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1 NPS-OR NAVAL POSTGRADUATE SCHOOL MONTEREY CALIFORNIA Optimizing Eectric Grid Design Under Asymmetric Threat (II) by Javier Sameron and Kevin Wood Nava Postgraduate Schoo Ross Badick University of Texas at Austin March 2004 Approved for pubic reease; distribution is unimited. Prepared for: U.S. Department of Justice Office of Justice Programs and Office of Domestic Preparedness 810 Seventh St. NW Washington DC ; under the aegis of the Nava Postgraduate Schoo Homeand Security Leadership Deveopment Program.

2 NAVAL POSTGRADUATE SCHOOL MONTEREY CA RDML Patrick W. Dunne USN Superintendent Richard Ester Provost This report was prepared for the U.S. Department of Justice Office of Justice Programs and Office of Domestic Preparedness 810 Seventh St. NW Washington DC ; under the aegis of the Nava Postgraduate Schoo Homeand Security Leadership Deveopment Program. Reproduction of a or part of this report is authorized. This report was prepared by: JAVIER SALMERON Research Assistant Professor of Operations Research KEVIN WOOD Professor of Operations Research ROSS BALDICK Professor of Eectrica and Computer Engineering University of Texas at Austin Reviewed by: LYN R. WHITAKER Associate Chairman for Research Department of Operations Research Reeased by: JAMES N. EAGLE Chairman Department of Operations Research LEONARD A. FERRARI Ph.D. Associate Provost and Dean of Research

3 REPORT DOCUMENTATION PAGE Form Approved OMB No Pubic reporting burden for this coection of information is estimated to average 1 hour per response incuding the time for reviewing instruction searching existing data sources gathering and maintaining the data needed and competing and reviewing the coection of information. Send comments regarding this burden estimate or any other aspect of this coection of information incuding suggestions for reducing this burden to Washington Headquarters Services Directorate for Information Operations and Reports 1215 Jefferson Davis Highway Suite 1204 Arington VA and to the Office of Management and Budget Paperwork Reduction Project ( ) Washington DC AGENCY USE ONLY (Leave bank) 2. REPORT DATE March TITLE AND SUBTITLE: Optimizing Eectric Grid Design Under Asymmetric Threat (II) 6. AUTHOR(S) Javier Sameron Kevin Wood and Ross Badick 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Nava Postgraduate Schoo Department of Operations Research 1411 Cunningham Road Monterey CA SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) U.S. Department of Justice Office of Justice Programs and Office of Domestic Preparedness 810 Seventh St. NW Washington DC REPORT TYPE AND DATES COVERED Technica Report 5. FUNDING NUMBERS 2002-GT-R PERFORMING ORGANIZATION REPORT NUMBER NPS-OR SPONSORING / MONITORING AGENCY REPORT NUMBER 11. SUPPLEMENTARY NOTES The views expressed in this thesis are those of the author and do not refect the officia poicy or position of the Department of Defense or the U.S. Government. 12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Approved for pubic reease; distribution is unimited. 13. ABSTRACT (maximum 200 words) This research extends our earier work to improve the security of eectric power grids subject to disruptions caused by terrorist attacks. To identify critica system components (e.g. transmission ines generators transformers) we devise bieve optimization modes that identify maximay disruptive attack pans for terrorists who are assumed to have imited offensive resources. A new mode captures the dynamics of system operation as a network is repaired after an attack and we adapt an earier heuristic for that mode s soution. We aso deveop a new mixed-integer programming mode (MIP) for the probem; a mode that can be soved exacty using standard optimization software at east in theory. Preiminary testing shows that optima soutions are readiy achieved for certain standard test probems athough not for the argest ones which the heuristic seems to hande we. However optima soutions do provide a benchmark to measure the accuracy of the heuristic: The heuristic typicay achieves optimaity gaps of ess than 10% but occasionay the gap reaches 25%. Research wi continue to refine the heuristic agorithm the MIP formuation and the agorithms to sove it. We aso demonstrate progress made towards a graphica user interface that aows performing our interdiction anaysis in a friendy environment. 14. SUBJECT TERMS Homeand Security Eectric Power Grids Network Interdiction 15. NUMBER OF PAGES PRICE CODE 17. SECURITY CLASSIFICATION OF REPORT Uncassified 18. SECURITY CLASSIFICATION OF THIS PAGE Uncassified 19. SECURITY CLASSIFICATION OF ABSTRACT Uncassified 20. LIMITATION OF ABSTRACT UL i

4 OPTIMIZING ELECTRIC GRID DESIGN UNDER ASYMMETRIC THREAT (II) by Javier Sameron and Kevin Wood Department of Operations Research Nava Postgraduate Schoo Monterey CA Ross Badick Department of Eectrica Engineering University of Texas at Austin Austin TX Abstract This research extends our earier work to improve the security of eectric power grids subject to disruptions caused by terrorist attacks. To identify critica system components (e.g. transmission ines generators transformers) we devise bieve optimization modes that identify maximay disruptive attack pans for terrorists who are assumed to have imited offensive resources. A new mode captures the dynamics of system operation as a network is repaired after an attack and we adapt an earier heuristic for that mode s soution. We aso deveop a new mixed-integer programming mode (MIP) for the probem; a mode that can be soved exacty using standard optimization software at east in theory. Preiminary testing shows that optima soutions are readiy achieved for certain standard test probems athough not for the argest ones which the heuristic seems to hande we. However optima soutions do provide a benchmark to measure the accuracy of the heuristic: The heuristic typicay achieves optimaity gaps of ess than 10% but occasionay the gap reaches 25%. Research wi continue to refine the heuristic agorithm the MIP formuation and the agorithms to sove it. We aso demonstrate progress made towards a graphica user interface that aows performing our interdiction anaysis in a friendy environment. ii

5 TABLE OF CONTENTS 1. INTRODUCTION OBJECTIVE AND SUMMARY OF WORK COMPLETED MODELING RESTORATION AND ALGORITHMIC IMPLEMENTATION PREVIOUS INTERDICTION MODEL WITHOUT RESTORATION INTERDICTION MODEL WITH RESTORATION HEURISTIC ALGORITHM FOR THE INTERDICTION PROBLEM WITH RESTORATION MIXED-INTEGER REFORMULATION OF THE INTERDICTION MODEL PRELIMINARY IDEAS LINEARIZING ADMITTANCE EQUATIONS DUALITY: CONVERTING THE MODEL INTO A SIMPLE MIXED-INTEGER PROGRAM LINEARIZING CROSS-PRODUCTS IN THE OBJECTIVE FUNCTION TEST CASES COMPARING SOLUTIONS WITH AND WITHOUT SYSTEM RESTORATION COMPARING HEURISTIC AND OPTIMAL SOLUTIONS VULNERABILITY OF ELECTRIC POWER GRID ANALYZER (VEGA) OVERVIEW VEGA GUI OVERVIEW OTHER ACTIVITIES THESIS STUDENTS OTHER REPORTS AND ACTIVITIES OTHER SOURCES OF FUNDING FUTURE WORK VALUE OF THE RESEARCH TO HOMELAND SECURITY APPENDIX A: SUMMARY OF PREVIOUS NOTATION APPENDIX B: LINEARIZATION OF CROSS-PRODUCTS INITIAL DISTRIBUTION LIST iii

6 1. INTRODUCTION This report documents the continuing research project entited Homeand Security Research And Technoogy Proposa (Optimizing Eectric Grid Design Under Asymmetric Threat) which is sponsored by the U.S. Department of Justice Office of Justice Programs and Office of Domestic Preparedness. This research extends our previous effort aimed at deveoping new optimization modes and methods for panning expansion and enhancements of eectric power grids that improve robustness to potentia disruptions caused by natura disasters sabotage and especiay terrorist attacks. The research reported here shows the progress made in different areas comprising modeing agorithms and their impementation testing and user interfaces. The document is organized as foows: Section 2 presents an overview of project accompishments to date. Sections 3 through 7 describe this year s activities in detai. In particuar Section 3 describes our mode of post-attack system restoration over time and an associated soution procedure. Section 4 focuses on a new mode representation as a standard mixed-integer program. Computationa resuts for these modes are presented in Section 5 incuding comparisons with earier resuts. Section 6 presents an overview of our new VEGA decision-support system which incudes database and graphica user-interface toos aong with an optimization modue. Other activities are summarized in Section 7. Section 8 presents an overview of the work intended for year We concude in Section 9 with the ist of criteria used to assess the vaue of our project to Homeand Security. 2. OBJECTIVE AND SUMMARY OF WORK COMPLETED Our project deveops new mathematica modes and optimization methods for robust panning of eectrica power grids focusing on security and reiabiity with specia emphasis on potentia disruptions caused by terrorist attacks. We refer to our previous proposa [Sameron and Wood 2002-I] and references therein for detaied background on the probem of eectric power-grid vunerabiity. A statement from the Committee on Science and Technoogy for Countering Terrorism [2002] succincty states the motivation for our work: The nation s eectric power systems must ceary be made more resiient to terrorist attack. This motivation has been further strengthened 1

7 by the backout on 14 August 2003 in the Northeast U.S. eectric power grid [U.S.-Canada Power System Outage Task Force 2003]. Athough the backout was not instigated by terrorists its cause appears to have been mutipe faiures of infrastructure eements in the transmission system and our current research addresses precisey such situations. We aso refer to our previous report [Sameron Wood and Badick 2003-I] where we estabish the mathematica foundations for the modes and agorithms that we have enhanced in the research reported here. The foowing miestones have been achieved as the resut of previous and current research (see proposas by Sameron and Wood [2002-I] [2002-II]). (Underined items identify the most recent contributions.) (1) Formuation of mathematica modes that represent the probem of optimay interdicting an eectrica power grid. A newer formuation incudes system restoration over time. (2) Deveopment of heuristics that identify highy disruptive attack pans to a specific eectric power grid given imited interdiction resources. Newer heuristics keep pace with the deveoping modes and have been adapted to incorporate system restoration over time. This agorithm has been impemented using the Genera Agebraic Modeing Language software [GAMS 1996]. (3) Incorporation of different measures of effectiveness any of which can be optimized: Short-term power disruption (MW); Short-term cost ($/MW) over a consumer sectors; Long-term energy disruption (MWh) incuding system restoration over time; and Long-term cost ($) over a consumer sectors incuding system restoration over time. (4) Deveopment of techniques to convert modes in (1) into standard mixed-integer programs that can be soved exacty. This means that not ony we can determine good attack pans as our heuristic approach (2) does but we can prove that these pans are optima. In turn this provides us with a precise measure of vunerabiity. We have impemented and soved the converted modes using GAMS [1996]. 2

8 (5) Soution to cases with up to 100 eectrica buses (drawn from the IEEE Reiabiity Test Data [1999-I] [1999-II]). (6) Presentations in the Homeand Security Leadership Deveopment (HSLD) seminars: Eectric Power Grids Vunerabiity CS4920 Nava Postgraduate Schoo (3 December 2002). Vunerabiity of Eectric Power Grids CS3660 Nava Postgraduate Schoo (18 June 2003). (7) Reports: First-year report [Sameron Wood and Badick 2003-I]. Research Paper [Sameron Wood and Badick 2003-II] accepted for pubication in IEEE Transactions on Power Systems. (8) Thesis students invovement: Major Dimitrios Stathakos Greek Army. An Enhanced Graphica User Interface for Anayzing the Vunerabiity of Eectrica Power Systems to Terrorist Attacks [Stathakos 2003] graduated in December LCDR Rogeio Avarez USN. Interdicting Eectrica Power Grids graduation expected in March (9) Graphica User Interface (GUI) Deveopment: We reaize the importance of enabing access to this type of anaysis to a number of potentia users who are not necessariy famiiar with the optimization arena. To bridge this gap we have initiated the design and impementation of the Vunerabiity of Eectrica Power Grids Anayzer (VEGA) system. VEGA is an integrated too comprising a graphica user interface (GUI) a supporting database (DB) and the aforementioned optimization toos. A preiminary Web page has been set up for this project [VEGA 2003]. VEGA 1.0 is the first prototype of this system. We next describe the items (1)-(9) above in more detai. 3

9 3. MODELING RESTORATION AND ALGORITHMIC IMPLEMENTATION 3.1 Previous Interdiction Mode without Restoration The mathematica mode we presented in our previous report [Sameron Wood and Badick 2003-I] attempts to maximize immediate eectric power shedding by (optimay) seecting a set of interdictions given imited resources. We refer to that report for a fu description of its formuation which can be shorty stated as the foowing Max-min (Mm) probem: (Mm) max min cp ' δ p s.t. gp ( δ) b p 0 Reca that in this mode an interdiction pan is represented by the binary vector δ whose k-th entry δ is 1 if component k of the system is attacked and is 0 otherwise. For a given pan k the inner probem (caed DC-OPF) is an optima power-fow mode [Wood and Woenberg 1996 p. 514] that minimizes generation costs pus the penaty associated with unmet demand together denoted by c 'p. Here p represents power fows generation outputs phase anges and unmet demand i.e. the amount of oad shed; c represents inearized generation costs and the costs of unmet demand. The outer maximization chooses the most disruptive resourceconstrained interdiction pan δ where is a discrete set representing attacks that a terrorist group might be abe to carry out. In this mode g corresponds to a set of functions that are noninear in (. p δ) gp ( δ) The inner probem invoves a simpified optima power-fow mode with constraint functions that are however inear in p for a fixed δ = δ. ˆ 3.2 Interdiction Mode with Restoration Overview Mode (Mm) provides ony a rough estimate of energy shedding and thus the true cost to society of an attack on a power grid. This is because (Mm) is based on the system capabiity after the initia return to service of non-damaged equipment foowing an attack disregarding mediumand ong-term effects. The ony case in which this is not important is when the outage duration of 4

10 a interdictabe components in the system is the same which seems unikey (e.g. interdicted substations wi in genera take much onger to repair than interdicted ines). We have extended mode (Mm) to hande the cost and timing of repairs which aows us to obtain interdiction pans seeking to maximize tota cost as the system is restored over time. Essentiay this mode measures tota weighted energy where weights represent costs of ost energy to various customer sectors and possiby other factors. This is accompished by using interdiction constructs to coupe instances of DC-OPF one for each system state that represents a stage or time period of system repair. In outine the mode is: (Mm ) max min D cp ' δ p t T t s.t. g ( p δ) b t t t p 0 t t T t T. Mode (Mm ) extends (Mm) to incorporate the houry cost of power fow cp ' t in each time period t mutipied by the period s duration in hours D t. Figure 1 shows the difference between potentia soutions provided by (Mm) and (Mm ). The mode coud be further extended to incorporate sub-time periods through oad duration curves but we have not yet expored this possibiity; a oads are hed constant over time. Foowing the notation and conventions in our previous report (see Appendix A) we next describe the fu mode that incorporates system restoration. We first need to introduce some additiona notation: T = set of periods for t T * * * ξ = L G B S * set of a (directy) interdictabe eements Dur() e = Duration (hours) of outage for eement e ξ if attacked D = Duration (hours) of time period t for t T t 5

11 β te = 1 if component e remains unrepaired in time period t after being attacked 0 if component e is repaired before time period t after being attacked for t T e ξ. Remark: In the above notation β Line t β Bus ti β Gen tg and Sub β ts denote te β when e= is a ine or e=i is a bus or e=g is a generator or e=s is a substation respectivey. Power Shed (MW) Power Shed (MW) measure of effectiveness measure of effectiveness time after the attack time after the attack Figure 1. The mode without restoration provides the optima interdiction pan according to instantaneous power shed after initia return to service of non-damaged equipment (eft). The mode with restoration over time accounts for energy disruption (right). The foowing agorithm constructs the set of time periods T based on the different outage durations for a interdictabe eements. In the course of the agorithm D and η te are aso t constructed: 6

12 Agorithm Construct Time Periods : Initiaization: % ξ ξ T t 0 m0 0; Whie % ξ : t t 1 β te t {} T T t m End Whie 1 if e % ξ 0 otherwise min { Dur( e) e % ξ} { ee % ξ Dur( e) mt} ξ = t D m m % ξ % ξ \ t t t 1 ξ t Due to different outage duration for the system components we need to estabish which components might be out of service during each time period. For exampe if a ine can be interdicted but it is not connected to an interdictabe bus then the ine is guaranteed to be in service after Dur() hours independent of whether it is attacked or not. At this point the foowing definitions are needed: ** L t = Set of ines that coud be out of service in period t foowing a direct or indirect interdiction. ** G t = Set of generators g that coud be out of service in period t foowing a direct or indirect interdiction. β te heps us define these sets precisey. In particuar: L ** t Line βt = 1 or Bus Bus βti = 1 for some i Li or if either: Sub Sub βts = 1 for some s Ls or Line Par βt = 1 for some L 7

13 Gen βtg = 1 or ** Bus g Gt if either: βt( i g) = 1 or Sub βtsi ( ( g)) = 1 Additiona definitions are: λ λ 1 if L 1 if L * ** L L t = λt = 0 otherwise 0 otherwise 1 if g G 1 if g G * ** G G t g = λt g = 0 otherwise 0 otherwise * I 1 if i I λi = 0 otherwise * S 1 if s S λs = 0 otherwise (Hereafter equations in boxes represent fina constraints in our modes.) The objective function of the interdiction probem with restoration (I-DC-OPF-R) becomes: max min δ Gen Line ( Pt Pt St θt) Γ D h P f S c (I-R.0) Gen t g t g i c t i t T g i c where as in the case without time periods we attempt to minimize power generation cost pus oad shedding cost but this time these terms are specified by time period and weighed by its duration in order to account for energy cost. δ sti represents the resource-constrained interdictions that a terrorist group might be abe to carry out and is the same as in the case without time periods. Or expicity: M δ M δ M δ M δ M Gen Gen Line Line Bus Bus Sub Sub g g i i s s * g G * L * i I * s S (I-R.1) A δ -variabes are binary (I-R.2) 8

14 Gen Line ( P P θ S ) Γ represents the time-dependent decision variabes of the inner t t t t DC-OPF-R (DC-OPF mode with restoration). These constraints (and the associated duas denoted as π with appropriate sub- and super-indices) are expicity stated as: Admittance equation for ine subject to possiby being out of service during period t: P = B ( θ θ )(1 λ β δ ) (1 β δ ) Line Line Line Line Bus Bus t to ( ) td ( ) t ti i * Bus i I Li - Sub Sub Line Line A βt s δ s βt δ t πt * Sub * Line s S Ls L L (1 ) (1 ) ( ) (IDC-R.1) Baance equation for the bus i in period t: P P P S = d i t ( π ) (IDC-R.2) Gen Line Line Ba tg t t tic ic ti g Gi o() = i d() = i c c Line capacity subject to possiby being out of service during period t: Line Line ** L0 t t π t P P t L ( ) P P (1 δ ) t L β =1 Line Line Line * Line t t - LCap t - Line Line Bus * Bus Bus LB t δi i βti = πti Line Line Sub * Sub t s s ts ( π ) P P (1 ) t i i I L 1 ( ) P P (1 δ ) t s s S L β = 1 - Line Line Line * Par Line LL t δ βt = πt Line Line ** t t - LS ts ( π ) P P (1 ) t L L 1 ( ) P P t L Line Line Line * Line LCap t δ βt πt Line Line Bus * Bus Bus t i i ti L0 t ( π ) P P (1 ) t L =1 ( ) P P (1 δ ) tii I L β = 1 Line Line Sub * Sub LS t δs s βts = πts Line Line Line * Par Line LL t (1 δ ) βt 1 ( π t LB ti ( π ) P P (1 ) t s s S L 1 ( ) P P t L L = ) (IDCR.3) 9

15 It is worth noting that for the each ine we need five constraints to specify its maximum capacity in period t and another five constraints for the minimum capacity. For exampe consider the case where a ine can be interdicted but it is not connected to an interdictabe bus ** or substation. Then L t for some of the first periods t (e.g. t=123) but since the ine is guaranteed to be back in service after Dur() hours ** L t for the rest of the periods (e.g. t=456). Assuming aso that the ine has no other ines in parae the constraints to be used wi be: Line Line Pt P t = 456 Line Line Line P P (1 δ ) t = 123 t On the other hand if the ine is connected to an interdictabe bus and the bus outage covers for exampe periods t=1...5 the constraints woud be: P P t Line Line t = 6 Line Line Line Pt P (1 δ ) t = 123 Line Line Bus Pt P (1 δi ) t = Generator g maximum output subject to possiby being out of service during period t: P P t g G ( π ) Gen Gen ** G0 tg g t tg P P (1 δ ) t g g G β =1 ( π ) Gen Gen Gen * Gen G tg g g tg tg P P (1 δ ) t g i( g) I β =1 ( π ) Gen Gen Bus * Bus GB tg g ig ( ) tig ( ) tg P P (1 δ ) t g s( i( g)) S β =1 ( π ) Gen Gen Sub * Sub GS tg g sig ( ( )) tsg ( ) tg (IDC-R.4) Demand shedding at bus i for customer c: S d i c ( π ) (IDC-R.5) Load tic ic tic 10

16 Variabe sign: P Gen tg P S θ Line t tic ti 0 t g unrestricted t 0 t i c unrestricted ti (IDC-R.6) In summary our interdiction mode with restoration (I-R) becomes: (I-R): max min (I-R.0) δ Gen Line Pt Pt St θt subject to: (I-R.1) (I-R.2) and (IDC-R.1) to (IDC-R.6) 3.3 Heuristic Agorithm for the Interdiction Probem with Restoration The agorithm that we use to sove (I-R) i.e. a probem without restoration is schematicay depicted in Figure 2. Sove the DC-OPF for the present grid configuration Based on present and previous fow patterns assign a Vaue to each interdictabe asset Maximize the Vaue of the Assets to be Interdicted (excuding previousy expored soutions) Figure 2: Interdiction agorithm framework (without restoration). This agorithm first soves DC-OPF assuming no attacks. The power-fow pattern is used to assign reative vaues to a the components of the power grid. These vaues are used to maximize 11

17 the estimated vaue of the assets to be interdicted whie ensuring that the resources required for the interdiction pan are not exceeded. With this interdiction pan we modify the right-hand side of DC-OPF mode (disregarding restoration) and obtain its soution. In this case we expect part of the oad to be shed. The process continues by finding aternative sets of vauabe assets to interdict that have not been identified at earier iterations and by evauating oad shedding for each of these interdiction pans. More detais on the agorithm can be found in Sameron et a. [2003-II]. We can adapt this agorithm to the mode with restoration by soving the new DC-OPF probem in the upper box in Figure 2 assuming that a δ -variabes have been fixed (say satisfying (I-R.1) and (I-R.2)). That probem denoted DC-OPF-R( ˆδ ) becomes: δ= δ ˆ DC-OPF-R( ˆδ ) : min Gen Line Pt Pt St t θ (I-R.0) subject to: (IDC-R.1) to (IDC-R.6) DC-OPF-R( ˆδ ) (caed sub-probem in the above agorithm for a specific interdiction pan ˆδ ) provides the joint power fow patterns for a number of system stages : one for each restoration period. Notice that DC-OPF-R( ˆδ ) decomposes into T sub-sub-probems each of which consists of an instance of DC-OPF with some subset of system components being out of service. Outaged components are determined by δ ˆ (interdictions) and by β which informs t e DC-OPF about the status of interdicted components in period t. In addition to this modification our heuristic agorithm aso redefines the concept of vaue which is used to determine which grid components appear more attractive for interdiction in each iteration. We maintain the same concept of vaue (denoted as a vector V) as in our previous work (again see detais in Sameron Wood and Badick [2003-I]) but noticing that these vaues must be mutipied by Dur() e (for a generic component e) in order to account for energy-based vaues. Therefore assuming the definitions of vaue from previous work the new definition of vaue for every generic component e is: V Restoration (e) = Dur(e) V No-Restoration (e) 12

18 With these vaues the master probem can find potentiay good interdiction pans δˆ that have not been expored yet. The master probem (MP-R) at a specific iteration τ is: τ MP-R( ˆ τ τ V ) : max V δ δ subject to: τ Eqs. (I-R.1) - (I-R.2) repacing δ with δ and: δ δ δ δ δ δ 1 g G i I Gen Bus * * g i i Bus δ 1 L L i I Line i i Line Line Par ' * * δ 1 L L L * * Sub δ 1 i I I s S Bus i s s * * Sub δ 1 L L s S Line s s * * (MP-R.1) ( ˆ δ δ ) ( ˆ δ δ ) Gen τ' Gen Line τ' Line g g * g G * L ˆGen τ ' ˆLine τ ' g = 1 δ = 1 δ ( ˆ δ δ ) ( ˆ δ δ ) 1 τ ' τ. Bus τ' Bus Sub τ' Sub i i s s * i I * s S ˆBus τ ' ˆSub τ ' i = 1 δs = 1 δ (MP-R.2) where ˆ τ is a set that contains the information on a previousy-generated interdiction pans. The first bock of constraints (MP-R.1) are vaid inequaities to ensure that a system component is not interdicted if it has been indirecty interdicted by an eement to which it is connected. (Remark: The vaid inequaities (MP-R.2) aso need to be adjusted when considering system restoration because they were based on arguments that ignored repairs over time.) The second bock (MP-R.1) are super-vaid inequaities that account for previousy generated soutions in the agorithm in order to aways examine aternative soutions. 13

19 Figure 3 sketches a framework for the new agorithm. Sove the DC-OPF-R for the given Interdiction Pan: Sove T DC-OPF probems Based on present and previous fow patterns assign an (energy-based) Vaue to each interdictabe asset MP-R: Maximize the Vaue of the Assets to be Interdicted (excuding previousy expored soutions) Figure 3: Interdiction agorithm framework (with restoration). A more detaied version of this agorithm (I-ALG-R) foows: I-ALG-R Input: Grid data; Interdiction data; τ max (iteration imit). Output: ˆδ * is a feasibe interdiction pan causing a disruption with cost * γ. If the agorithm exits because optima. Initiaization: τ MP( ˆ τ V ) is infeasibe then a feasibe pans have been enumerated and ˆδ * is therefore - ˆ1 ˆGen1 ˆ Line1 ˆ Bus1 ˆSub1 δ ( δ δ δ δ ) ( 0000) (initia attack pan). δ ˆ δ ˆ1 ˆ1 { δ }. * - ˆ1 (best pan so far) and * - γ 0 (cost of the best pan so far). - τ 1. Subprobem: - Sove DC-OPF-R( ˆτ δ ) for objective vaue ( ˆτ γ δ ) and soution ˆ Gen P ˆτ = ( P ˆ Line τ P τ Sˆ τ θˆ τ ). * - If ( ˆτ * γ δ )> γ then ( ˆτ * γ γ δ ) and δˆ δ ˆτ. max - If τ = τ then Print ( ˆδ * * γ ) and hat. 14

20 Master Probem: - Compute estimated vaues: V τ ( V Gen τ V Line τ V Bus τ V Sub τ τ ) 1 Gen τ Line τ Bus τ Sub τ ( ) τ - Sove MP- R( ˆ τ V ) for ˆ t1 δ. V V V V τ τ = 1 ˆ * τ - If MP-R( ˆ τ * V ) is infeasibe then Print (δ γ ) and hat. ˆ ˆ { δ ˆ }. τ 1 τ τ τ τ 1. - Return to Subprobem. 4. MIXED-INTEGER REFORMULATION OF THE INTERDICTION MODEL 4.1 Preiminary Ideas We woud prefer to convert the mode (I-R) into a standard (minimizing or maximizing) mixed-integer program (MIP) because a weath of techniques exist to sove such modes efficienty. (I-R) possesses severa features that make this conversion difficut however:. (I-R) is as a max-min probem not a simpe minimization or maximization. This difficuty can be overcome by duaizing the inner minimization (DC-OPF-R). This converts (I-R) into max-max probem i.e. a simpe maximization. However as we wi see ater this conversion eads to other difficuties that must be overcome. (I-R) is highy non-inear due to the presence of mutipe products of variabes associated with the admittance equation (IDC-R.1). The foowing two ideas enabe us to convert (I-R) into a MIP: (a) Dropping equaity non-inear constraints: Consider an admittance equation for a generic ine with potentia interdiction represented as a non-inear equaity of the form: P= B ( θ θ )(1 δ )(1δ ) a b

21 aong with capacity constraints: P P δ (1 1) and P P δ 2 P P δ (1 ) (1 1) and P P δ 2 (1 ). Here P θ a and θ b are continuous decision variabes representing power fow on the ine and phase anges at buses a b respectivey; δ 1 and δ 2 are binary decision variabes representing two possibe ways to interdict the ine say attacking the ine directy and attacking one of the buses the ine is connected to. When both δ 1 and δ 2 are 0 the admittance equation and capacity constraints become: P= B( θ θ ) P P P P which is the usua power fow admittance equation and a b capacity constraint. On the other hand if either δ1or δ 2 equas 1 indicating an interdiction has occurred the above equations become: P = 0 P 0 P 0 which is the desired effect there is no power fow on the ine whie θ a and θ b may vary independenty now. Remark: It does not suffice to set P 0 and P 0 using the capacity constraints ony. The reason is that athough this woud impy P = 0 the admittance constraint woud become 0 = B ( θ a θ b ) forcing θa = θb which is not necessariy optima. To avoid the noninearities in the admittance equation we can estabish two constraints that enforce P= B ( θ a θ b ) when a δ -variabes are 0 and drop this constraint when any of the δ -variabes is 1. Let θ ab be an upper bound on the absoute vaue of the maximum phase ange difference define: P M = P Bθ ab then PB( θ θ ) M( δ δ ) ( ) ( δ ). (L.I) a b 1 2 = B( θa θb)(1 δ1)(1 δ2) P B θa θb M δ1 2 Notice that when both δ 1 and δ 2 are 0 the two inear inequaities (L.I) yied precisey PB( θa θb) = 0. If either δ 1 or δ 2 equas 1 the upper and ower bound imits on the 16

22 constraints are sufficienty arge that they can never bind i.e. the origina constraints vanish as they shoud. (b) Linearizing cross-products: In the deveopment that foows we encounter a number of cross-products of the form δπ where δ is a 0-1 variabe representing interdiction and π is a continuous non-negative or non-positive variabe representing the dua variabe for a ine capacity constraint ike those in (IDC-R.3). For simpicity et us assume that π 0 and that an upper bound π π is known. The π bound can be estabished by anayzing the maximum benefit per unit that we coud obtain by increasing the ine capacity. Assuming that the argest penaty for faiing to meet the demand max f ic ( ic ) is greater than the maximum generating cost ( max h g ) a bound that woud work in g most cases is π = max f ic. A more conservative but generay vaid bound is ic π = 2 max f min f. ic ic ic ic If we define a new continuous variabe v form as: = δπ we can represent the cross product in inear v πδ v π δπ v π π(1 δ ) 0 π π v 0 (L.II) 17

23 The foowing diagram shows the vaidity of this transformation: v 0 v π v = 0 δ = 0 v π π v πδ π 0 π π v π v 0 v π π(1 δ) v π 0 π π v π v 0 v = π δ = 1 v π π 0 π π v 0 [ 0 π ] [ 0 π ] 4.2 Linearizing Admittance Equations Using the inearizing inequaities (L.I) we can convert the admittance equations P = B ( θ θ )(1 λ β δ ) (1 β δ ) Line Line Line Line Bus Bus t to ( ) td ( ) t ti i * Bus i I Li - Sub Sub Line Line A βt s δ s βt δ t πt * Sub * Line s S Ls L L (1 ) (1 ) ( ) (IDC-R.1) into the foowing inear expressions: P B M t - Line Line Line Line Bus Bus Sub Sub Line Line A t ( θto ( ) θtd ( ) ) ( λ βt δ βti δi βts δs βt δ ) ( πt ) * Bus * Sub * Line i I Li s S Ls L L P B ( θ θ ) M ( λ β Line Line Line t to ( ) td ( ) t δ β δ β δ β δ ) t ( π ) Line Bus Bus Sub Sub Line Line A t i i t s s t t * Bus * Sub * Line i I Li s S Ls L L (IDC-R.1) The revised interdiction mode with restoration (I-R ) becomes: (I-R ): max min δ P Gen Line t P t S t θ t (I-R.0) subject to: (I-R.1) (I-R.2) (IDC-R.1) (IDC-R.2) to (IDC-R.6) 18

24 Accordingy the inner power-fow probem can be caed (DC-OPR-R ): (DC-OPF-R ): min P Gen Line t P t S t θ t (I-R.0) subject to: (IDC-R.1) (IDC-R.2) to (IDC-R.6) where we assume a given interdiction pan δ = ˆ δ. 4.3 Duaity: Converting the Mode into a Simpe Mixed-Integer Program If we take the dua of the inner mode in (I-R ) i.e. the dua of (DC-OPF-R ) we obtain: Mode (D-I-R ): - - L L Line A A B Bus A A maxmax M λ βt δ ( π t π t ) βti δi ( π t π t ) δ π t * Bus i I L i - - Sub Sub A A L Line A A βts δs ( π t π t ) βt δ ( π t π t ) * Sub * Line s S Ls L L ( d ) π P Ba G G0 G Gen G ic t i g t g g g t g i t c t ** g G * Gen g G t βt g = 1 G Bus GB G Sub GS g i( g) t g g s( i( g)) t g * Bus gi ( g) I t βti ( g) = 1 * Sub gs (( i g)) S t βts ( i( g)) t ** L π P (1 δ ) π P (1 δ ) π P (1 δ ) π L L0 L0 L Line π t π t P δ * Line L t βt = 1 - L L t π t P ( ) (1 ) ( π ) - - L Bus LB LB L Sub LS LS P δi π ti π ti P δs π ts π ts * Bus Bus * Sub Sub i I Li t βti = 1 s S Ls t βts = 1 (1 ) ( ) (1 ) ( ) - L Line LL LL P (1 δ ) ( π t π t ) * Line Line L L t β t = 1 ict d π Load ic tic subject to: Dua constraints for power generation: (D-I-R.0) π (1 λ ) π λ π λ β π λ β π D( th ) tg ( P ) Ba G0 G0 G G I Bus GB S Bus GS G t i( g) t g t g g t g i( g) t i( g) t g s( i( g)) t s( i( g)) t g g t g (D-I-R.1) 19

25 Dua constraints for power fow on ines: - - LS LS LL LL ( πts πts ) ( πt πt ) ( ) A A L0 L0 L0 L Line LCap L Line LCap LB LB t t (1 t )( t t ) t t t t ti ti * Bus Bus i I Li βt i = 1 π π λ π π λ β π λ β π π π * Sub Sub * Par Line s S Ls βt s = 1 L L βt = 1 π π = 0 t ( P) Ba Ba L to ( ) td ( ) t Dua constraints for power shedding: (D-I-R.2) π π Dt ( ) f tic (S ) (D-I-R.3) Ba Load ti tic ic tic Dua constraints for phase anges: A ( ) ( ) - A A - A π π π π θ (D-I-R.4) B B = 0 t i ( i) t t t t t o( ) = i d ( ) = i Dua variabe sign: π Ba unrestricted - A L 0 LCap LB LS LL π π π π π π A L 0 LCap LB LS LL 0 0 π π π π π π 0 G0 G GB GS π π π π 0 π Load (D-I-R.5) Interdiction resource (same as (I-R.1) and (I-R.2)): M δ M δ M δ M δ M (I-R.1) Gen Gen Line Line Bus Bus Sub Sub g g i i s s * g G * L * i I * s S A δ -variabes are binary (I-R.2) 20

26 In outine this mode is: (D-I-R ): max min π δ subject to: (D-I-R.0) (I-R.1) (I-R.2) (D-I-R.1) to (D-I-R.5) 4.4 Linearizing Cross-products in the Objective Function The objective function (D-I-R.0) contains cross-products of the form δπ. Using the inearizing technique (L.II) from Section 4.1 we can inearize the objective function as foows: - - L Line A A Bus BA BA max max M λ βt ( vt vt ) βti ( vt vt ) δ π v * Bus i I L i - - Sub SA SA Line LA LA βts ( vts vts ) βt ( vt vt ) * Sub * Line s S Ls L L ( ) Ba G G0 G G G dic π t i Pg π tg Pg π tg vtg it c t ** * Gen g G g G t βt g = 1 ( ) P ( π v ) P ( π v ) G GB GB G GS GS g t g t g g t g t g * Bus gi ( g) I t βti ( g) = 1 * Sub gs (( i g)) S t βts ( i( g)) = 1 - L0 L0 P L ( ) ( )-( π t π t P π t π t v L ( t vt )) t ** L * Line L t βt = 1 * Bus Bus i I Li t βti = 1 * Sub Sub s S Ls t βts = 1 - (( π π ) ( )) P v v L LB LB LB LB ti ti ti ti - (( π π ) ( )) P v v L LS LS LS LS ts ts ts ts ( ) - L LL LL LL LL Load ( πt π t ) ( vt vt ) dicπ tic i c t P * Line Line L L t βt = 1 (D-I-R.0) 21

27 In addition we require a set of constraints of the form: v πδ v π v π π(1 δ) 0 π π v 0 (D-I-R.1) for every inearized cross-product. These constraints are specified in detai in Appendix B. The fina mode in outine form is: (D-I-R ): max δπ v (D-I-R.0) subject to: (I-R.1) (I-R.2) (D-I-R.1) to (D-I-R.5) (D-I-R.1) Mode (D-I-R ) is the cumination of a the inearizations and duaizations described previousy. It exhibits mutipe advantages: The most important one is to represent the interdiction probem as standard compact mixed-integer program (MIP). That means that any of the generic optimization techniques avaiabe for soving bounding or approximating the soution to a MIP are appicabe to this mode. Hereafter we refer to mode (D-I-R ) as (I-MIP) i.e. Interdiction probem in MIP form. 5. TEST CASES We present a summary of resuts to show: (a) the benefit of incorporating system restoration into our anaysis and (b) the potentia of the MIP reformuation which can be soved exacty. 22

28 For item (a) we wi compare the enhanced heuristic with system restoration (I-ALG-R) with the former heuristic (I-ALG). For item (b) we wi compare the heuristic soution provided by (I-ALG-R) with the exact soution obtained by soving the MIP reformuation (I-MIP). Our tests are carried out on the same set of IEEE reiabiity test networks ( One and Two Areas) described in our previous report. Assumptions regarding outage duration are summarized in the foowing tabe: Grid Component Interdictabe Resources M Outage Duration (h) (no. of terrorists) Lines (overhead) YES 1 72 Lines (underground) NO N/A* N/A* Transformers YES Buses YES Generators NO N/A* N/A* Substations YES *Not Appicabe. Data for outage durations (i.e. repair or repacement times) are based oosey on IEEE [1999-I]. Outage duration for transformers is 768 hours. For overhead ines instead of the 10 or 11 hours used in IEEE [1999-I] we are more conservative and assume 72 hours. This is justified because (a) we expect more damage to resut from the intentiona destruction of a ine this woud probaby invove the destruction of one or more towers [Miami Herad 2002] than the average time needed to repair damage from common natura causes such as ightning and (b) if n ines and other grid eements are attacked tota repair time may be onger if fewer than n repair teams are avaiabe. We aso assume that a arge substation requires 768 hours for repair but buses for which IEEE [1999-I] provides no data require 360 hours. 5.1 Comparing Soutions with and without System Restoration Athough in both cases beow I-ALG finds better short-term disruptions (compare power shed) over the first 72 hours foowing the attack it is cear that ong-term effects are better captured by I-ALG-R: Case/Agorithm RTS-One-Area I-ALG RTS-One-Area I-ALG-R Directy Interdicted Components. Resource: M=6 Time Period Power Shed (MW) Energy Shed (MWh) Lines: A11 A20 A21 A25-1 A27 A h Tota: Lines: A h Transformers: A h Substations: Sub-A2 Tota:

29 Case/Agorithm RTS-Two-Areas I-ALG RTS-Two-Areas I-ALG-R Directy Interdicted Components. Resource: M=12 Time Period Power Shed (MW) Energy Shed (MWh) Lines: A20 A21 A27 A33-1 A35-1 AB h B20 B21 B25-1 B27 B33-1 B34 Tota: Substations: Sub-A1 Sub-A2 Sub-B1 Sub-B Tota: Athough our goa is focused on ong-term disruption we recognize that the optima soution provided by the I-ALG mode is sti an insightfu measure of vunerabiity in the anaysis of short-term effects. 5.2 Comparing Heuristic and Optima Soutions First Goa: Optima short-term disruption. In this case we are comparing I-ALG with a specia version of the I-MIP mode in which there is ony one period t = 1 the duration of this period is D 1 = 1 hour and the duration outage for a interdictabe eements e ξ is Dur() e = 1 hour. From the next two tabes the heuristic soution for optima short-term disruption (provided by I-ALG) is of good quaity when compared to the best possibe provided by (I-MIP): Case/Agorithm RTS-One-Area I-ALG RTS-One-Area I-MIP Directy Interdicted Components. Resource: M=6 Power Shed (MW) Lines: A11 A20 A21 A25-1 A27 A Lines: A11 A20 A21 A25-2 A27 A Case/Agorithm RTS-Two-Areas I-ALG RTS-Two-Areas I-MIP Directy Interdicted Components. Resource: M=12 Lines: A20 A21 A27 A33-1 A35-1 AB2 B20 B21 B25-1 B27 B33-1 B34 Lines: A21 A25-1 A27 A33-1 B18 B21 B25-1 B27 B33-2 Power Shed (MW) However this quaity deteriorates when interdiction resource increases: Case/Agorithm RTS-Two-Areas I-ALG RTS-Two-Areas I-MIP Directy Interdicted Components. Resource: M=24 Buses: Substations: Sub-A1 Sub-A2 Sub-B1 Sub-B2 Lines: A21 A33-1 A34 B21 B27 B33-1 Buses: Power Shed (MW)

30 This shows the MIP optima soution may outperform the soution provided by the heuristic agorithm by 25% in some cases. Second Goa: Optima ong-term disruption. In this case we are comparing I-ALG-R with the I-MIP configured for a tota horizon of 768 hours which is the onger time to repair for any component in our test cases. Again the first two tabes beow show that the heuristic soution for optima ong-term disruption (provided by I-ALG-R) is acceptabe for the cases tested. Case/Agorithm RTS-One-Area I-ALG-R RTS-One-Area I-MIP Directy Interdicted Components. Resource: M=6 Time Period Power Shed (MW) Energy Shed (MWh) Lines: A h Transformers: A h Substations: Sub-A2 Tota: Lines: A h Transformers: A h Substations: Sub-A2 Tota: Case/Agorithm RTS-Two-Areas I-ALG-R RTS-Two-Areas I-MIP Directy Interdicted Components. Resource: M=12 Time Period Power Shed (MW) Energy Shed (MWh) Substations: Sub-A1 Sub-A2 Sub-B1 Sub-B Tota: Lines: A23 B h Transformers: A7 B h Substations: Sub-A2 Sub-B2 Tota: In the case beow the soution provided by the heuristic for a probem with a arger interdiction resource vaue sti exhibits acceptabe quaity. Case/Agorithm RTS-Two-Areas I-ALG-R RTS-Two-Areas I-MIP Directy Interdicted Components. Resource: M=24 Time Period Power Shed (MW) Energy Shed (MWh) h Buses: h Substations: Sub-A1 Sub-A2 Sub-B1 Sub-B2 Tota: Lines: A30 A h Transformers: A7 B h Buses: h Substations: Sub-A2 Sub-B2 Tota: Vunerabiity of Eectric Power Grid Anayzer (VEGA) 6.1 Overview VEGA is an integrated decision-support system comprising a GUI a reationa database management system (RDBMS) an optimization modue and an administration program that 25

31 contros a of those components (Figure 3). VEGA 1.0 is the first version of this system and has been buit on the Microsoft (MS) Windows 2000 operating system [Microsoft 2003]. This section provides an overview of VEGA. Figure 3. VEGA decision support system. The utimate goas of VEGA are to enabe access to our anaytica techniques for a wide range of potentia users incuding students in the HSLD curricuum and to bridge the gap between the underying mathematica optimization methods and the decision-makers. A preiminary Web page has been set up for this project [VEGA 2003]. The GUI and database (DB) are key to organizing panning data reducing cerica error through embedded vaidations competing missing detais fitering information according to user s needs and dispaying mutipe scenarios with their resuts for comparison purposes. The GUI is aso key for demonstrating the potentia that optimization techniques have for panning interdiction and interdiction defense. The GUI heps prepare power-network data for anaysis and then dispays anaytica resuts by enabing easy navigation through customized tabes and graphics containing probem data and resuts. This gives a user easy access to the mathematica anaysis of a probem even if the user in not an expert in mathematica modeing and optimization. 26

32 The VEGA optimization modue performs the mathematica anaysis of the probem independenty of the GUI. The purpose of the GUI is to prepare the case data to be anayzed and to retrieve and dispay the optimization resuts in a user-friendy fashion. VEGA 1.0 s core is an optimization mode that assesses the maximum possibe disruption a network might experience from a terrorist attack. Naturay this core can work as independent entity and its operation is in fact transparent to the user of VEGA. The administration program and the GUI are impemented in the MS Visua Basic (VB) 6.0 programming anguage [Microsoft 1998] supported by a RDBMS impemented with MS Access 2000 [Microsoft 2003]. The underying optimization modue is impemented using GAMS [GAMS 2003 Brooke et a. 1996]. Data transfer and synchronization of the GUI with GAMS are performed by means of pain ASCII fies because GAMS is not avaiabe as a caabe or dynamic ibrary; therefore GAMS executes as an externa program. 6.2 VEGA GUI Overview The front-end appication responsibe for the VEGA GUI uses a Windows-based methodoogy that faciitates for the user: a. Network data input; b. Other data input incuding possibe scenarios optimization parameters etc.; c. Anayzing resuts provided by the optimization mode; d. Graphica dispay of the network input data and output resuts; and e. Administration of mutipe cases with severa scenarios per case. 27

33 Figure 4. Exampes of VEGA data tabes. The GUI in VEGA 1.0 uses VB tabes in order to import data from the database and edit the records associated with a probem. Figure 4 shows an exampe of data tabes for Buses and Generators. Upon competing necessary data entry the user can invoke the optimization modue to produce optima or near-optima interdiction pans. These pans can be dispayed in tabuar form (Figure 5) or in graphica form (Figure 6). 28

34 Figure 5. Interdiction pan shown in tabuar form. Figure 6. Graphica representation of overa resuts by scenario (eft) and by scenario and time (right). One important recent accompishment has been the enhancement of a modue in the VEGA GUI caed the One-Line Diagram (OD) GUI. ODs are used by eectrica engineers to represent eectric power grids graphicay. The VEGA 1.0 (see VEGA [2003]) OD GUI had many imitations that have been overcome through the thesis research of an NPS student [Stathakos 2003]. Figure 7 shows one of the new OD representations. 29

35 Figure 7. One-Line Diagram for an eectric network under interdiction. The above snapshots are intended to give an overview of the VEGA GUI ony and a fu report on its capabiities wi be provided in a separate future report. 7. OTHER ACTIVITIES 7.1 Thesis Students To date two students have devoted their Master s thesis research to our project Major Dimitrios Sthatakos (Greek Army) and LCDR Roger Avarez (USN). Major Sthatakos who graduated in December 2003 enhanced the one-ine diagram (OD) interface in our VEGA GUI. The OD GUI represents the detais of power fows and interdictions graphicay (Figure 7). Major Sthatakos created a highy fexibe OD GUI by repacing a prototypic OD GUI (based on standard Visua Basic objects) with an advanced OD GUI based on state-of-the-art ActiveX contros. For more information see Sthatakos [2003]. 30

36 LCDR Avarez is currenty working on the MIP representation of our mathematica modes (see Section 4) and on their sovabiity. He has been instrumenta in competing many of the refinements to the MIP described in this report and he is exporing aternative soution techniques for the MIP with specia focus on Benders decomposition. He is expected to graduate in March 2004 [Avarez 2004]. We continue to seek the invovement of NPS students in our project. 7.2 Other Reports and Activities In addition to our previous report [Sameron Wood and Badick 2003-I] our first year s work has yieded a refereed pubication: Anaysis of Eectric Grid Security Under Terrorist Threat [Sameron Wood and Badick 2003-II] which wi appear in IEEE Transactions on Power Systems. The feedback from the four reviewers was highy positive. Aso our work was presented in the CS3660 seminar (Critica Infrastructure Protection) as part of the recenty created HSLD curricuum at NPS. 7.3 Other Sources of Funding The Department of Justice has been our soe source of funding to date. We are seeking additiona research support from the Department of Energy (proposa submitted) and the Nationa Science Foundation (proposa in preparation). 8. FUTURE WORK The major chaenge we face in the present year is to obtain actua U.S. power-grid data for testing and vaidating our methodoogy. In doing so we need to continue the deveopment of techniques to sove the exact modes (MIPs) for those cases and other reaisticay sized probems. (See the proposa Sameron and Wood [2003] for more detai.) Currenty we are acquiring data sets for different areas in the U.S. North American Eectric Reiabiity Counci system and we are adapting and extending those data for our purposes. We have had success in formuating our modes as MIPs but additiona deveopment and experimentation is needed in order to be abe to sove them efficienty. In particuar we wi be investigating specia bounding techniques and speciaized cutting-pane techniques (Geoffrion [1972] Israei and Wood [2002]). 31

37 During the coming year we wi aso: Continue to work on the VEGA GUI; Extend our modes and agorithms to capture the dynamics of oad variation over time. This requires extension of the VEGA optimization modue as we as the VEGA database and GUI (at the eves of data and resut management and of graphica representations); and Initiate work on trieve modes to identify optima protection pans for power grids. 9. VALUE OF THE RESEARCH TO HOMELAND SECURITY The ca for proposas that this research addresses asks how our research adds vaue to the Homeand Security effort. We respond as foows: Simuation software for Homeand Security (HLS): - Attacks on critica infrastructure: Power Grids Deiverabes (this document): - Modes and agorithms (as presented) - Case studies (as presented) - Software (optimization agorithms and GUI under deveopment) - Pubications (as presented) By criterion used to fund the project: - This research addresses an important probem in HLS - This research adds to the body of HLS knowedge - This research is interdiscipinary - This research is nove and usefu - This research invites non-nps coaborators - Principa Investigators (PIs) have a reputation in the proposed fied of study - PIs wi try to get students invoved in this research and produce theses - Resuts wi be pubishabe - Resuts wi be usefu in teaching HLS courses - PIs beieve the budget is in ine with the resuts 32