric linear control of standardgeneration, for example of the proportional type. • We illustrated algorithmic utility and efficiency of ournewly suggested algorithm on example of the IEEERTS-96 grid with 10%, 20% and 30% of added re-newable generation. Main conclusions of our numericaltests are: (a) Addition of renewables may lead to signif-icant increase of the probability of failures (destructive Note that this observation does not violate the N − N edges of the graph removed) simply becausethe condition is enforced only at the equilibrium point. effect), but it may also help to make the network tobecome prone to failures (constructive effect), depend-ing on quantitative details of how the grid extension isdone. (b) The instanton configurations represent globalcorrelations in the graph, which shows itself in the factthat the (single) overloaded line is typically relativelyfar from the over-producing renewable sites. (c) Ouralgorithm is very efficient computationally in providingthe exact assessment of the gain (or loss) associatedwith addition of renewables.We plan to continue this work on discovering extremestochastic events in the power grids efficiently. Our main goalhere is to design a reliable predictive tool, extreme stochasticevent toolbox , capable to provide awareness (fast guidanceto utility operator) in terms of predicting dangerous extremestochastic events associated with the renewable generation.More specifically, this work will be continued along thefollowing lines. • First of all, we will be testing scalability of the approachfor larger, continental scale, transmission networks. Ourtask here is to approach complexity which would scalelinearly with the size of the system. (This may beachieved, e.g. by replacing standard convex optimizationsolvers, by their linear scaling and distributed proxies.) • Our modeling of wind statistics needs to be morerealistic. We plan to apply our algorithm to wind datataken from a realistic forecast, e.g. of the type availableat [21], in particular accounting for long spatial (usu-ally, at least hundreds of kilometers long) correlationsbetween different sites. Our future test beds will includemodels of ERCOT (with actual or planned wind farmsin Western Texas), as well as models of other wind-richparts of US power grids. • We will incorporate into the (so far static) schemedynamic effects and actual (time series) measurementsof the wind intensity. This will require extending theinstanton approach to account for temporal Lagrangiancorrelations in the cost function, integrated in timeover pre-history, and enforcing the transmission andgeneration conditions not only instantaneously but alsoover the (dicretized) time horizon. • Our model of proportional control is convenient foranalysis, but it matches the details of droop controland of regulation response only schematically. We willmodify the model to more fully represent these actions,in particular accounting for generation limits. • We envision incorporating the instanton analysis intocontrol schemes, in particular in the tertiary (balance)control, for example penalizing top instanton configu-rations (and their vicinities) in the modified optimumpower flows. Another interesting control option is tofit the instanton framework into an adversarial process,whereby any given control action can be matched by acorresponding instanton. This yields an iterative process(with each iteration roughly like the problem solved inthe paper), which produces robust control actions andignificant events, at the same time. • We will also work on extending the algorithm to thecase of AC power flows (which is especially importantin terms of accounting for additional issues related tovoltage variations), e.g. utilizing new advances in relatedoptimization techniques [22]. We will consider otherschemes of transmission and voltage control, in particu-lar related to demand response [23], line switching [24],controlling DC tie-lines, capacitor banks, phase shiftersand related [10]. We also plan to account for other typeof stochastic issues, e.g. related to dynamic stability [25]and voltage collapse [26].Finally, once the extreme stochastic event toolbox is devel-oped, we envision using it (as a black box) for developingnew planning and control schemes for smart grids of the fu-ture, for example in the spirit of the general approach of [27].We envision using this toolbox to solve related problems ofdiscovering interdiction attacks on power grids [28], [29],and analyzing, controlling and preventing cascading failures[30], [31]. VI. A
CKNOWLEDGEMENTS
We are thankful to D. Bienstock and P. Parrilo for veryvaluable and useful comments on theoretical and algorithmicaspects of the underlying optimization problem, to threeanonymous referees for their remarks and comments whichhelped us to improve the presentation quality and discussionof limitations of our approach/technique, and to the partic-ipants of the “Optimization and Control for Smart Grids”LDRD DR project at Los Alamos and Smart Grid SeminarSeries at CNLS/LANL, and especially to S. Backhaus, R.Bent and K. Turitsyn, for multiple fruitful discussions.R
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