Statistically Characterizing the Electrical Parameters of the Grid Transformers and Transmission Lines

Statistically Characterizing the Electrical Param eters of the Grid Transformers and Transmission Lines * Mir Hadi Athari Departm ent of Electrical and Com puter Engineering Virginia Comm onwealth Univ ersity Richmond, VA, USA Email: atharih@v cu.edu Zhifang Wang † Departm ent of Electrical and Com puter Engineering Virginia Comm onwealth Univ ersity Richmond, VA, USA Email: zfwang @vcu.edu Abstract — This paper presents a se t of validation metrics for transmission network parameters th at is applicable in both creation of synthetic power system test cases and validation of existing mode ls. Using actual data from tw o real w orld power grids , statistical analyses are performed to extract some useful statistics on transformers and transmission lines electric al parameters including per unit reactance, M VA rating, and t heir X/R ratio. It is found th at c onversion of per un it reactance calculated on syste m common base to transformer ow n power base will significantly stabilize its range and remove the correlation be tw een per unit X and MVA r ating. This is fairly consistent for transform ers w ith different volta ge levels and sizes and can be utilized as a strong validation metric for synthetic models . It is found t hat transmission lines exhibit different statistical properties than transformers w ith different distribution and range for th e parameters . In addition, statistical analysis shows that t he empirical PDF of tra nsmission network electrical parameters can be approxi mated w ith ma the matical distribution functions w hich w ould help a ppropriately characterize them in synthetic power n etworks. Kullback-Leibler divergence is use d as a measure for goodness of fit for approximated distributions. Index Terms — Transmission network parameters, synt hetic grid models, statistical analysis, distribution fitting I. I NTRODUCTI ON Synthetic po wer net works a re e merging as a p otential solution for the lack of test cases for per formance e valuation in po wer s ystem research and d evelopment. Ge nerally, access to real data in critical infrast ructure like po wer net works is limited d ue to confidentiality requirements. Utility co mpanies and regulatory agencies don’t s hare such data and s trictly limit access to actual po wer systems data for public and researchers due to their sensitivity. On the other hand, it is important that new concepts and algorithms developed by researchers be evaluated in relatively large and complex networ ks with the same characteris tics as act ual grids so that t hey ca n be reproducible b y peers. For example, authors in [ 1] – [3 ] have developed a new storage management and energy man age ment algorithms which enable a b idirectional power flo w from microgrids to power networks that need evaluatio n with realistic grid top ology. Since Synthetic po wer net works a re entirely fictitious but with the same character istics as realisti c networks, they can be freely pu blished to the public to facilitate advancement of ne w technologies in po wer systems. * This paper w as prepared as a re sult of work sponsor ed by the Advance d Research Projects Agency-Energy (ARPA-E), U.S. Departme nt of Energy , under Awar d Number DE-A R0000714. † Corresponding au thor. Email: z fwang@vcu.edu Development o f e fficient synthetic po wer system models requires that their size, complexity, and electrical a nd topological charac teristics match those of rea l po wer grid s. Power networks are complex in frastruc tures with various components. In addition to top ological characteristics o f po wer networks, they include seve ral co mponents with different electrical characteristics s uch as different types of transformers, switched shunt reactive pow er compensation, remote tap changing b us voltage reg ulatio n, etc. Develop ment of synthetic po wer networks with t he same co mplexity that ca n simulate the exact b ehavior of actual grids needs a comprehensive stud y of different co mponents from both electrical and to pological per spectives. Also , increasing le ve l of renewable generation i n po wer systems has introd uced an unprecedented level of uncertainty into grids [4].I n the literature, many studies are dedicated for characterizing actual power networks mainly from t opological p erspectives such as ring-structured power g rid developed in [5] and tree structu red power grid model to add ress the po wer system r obustness [6], [7]. Small world app roach described in [ 8] served as a reference for the works of [9] – [11 ] to develop an approach for generating tr uly synthetic tr ansmission line topologies. A random topo logy po wer net work model, called RT - nestedSmallWorld , is proposed in [10] based on comprehensive studies on the electrical top ology of so me re al world po wer grids. T he impacts of d ifferent bus t ype assignments in synthetic power networks on grid vulnerability to cascading failures are in vestigated in [ 12]. In [13] the authors presented a substation place ment method and transmission lines assignment fro m real energy and population data based o n methodology introduced in [14], [15]. The proposed methodology employs a clustering technique to en sure t hat s ynthetic substations meet reali stic proportions of load and generation. However, the authors will continue to au gment test cases by adding additio nal complexities such as transmission network electrical parameters assignment. In another study, t he au thors performed a stat istical anal ysis on transmission line capacit y regarding both topology and electrical parameters. However, all the se studies focus mainly on topolo gy-related para meters of trans mission lines and ignore electr ical parameters s uch as impedance of trans mission lines and trans formers. Review of the literature on synthetic grid modeling reveals that there is a need for statistical studies to characteri ze electrical parameters o f transmission net work to b e used in synthetic grid models. In this paper, we mainl y focus on the statistical anal ysis of transfo rmers a nd trans mission lines electrical parameters such as per unit impedance, nominal capacity and X/R ratio. T he goal of this p aper is to a) p rovid e a well- defined “rules” for transmissio n network par ameters as potential validatio n metrics for existing s ynthetic grid mode ls and b) to pr ovide guidelines on how to acc urately configure them in synthetic models. A very large sample o f actual operating tra nsfor mers and tr ansmission lines fro m t wo real world p ower systems is used to extract the statistical characteristics of t heir parameters. The r est of the p aper is o rganized as follows. Section II presents t he statistical analysis on transfor mers electric al parameters. Sectio n III discu sses the statistics of tr ansmission lines para meters and finally some concludin g rem arks and future work direction will b e presented in sectio n IV . II. G RID T RANSFO RMERS Generally, in power systems branches are referred to transmission lines or transformers bet ween two buses in the network. Also, in some cases shunts are considered in the branch category. In this paper we first perform some statistical analysis o n tran sformers electr ical parameters e xtracted from two real world po wer systems. Next, transmission lines from the same networks will be studied to extract som e statistics for their critical para meters. A. P er unit impeda nce using the system MVA base o r transformer ’s power ratin g? In pow er system analysis the use of per unit system to express the system quantiti es a s fracti ons of a define d base unit quantity is common. This is im portant especially for transform ers as the voltage level is diff erent for their te rminals and per unit sy stem simplifies transf ormer calcula tions. An other advantage for this expression is that similar types o f apparatus l ike transform ers will have the im pedances lying within a narrow numerical range when expressed as a per -unit fraction of the equipment rating, even if the unit size varies widely. However, per un it impedances of power gr id components are usually co nvert ed to new values using a common system-w ide base for application in power system analysis like power flow or econom ic power flow calculati ons. This co nversi on depends on reference voltage base for d iffere nt zones in th e sy stem and a pred efined u nique power base for the entire sy stem acco rding to the follow ing sim ple equation:         󰇧       󰇨   󰇧       󰇨  where    ,    ,    a re g iven per unit impedanc e, voltage base, and power b ase for each apparatus and    is the new per unit impedance calculated using    and    . Usually , the voltage b ase values are selected the same as the nomin al voltage of trans form er terminals for each zone to simplif y the calculations. Therefore , the conversion formu la for per unit impedance can be ex pressed as         󰇧        󰇨  In the p ower grids the use o f different voltage levels is a common pra ctice to decre ase the po wer loss through transmis sion lines. Thus th ere are t ransform ers with d iffere nt turn ratios to couple the areas with different voltage levels. In this study , the transform ers are grouped into diffe rent categori es based on their hig h voltage term inals. T his is because as the nomin al voltage level incre ases the trans former size gets larger, so studying them in groups based on voltage level seem s reasonabl e for extracting validation metrics. The p urpos e of statistic al experim ents in this study is to identify several validation metrics for transform ers p aramete rs including their impedances to help validat e synthetic power networks. This would be even more helpful if the range for diffe rent paramete rs can be specif ied fo r ty pical power system components. The fi rst experim ent tries to find the relationship betw een MVA rating of transformers and their per unit impedances. These analyses are performed on both p er u nit values in system base, an d converted va lues to transf ormers ow n MVA ratings. The original p ower sy stem data used in this s tudy off er transform er impedance in per unit calcul ated based on the c ommon base f or the system. Fig. 1 sh ows the scatte r p lot of transformers per unit reactance (X) and MVA rating for the original and convert ed per un it reactance of tran sformers. Note tha t although transform ers with high voltage terminal of 1 15 kV are select ed for this comparison, the results are fairly consistent for oth er voltage l evels as shown in fig. 2. (a) (b) Figure 1. Scatter plot of per u nit reactance versus MVA rating of transformer for a) system commo n base and b) conver ted to transforme r own MVA rating. The scatter plot for per unit reactance on system common base shows a descen ding trend as the size o f transform er increases w hich means there is relatively large correlation coefficient between the two as show n in fig. 1 (a). In this case, the per unit reactanc e values span from nearly 0 to 2.75 p.u which is relatively large range for this parameter. However, when w e consider the same scatter plot for converted per unit reactance to transform er o wn MVA rating, this range nar rows down to [0, 0.5] p.u puttin g at least 80% of th em with in even a narrow er rang e of [0.05 , 0. 2] p.u. In addition, almost zero correlati on coeffici ent means that this range is independent of transform er size and voltage level. The same scatt er plots f or converted va lues of p er unit reactance versus MVA rating of transform ers for other volt age levels are depicted in fig.2. It is f ound th at p er unit rea ctance of transform ers in pow er system s regardless of their siz e lie within a narrow r ange wh en calculated o n their own p ower base and statistics reflect w hat is kn own fr om engin eering practi ce. This can be a potential validati on metric for sy nthetic power netw orks transf ormers along with othe r statis tical measure s such as th eir proba bility distribu tion. 138 kV 230 kV Figure 2. Scatter plot of per un it reactance ver sus MVA rating of transformer fo r 138 and 230 kV tra nsforme rs. B. Tran sformer parameter di stribution Transform er pa rameters statistics are derived using over 30000 actual pow er transform ers. The database includes different types of t ransform ers such as fixed st ep down and step up transform ers, three win ding transformers , On -load Tap Changer (OL TC) transf ormers, an d autotr ansformers . A negative im pedance often occu rs in the sta r m odeling of a th ree winding transformer due to how the leakage reactance is measured/m odeled [16]. Also, Netw ork equivalenc ing m ethods can create neg ative impedanc es which can affect the statisti cs of transm ission netw ork param eters. To avoid such s cenario, data are filtere d by       to exclude ab normal tr ansform er p arameters fr om sam ples. Also, due to lack of detaile d informati on on some transfo rmers, their MVA rating s are repo rted with either ver y large or zero values . These transform ers too are exclu ded from samples to have accurate statistics . The probability distribution of transform er parameters is another measure that can b e us ed along with param eter rang e as validation metric in synthetic power networks. The probability distributi on of a random variable, say transformer per u nit reactance , is a function that describes h ow lik ely w e can o btain the different possib le valu es of th e ran dom v ariables. Usin g th e database o f real transformer d ata, we can get the empirical cumu lative density function (CDF) of each p aramete r that can give us the empirical probabili ty density function (PDF ). Next, to provi de a m ore sy stematic a pproach f or gener ating sy nthetic models, we try to fit approxi mated distri bution functions to empirical PDFs. The go odness of th is fit can be measure d with Kullback-L eibler div ergence. 1) Kullback-Leibler Dive rgence In probabil ity theory and info rmation the ory, th e Kullback – Leibler (KL) divergence, also called discrimin ation inform ation, is a m easure of the d ifferen ce between two probability distributions P and Q. It is not symmetric in P and Q. In a pplications, P ty pically rep resents the "true" dis tributi on of data, obse rvations, or a precisely calculat ed theoret ical distributi on, while Q typically accounts for a theory , model, descripti on, or approxim ation of P [17]. Sp ecifical ly, the KL divergenc e from Q to P, den oted   󰇛   󰇜 , is the amount of inform ation lost w hen Q is used to app roximate P. Fo r discret e probability dist ributions P an d Q, the KL divergence from Q to P is defin ed to be [18]   󰇛    󰇜    󰇛  󰇜   󰇛 󰇜  󰇛 󰇜   In words, it is the expectati on of the logarithm ic difference between the probabiliti es P and Q, where the expectation is taken using the probabiliti es P. T heref ore, smaller values for the divergenc e represen ts more accurate fit for the empirical PDF of transf ormer par ameters. 2) Transformer per unit rea ctance distribution Three different volt age levels, 115, 138, and 230 kV are selected t o r eport in this stu dy. T ransform ers are grou ped base d on their high voltage term inal and categorized into three volt age levels. For tran sformer per unit reactance, the conv erted per unit values to transf ormer power base is use d to identify the distributi on o f per unit reactance. Fig. 3 shows the empiric al PDF of transform er per unit reactan ce for different voltage levels. As found earlier in this paper, converted values of pe r unit reactan ce lie w ithin a narr ow range. According to KL divergenc e measure, it is found that the distributi on of per unit reactance can b e approximated with t Location-Scale (TLS) distributi on with three parameters as show n in the followin g distributi on functi on:  󰇛        󰇜   󰇡     󰇢     󰇡   󰇢 󰇯   󰇡     󰇢   󰇰 󰇡    󰇢  where  is the gamma function,  is the location par ameter,  is the scale parameter, and  is the shape parameter. The mean of the T LS distribution i s  and is onl y de fined for    and the variance is       and is only defined for    . Note that if rand om variab le  has a T LS distribution with parameters  ,  , and  , then    has a Stude nt’s t - distribution with  degrees of freed om. In probability and statistics, Student's t-distribut ion (or sim ply the t-distribution) is any member of a family of conti nuous probability distributions that arises w hen estimatin g the m ean o f a normally distribut ed population in situa tions where the sa mple size is s mall. 115 kV 138 kV 230 kV Figure 3. Empirircal PDF and T LS-fit of per unit re actance for 115, 138, and 230 kV tra nsformers. Table I shows the media n, mean, minimu m and maximum range, and the percentage o f per unit reactance l ying within [0.05, 0.2] p. u range for the two real world po wer grids. TABLE I . P ER UNIT REACTANCE ST ATISTICS FOR 115, 138, AND 230 K V TRANSFORMERS Transformer Per un it reactance Voltage Levels (kV) Median Mean Range % at [0.05, 0.2] 115 0.1291 0.1363 [3.92e-4, 1.0162] 81.88 138 0.1246 0.1381 [1.00e-4, 1.26] 82.01 230 0.1260 0.1392 [2.47e-4, 1.08] 87.33 3) Transformer Capa city Distribution Another key para meter o f a trans former is its capacit y o r MVA rati ng. For the set of data from real world power grids , there are trans formers with di fferent sizes from couple MV A to +1000 MVA. Also, d ue to the lack of detailed infor mation in some cases, the MVA rating of some transformers are set to a very large or s mall values. To exclude such cases, in addition to identifying the full range of trans former MV A rating, an 80% range centered at the median is d efined to get rid o f “extreme values” on bo th upper and lower bound s. T his will give us a m ore useful range where most transfor mers fall in. Table II shows t he median, mean, mini mum a nd maximum range, and 80% range for tr ansformers MV A ratings. TABLE II . MVA RATING STATISTICS FOR 115, 138, AND 230 K V TRANSFORMERS Transformer MVA rating Voltage Levels (kV) Median Mean Range 80% range 115 53 71.30 [3, 384] [22, 140] 138 83 117.24 [3.3, 616] [39, 239] 230 203 246.61 [10, 1380] [62.5, 470] Fig. 4 dep icts the e mpirical PDF of transfor mers MVA rating and the ap proximate d fit distribution for 115 kV transformers. No te tha t the results for 13 8 kV a nd 230 kV transformers w ill be presented later in a table. A ccording to the KL divergence, tra nsformers ca pacity i s appro ximated with Generalized Extreme Value (GE V) distribution with the minimum   value where its CD F is represented b y (5)  󰇛        󰇜   󰇭    󰇛   󰇜     󰇮  where  is location para meter,  is scale parameter, and    is shape para meter. Using this mathematica l distribution, on e can generate reasonable val ues for transformer cap acities in a given synthetic grid model. 4) Transformer X/R distribu tion The thir d im portant parameter of t ransform ers is the ratio of their per unit react ance to per unit resistan ce. Using such ratio, one can estimate the v alue of per unit resistance giv en the range and distri bution of per unit reactance of the transform er. These tw o param eters fo rm the real and imaginary parts o f transform er impedance that is necessary for power flow analysis in synth etic pow er netw orks. Table III shows the median, mean, m inimum and maximum range, an d 80% range f or trans formers MVA ratings. The 80% range is determined using the same approach as used in MVA rating det erminati on. Figure 4. Empirircal PDF and GEV-fit of MVA rating f or 115 kV transformers. TABLE III . X/R RATIO STATISTICS FO R 115, 138, AND 230 K V TRANSFORMERS Transformer MVA rating Voltage Levels (kV) Median Mean Range 80% range 115 25.39 37.83 [0.0577, 5.41e 3] [16.2 , 47.5] 138 29.58 39.73 [0.2033, 1.92e 3] [19.1, 54] 230 44.37 65.77 [0.1786, 4.03e 3] [25, 84] Fig. 5 sho ws t he e mpirical and approximated distribution for 115 kV transformers. Again, it is found th at the GEV distribution can fit the data b est accord ing to KL diver gence measure. So me ver y small X/R ratios co me fr om autotransformers, and the ballp ark is that if t he r atio is less than 4 to 1, it is an autotransfor mer. Figure 5. Empirircal PDF and G EV-fit of X/R ratio for 115 kV transformers. Table IV presents t he estimated parameters of the b est fitting functions for transfor mer MVA rating a nd X/R ratio. III. T RANSMI SSION L INES Transmission line parameters statistics are der ived using over 5 0000 lines from real p ower syste ms. Transmission lines are categorized based on their no minal voltage level which ranges fro m 0 .6 to 765 kV. Here we stud y lines with nominal voltage levels o f 115, 138, and 230 kV. We studied per unit reactance, X/R ratio, and line capac ities a s three critical parameters o f trans mission lin es to provide several validation metrics and guidelines for s ynthetic grid modeling. A. Tran smission line p er unit reactan ce distribution Fig. 6 shows the e mpirical PDF of tra nsmission line p er unit reactance a nd t he app roximated fit dis tribution for different voltage levels. TABLE IV . T HEORETICAL DISTRIBUTION FUN CTIONS ESTIMA TED PARAMETERS FOR TRANSFORMER S MVA RATING AND X/R RATIO Estimated Param eters MVA rating statistics 115 kV 138 kV 230 kV   = 0.1295   = 0.0990   = 0.1148  = 41.08  = 66.82  = 154.79  = 27.38  = 42.31  = 105.61  = 0.3732  = 0.4166  = 0.2433 X/R ratio statistics   = 0.0918   = 0.0949   = 0.0984  = 22.29  = 25.88  = 37.79  = 10.70  = 12.34  = 19.67  = 0.2135  = 0.2167  = 0.2594 115 kV 138 kV 230 kV Figure 6. Empirircal PDF and Exponential-fit o f per unit reacta nce for 115 , 138, and 230 kV tr ansmission line s. It is found that for all three voltage levels, p er unit reactance is mostly less than 0.02 p.u. and the density drops exponentially as reactance increases. According to the KL divergence, trans mission line r eactance is appr oximated with Exponential distrib ution w ith the mini mum   value where its PDF is represented by (6)  󰇛    󰇜        Using this mathematical d istribution, one can generate reasonable values for transmission li ne p er unit reactance i n a given synthetic grid model. N ote that, t he distribution o f per unit reactance for trans mission lines is v ery different from T LS distribution for those of transfor mers. T his i s bacuse o f p er unit conversion for transfore mrs a nd implies t hat in or der to have more stablized range for t ransmission lines reacta nce, it is better to study t heir act ual d istrib uted reactance (  km). This will be presented in our next c omprehensive st udy. B. Tran smission line ca pacity distribution Transmission line capacity is a critical parameter in variou s analysis s uch as optimal po wer flow (OP F) analysis, contingency anal ysis, and p ower grid expan sion pla nning. Therefore, here we studied the distribution of line capacit y for different v oltage levels to identify a useful guideline an d range for actual ca pacities in the real grids. Fig. 7 sho ws the empirical PDF of transmission l ine capac ity and t he approximated normal distrib ution with best estimated parameters based o n   for three different volta ge level s. Note that, unlike transfor mers the distribution o f MVA rating for transmission lines is approximated with nor mal distribu tion with higher mean val ues for each voltage leve l. 115 kV 138 kV 230 kV Figure 7. Empirircal PDF and Normal-fit of line capacity for 115, 138, an d 230 kV transmiss ion lines. C. Transmission line X/R ratio d istribution The third important par ameter of transmission lines is the reactance to resistance ratio . Using such ratio, one can estimate the value of per un it resistance given the range and distribution of per unit reactance of the line. These tw o parameters for m the rea l a nd imaginary parts of transfor mer i mpedance that is necessary for po wer flow analysis in synthetic power networks. Fig. 8 shows the empirical PDF of transmission line X/R ratio and the approximated d istribution with be st estimated parameters based on   m eas ure for three different voltage levels. It is found th at normal distributio n is th e best fit for this para meter based on t he empirical P DF derived from actual data from two power grids. As shown in Fig. 8, the X/R ratio of transmissio n lines for each voltage level is smaller than that of transfor mers. Also, note that for both transformers and transmission lines, this ratio gro ws as the voltage level increases. 115 kV 138 kV 230 kV Figure 8. Empirircal PDF and Normal-fit of line X/R ratio for 115, 1 38, and 230 kV tra nsmission lines. IV. C ONCLUSION AND F UTU RE W ORKS Statistic al analysis on transform ers and transm ission lines electric al param eters such as per unit reactance, MVA rating, and X/R ratio is performed in this study to provide both validation metrics and guideli nes for generating synthetic grid models. A large sample of real data on trans formers and transmis sion lines from two real-w orld power systems is used to obtain statistics for the electrical parameters. First, a comparis on made between per unit reactance calculated using system co mmon base and values calculated using transform er power base to decide which metric provides more stabilized range for per unit reactance of transfo rmers. It is found that using per unit reactance calculate d based on transfo rmer own MVA rating will give us a consis tently stabilize d range for per unit X over differ ent voltage levels . Next, using Kullback- Leibler diverg ence, w e tried to fit approxim ate dis tributi on functions on em pirical PDFs for elect rical parameters of branches. It is found that tran smission lines exhib it differen t statistic al prope rties than trans formers. Th e distri bution of X/R ratio for transm ission lines is appr oximated w ith norm al distributi on as opposed to the GE V distributi on of this paramete r for transf ormers. Also, this ratio is larger for transform ers compa red to t ransmis sion lines. It is also found that transf ormers/t ransmiss ion lines of th e higher voltage levels tend to h ave highe r power rating s and X/R rati os. Our analy ses provide a list of well-defined rules for validati on purpose in synthetic grid m odels. In addition, obtained fit distributions can be use d to configure electrical param eters of transmission netw ork in synthetic grid modeling. For future work, a comprehens ive stu dy w ill cove r m ore voltag e lev els in statisti cs derivati ons. We will also stu dy the interdependen ce of different electric al param eters of t ransmission network on voltage level. R EFERENCES [1] M . H. Athari a nd M. M. Ardehali, “Operational performance of energy storage as function of electricity p rices for on -grid h ybrid renewable energy system by opti mized fuzz y log ic controller ,” Re new. Ener gy , 2016. [2] H. Moradi, A. Abtahi, and M. Esfahanian, “Optimal energy management of a smar t residential combined heat, cooling and power,” Int. J. Tech. Phys. Probl. E ng , vol. 8, pp. 9 – 16, 2 016. [3] H. Moradi, A. Abtahi, and M. Esfa hanian, “Optimal operation of a multi - source microgr id to achieve cost and emi ssion ta rgets,” in 2016 IEEE Power and Ener gy Conference a t Illinois (PECI) , 2016, pp . 1 – 6. [4] M. H. Athari and Z. Wang, “Modeling the Uncertainties in Renewable Generation and Smart Grid Loads for the Study of t he Grid Vulnerability ,” 2016 IEEE Power E nergy Soc. Innov. Smart Grid Tech nol. Conf. , 2016. [5] M. Parashar, J. S. Thorp, and C. E. Seyler, “Continuum Modeling of Electrome chanical Dy namics in L arge- Scale Pow er Systems,” IEEE Trans. Circuits Syst. I Regul. Pap. , vol . 51, no. 9, pp. 1848 – 1858, Sep. 2004. [6] B. A . Carre ras, V. E. Ly nch, I . Dobson, and D. E. New man, “Cri tical points and transitions in an ele ctric power transmission model for cascading failure b lackouts,” Chaos An Interdiscip . J. N onlinear Sci. , vol. 12, no. 4, p. 98 5, 2002. [7] M. ROS AS -CASALS, S. VALVERDE, and R. V. SOL É, “TOPOLOGI CAL VULNERABILI TY OF THE EUROPEAN POWER GRID UNDER ERRORS AN D ATTACKS,” Int. J. Bifurc. Chaos , vol. 17, no. 7, pp. 2 465 – 2475, Jul. 2007. [8] D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small -wor ld’ networ ks,” Nature , vol . 393, no. 6684, pp. 440 – 442, Jun. 1998. [9] Z. W ang, R. J. Thomas, and A. S caglione, “Generating Random Topolo gy Power Grids,” in Procee dings of the 41st Annual Hawaii International Conference on S ystem Sciences (H ICSS 2008) , 2008, pp. 183 – 183. [10] Zhifang Wa ng, A. Scaglione, an d R. J. Thomas, “Generating S tatistically Correct Random Topolo gies for Testing Smart Grid Communication an d Control Netw orks,” IEEE Trans. Smart Grid , vol . 1, no. 1, pp. 28 – 39, Jun. 2010. [11] Z. Wang, A. Scag lione, and R. J. Tho mas, “The Node Degree Distrib ution in Power Grid and I ts Topology Robustness under Ra ndom and Sel ective Node Removal s,” i n 2010 IEEE International Conference on Communications Workshops , 2010, p p. 1 – 5. [12] Z. Wang a nd R. J. T homas , “On Bus Type Assignments in Random Topology Powe r Grid Mo dels,” in 2015 48th Hawaii Internationa l Conference on S ystem Sciences , 2015, pp. 2671 – 2679. [13] A. B . Birchfield, T. Xu, K. M. Gegner, K. S. Shetye, and T. J. Overby e, “Grid Structural Characteristics as Validation Criteria for Syntheti c Networks,” IEEE Tra ns. Power Syst. , pp. 1 – 1, 2016. [14] K. M . Gegne r, A. B . Bi rchfield, Ti Xu, K. S. Shetye, and T. J. Overbye , “A methodology for the creation of geographically realistic synthetic power flo w mode ls,” in 2016 IEEE Power and Ener gy Conference at Illinois (PECI) , 20 16, pp. 1 – 6. [15] A. B . Birchfield, K. M. Geg ner, T. Xu, K. S. Shety e, and T. J. Overby e, “Statistical Considerations in the Creation of Realistic Synthetic Power Grids for Geomagne tic Disturbance Studi es,” IEEE Trans. Power Syst. , pp. 1 – 1, 2016. [16] J. D. Glover, M. S. Sarma, and T. Overbye, Power Syst em Analysis & Design, SI Vers ion . Cengage Le arning, 2012. [17] S. K ullback and R. Le ibler, “ On information and sufficie ncy,” Ann. Math. Stat. , vol. 22, no. 1 , pp. 79 – 86, 1951. [18] D. MacKay , Information theory, inference and learning a lgorithms . Cambridge unive rsity press, 2003.