Inference, Estimation, and Prediction for StableOperation of Modern Electric Power Systems

Samuel Chevalier

Thesis Advisor: Luca Daniel (luca@mit.edu), MIT Professor of EECSCommittee Chair: Themis Sapsis (sapsis@mit.edu), MIT Associate Professor of Mechanical EngineeringCommittee Member: Steven Leeb (sbleeb@mit.edu), MIT Professor of EECS and Mechanical EngineeringCommittee Member: Petr Vorobev (p.vorobev@skoltech.ru), Skoltech Assistant Professor of Energy ScienceCommittee Member: Konstantin Turitsyn (turitsyn@mit.edu), D. E. Shaw Group

AbstractTo keep pace with the emerging social-ecological disruptions and technological progressions of moderns times, electrical

power systems must continually adapt. Of the many challenges posed by these shifting landscapes, ensuring the continuedstability of the system’s (a) electromechanical oscillations, (b) small-signal dynamics, (c) transient response, and (d) voltageprofile are of utmost importance. Without guarantees for system stability across these facets of operation, the system may notonly experience sever failure, but its owners and operators will be unable to address the higher level challenges associatedwith energy sustainability, efficiency and affordability. In order to address these challenges (a)-(d), this thesis develops a setof analytically rigorous yet practically oriented methods for ensuring stable power system operation. These methods leverageinference, estimation, prediction and analysis techniques from a variety of mathematical and engineering communities.

With the advent of Phasor Measurement Units (PMUs), power system operators have become increasingly aware of a varietyof oscillatory events which can compromise system stability. Forced oscillations (FOs), which are the result of extraneouscyclical perturbations, are especially troublesome since model based approaches cannot be used to predict their occurrence. Inorder to locate the source of these FOs, this thesis develops an equivalent circuit transformation for electrical power systems.By constructing the admittance, which is quantified as a frequency response function (FRF), associated with various gridcomponents, FOs are shown to appear in the equivalent circuit as current injections. A probabilistic framework, via Bayesiananalysis, is set up to locate the most probable source of these injections. As an extension, rigorous passivity analysis is appliedto the FRF associated with the full network model. This allows for the interpretation, analysis and improvement of anotherpopular FO source location method known as the Dissipating Energy Flow.

This frequency domain passivity framework is further leveraged, along with semidefinite programming techniques, in order togenerate certificates for stable operation of microgird power systems. By constructing (or measuring) the input-output frequencyresponse of various grid components, the components are analyzed at each separate frequency bin. Thus, a family of “passivitytransformation” matrices is generated; this family of matrices has the capability of guaranteeing the overall stability of anarbitrary interconnection of the considered components.

The proposed FRF framework may also be used for inference of wide-area system dynamics from online PMU measure-ments for improved transient stability simulation techniques. Because the structure of the inference problem is inherentlyunderdetermined, this thesis sets up a custom Frequency Domain Vector Fitting (FDVF) problem in order to test and show thatexternal dynamics may be uniquely and accurately inferred from ambient system perturbations. The problem is complicatedby measurement noise and by stochastic excitation coming from both the known-internal and the unknown-external systems.Null space analysis is performed to show system operators the unobservable subspace of the inference result, and mathematicalconsiderations are made for leveraging both prior expectations and the system’s known structural realities.

While performing dynamical inference at the transmission level is essential for ensuring stable dynamics, online estimation(or inference) of the state (i.e. complex voltage at each node) of the component distribution systems is also vitally important. Inthis thesis, we formulate a numerically expedient distribution system state estimation (DSSE) framework via Bayesian analysis.This DSSE framework incorporates a variety of real-measurement, virtual-measurement, and pseudo-measurement inputs, andit is formulated to use smart meter data in order to construct a series of novel linear constraints which are applied to thethree phase power flow problem. The reduced solution space for the constrained power flow problem will allow for significantcomputational benefits along with a solution of higher certainty. Using unscented transformations, the posterior distributionfor the DSSE problem is ultimately constructed and its probability is maximized. By leveraging statistical data from historicalsystem measurements, system operators are given a mechanism for rigorously quantifying how time lag between the smartmeter measurements and the estimation affects the ultimate uncertainty of the solution.

Index TermsBayesian analysis, critical infrastructure, forced oscillations, inference methods, inverse problems, passivity, phasor

measurement unit (PMU), power system dynamics, semidefinite programming, smart meter, state estimation, vector fitting.

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I. INTRODUCTION AND MOTIVATION

MODERN electric power systems (EPSs) are an engi-neering marvel, representing 140 years of incremental

advancement in design, construction, and operation. In orderto keep pace with the technological advances and social-ecological challenges of each new generation, these systemsrequire continual updating and modernization. This task ismade all the more challenging due to the fact that EPSsoperate on highly disparate timescales; such timescales rangefrom the microsecond effects of electromagnetic transientsand power electronic switching to the decade-scale timelinesassociated with expansion planning and generation capacityprocurement [1].

Over the last two decades, a variety of factors have leadto the severe disruption of EPSs’ conventional operatingparadigm. The primary factors contributing to this disruptioninclude the following:

• a dramatic increase in grid-scale renewable energy pene-tration and a corresponding decrease in traditional fossil-fuel based generation [2], [3];

• the advent of new Smart Grid technologies [4];• the increased popularity and reliance on microgrid sys-

tems [5], [6];• the increasing electrification of the transportation sec-

tor [7], [8];• and the distributed energy resources (DERs) which allow

traditional consumers to act as “pro-sumers” [4], [9].Although each of these factors represents an opportunity forthe enhancing efficacy and sustainability of EPSs, they alsorepresent new challenges associated with maintaining systemstability. The term stability is often used loosely in theliterature, but the IEEE/CIGRE Joint Task Force on StabilityTerms and Definitions indicates that power system stabilityrefers to “the continuance of intact operation following adisturbance” [10]. The disturbance type may range anywherefrom typical ambient load switching to a severe line fault.

Although many of the stated emerging factors present newchallenges for maintaining stable EPS operation, the roll-out of new Smart Grid technologies is accompanied by newopportunities for system operators to significantly enhancesystem stability. In particular, the deployment of Phasor Mea-surement Units (PMUs) [11], micro-PMUs (µ-PMUs) [12] andAdvanced Metering Infrastructure (AMI) [13] are increasingsystem observability in considerable ways. PMUs are deviceswhich can provide time synchronized measurements of avariety of grid signals at high sample rates (30 to 60 Hz); thesemeasurements are aggregated at Phasor Data Concentrators(PDCs) [14] and then relayed back to the system operators innear real-time. Typically, PMUs are capable of measuring volt-age and current phasors (magnitude and phase), frequency andrate of change of frequency. Active and reactive power flowsand injections are computed directly from the voltage andcurrent data. While PMUs are typically located at high voltagetransmission substations, µ-PMUs are placed at distributionfeeders and, potentially, throughout the distribution network.Due to smaller phase angle differences between nodes and

the higher degree of process noise (load switching), µ-PMUsare engineered to have a higher accuracy than transmissiongrade PMUs [12]. AMI is defined as the “integrated systemof smart meters, communications networks, and data manage-ment systems that enables two-way communication betweenutilities and customers” by the DOE [13]. The smart meterdevices measure load usage at the customer interface. Theyare typically capable of relaying active and reactive powerusage along with voltage magnitude, current magnitude andbasic power quality data back to utilities in 5, 15, 30, or 60minute intervals. As of 2018, approximately 86.8 million smartmeters had been installed across the US [15].

The massive amounts of real-time data being generated byPMUs, µ-PMUs and smart meters not only provide directobservability of previously unobservable system facets, butthey also allow for the formulation of previously unimaginedinverse problems. In formulating an inverse problem, theinherent objective is to reconstruct the particular “model”which generated some set of observed measurements [16],[17]. This thesis focuses on formulating inverse problemswhich either exploit these newly available data sets in newways or leverage simulated data which will be informed bythese new data streams. As posed, the solutions to theseinverse (or inference) problems will yield so-called actionableinformation. This information will allow system operators tomake dispatch and control decisions which ultimately enhancethe stability of the network.

The inherent goal of solving an inverse problem is modelreconstruction. In this thesis, a variety of different inverseproblems are formulated. Therefore, the term “model” inmodel reconstruction will simultaneously refer to parameterestimation, source identification, network state estimation, sys-tem identification, and more. In the following subsections, a setof four inverse problems are outlined: (a) locating the sourcesof forced oscillations, (b) determining passivity transforma-tions which allow for stability certification, (c) inferring widearea dynamics, and (d) estimating the state of a distributionsystem network. For each problem, a corresponding literaturereview is provided, and the potential impacts of the problemare outlined. In Section II, the proposed technical contributionsof this thesis, in each respective category, are outlined.

A. Forced Oscillations

With recent widescale deployment of PMUs across the UStransmission grid [18], system operators are becoming keenlyaware of the pervasive presence of low frequency oscillations.Generally, low frequency oscillations are either natural modes,attributed to poorly tuned control settings and large powerflows across weak tie lines, or forced oscillations (FOs), whichare caused by extraneous disturbances. FOs generally refer toa system’s response to an external periodic disturbance [19].Such external inputs may be related to a broad range ofcauses [19], [20], such as faulty controllers, turbine vibrations,or cyclical loads [21]–[23]. The appearance of FOs reducesthe quality of electric power and has potential detrimentaleffects on various equipment [21], [24]. More importantly,whenever a disturbance occurs at the frequencies close to one

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of the natural system modes, a resonance condition may leadto significant amplification of amplitude: a relatively smallperturbation on one bus can cause rather large power swingsin different locations around the system. An example of thiseffect is the 2005 WECC disturbance where a reasonably small20MW oscillation at the Nova Joffre co-generation powerplant in Canada resonated with one of the main inter-areamodes resulting in a 200MW power oscillation on the Oregon-California intertie [24].

Accordingly, there is a need in the EPS community forthe development of methods which are capable of usingon-line PMU data to trace the source of a FO. Locatingthe sources of FOs remains a challenging task due to theirsporadic nature, speed of propagation, and inability to bepredicted by the system operators’ dynamical models. It isaccepted that designing control methods for damping of forcedoscillations is impractical [25]; instead, disconnection of theidentified source with subsequent investigation of the causesof the disturbance is the main solution. A variety of sourceidentification techniques have been developed with varyinglevels of success [26]–[32]; many are outlined in a recentliterature survey [33] where the main requirements for suchmethods are also stated. A set of test cases for validatingdifferent source location methods is presented in [34]. Thesecases were developed in coordination with IEEE Task Forceon Forced Oscillations, and they allow for a standardizedexamination of all source detection algorithms.

In [26], eigenvalue decomposition of the linearized system’sstate matrix is used in conjunction with the FO’s measuredcharacteristics to perform source location identification. Theauthors of [35] employ machine learning techniques, via mul-tivariate time series analysis, to perform source identification;all off-line classifier training is based on simulated data. Afully data driven method, which employs convex relaxationto optimally locate sparse FO sources, is introduced in [36].Due to the characteristically narrow bandwidth of FOs, otherauthors have embraced frequency domain techniques. In [27],the pseudo-inverse of a set of system transfer functions aremultiplied by a vector of PMU measurements to yield a FOsolution vector.

An important class of source location methods, which aretermed the hybrid methods in [33], leverage both a knownsystem model and measured PMU data. Demonstrated in [37]and [38], these methods use measured PMU signals as inputsfor an EPS model. After simulating this model, the timedomain model outputs are compared with their correspondingmeasured PMU signals. Significant deviation between themodel predictions and the PMU measurements may indicatethe presence of a FO. These types of methods are also usedfor model validation. Model based source location algorithmsincorporate the unfortunate drawback of solution accuracybeing constrained by the accuracy of the model parametersused in the analysis. Purely data driven approaches, on theother hand, do not leverage known system structure anddynamics. Others have applied Bayesian analysis to EPSs inpast. For example, [39] used a Bayesian particle filter forpower plant parameter estimation, and [40] solved a maximum

a posteriori (MAP) optimization problem in the time domainto perform EPS parameter identification.

Of the many source location techniques currently availablein the academic marketplace, the so-called Dissipating EnergyFlow (DEF) method has enjoyed some of the most successfultesting results, both in simulation environments [25] and inreal-time applications [41] in the ISONE and WECC networks.The method was originally developed by Chen et al. [42] as theTransient Energy Flow method, but its underlying mathematicsleverage the Lyapunov functions from [43]. The DEF method,which was developed under the assumptions of a losslessnetwork and constant power loads, tracks the system-wide flowof so-called “dissipating energy” in order to locate the FOsource. One of the main advantages of this method is that ittracks the dissipating energy flow in all lines where PMU datais available, thus being naturally model independent.

The primary challenge to reliable DEF performance is thecontribution of dissipating energy from non-FO sources, suchas lossy transmission lines, lossy or negatively damped loads,and generator dynamics dominated by non-passive controllers.This phenomena has been evidenced in simulation [32] andin real application at ISONE. An actual example of thiscontribution may be found in Fig. 1 of [44]. If these con-tributions are large enough, the FO source can appear to be adissipating energy sink and the DEF method can fail. Despiteits inadequacies, the DEF’s excellent performance in real-timeapplication at ISO New England strongly implies that furtherresearch should be performed in order to more systematicallycharacterize the method. Shortcomings of the DEF methodhave been analyzed in [45], and [44] has recommended usingpassivity theory to interpret the method from a new math-ematical perspective, but no theoretical methods have beendevised for testing how the DEF method will perform in anarbitrary network. Such testing is essential in order to ensurethat the DEF can perform adequately in new environments,such as in microgrids where “R/X” line ratios are high andvoltage control is fast, or in networks that have particularlyresistive load pockets. To make such predictions, a systematicframework is needed in order to thoroughly study the DEF.

B. Decentralized Stability Criteria for Microgrids

In academia, industry, and defense, microgrids are becomingan increasingly popular topic [46]. According to the DOE,a Microgrid may be defined as “a group of interconnectedloads and distributed energy resources within clearly definedelectrical boundaries that acts as a single controllable entitywith respect to the grid. A microgrid can connect and dis-connect from the grid to enable it to operate in both grid-connected or islandmode” [5]. Advances in power electronictechnologies have lead to a significant decrease in renewableenergy generation costs: this has inspired discussions aboutsplitting the existing distribution grids into autonomous sys-tems. Subsequently, there has been significant progress in thedevelopment of control architectures for power electronics-interfaced generation, further allowing for flexible microgridoperation [47], [48].

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It was quickly realized that control methods which werestandard for large scale EPSs have rather limited applicationsin microgrids due to stability constraints [49]. Moreover,modelling approaches (e.g. modelling based on time-scaleseparation) routinely used for conventional EPSs appeared tobe inadequate for microgrids, which demand new modelingtechniques [50]. Due to the characteristic differences betweenlarge scale conventional power grids and microgrids, newapproaches and methodologies are necessary for certifyingmicrogrid stability. Relevant stability definitions and state ofthe art methodologies are reviewed in [51] by the IEEETask Force on Microgrid Stability Definitions, Analysis, andModeling.

It is generally possible to employ full-scale dynamic model-ing for stability analysis of microgrids, directly calculating theeigenvalues of the state matrix for any specific operating point.This is done in [49], where the stability of a droop controldominated microgrid is analyzed via root locus. However,such an approach assumes the full knowledge of systemconfiguration which is much less likely for a microgrid thanfor a conventional EPS. Certifying the stability of microgridsystems can be challenging due to the lack of information onexact values of system parameters. Moreover, performing full-scale stability analysis for every possible microgrid configura-tion is most likely economically, technically, and numericallyinconvenient. Authors in [52] developed a low-dimensionalmodel for inverter-based microgrids which allowed for thepinpointing of the main sources of instabilities and pavedthe way towards development of completely decentralizedinterconnection rules for such systems. However, the meth-ods still rely on rather specific dynamic models of systemcomponents (namely, droop-controlled inverters) and assumedat least partial knowledge about the system configuration.Moreover, stability certificates formulated in [53], while beingdecentralized, depend not only on the settings of the systemcomponents, but also on their interconnection.

The celebrated concept of dissipative dynamic systems [54],[55] allows for formulation of stability certificates for an entiresystem through the separate consideration of its components:if every component of the system is dissipative, then the wholesystem is also dissipative, and therefore stable, irrespective ofthe way components are interconnected. A specific form ofdissipativity, known as passivity [56], has allowed the formu-lation of rather simple, although not always easily realisable,constraints on input admittances of power system components[57]–[59]. The advantage of such an approach is that inputadmittances of individual components do not have to be knownfrom a model, but can simply be measured. However, it is notstraightforward to apply the method to components that are notpassive and cannot be made so by simple adjustments of theircontrol settings. Researchers have advocated for a passivity-based approach for microgrid stability certification. In [60],an active stabilization (control) strategy was proposed whichwould enforce passive terminal behaviour of all interfacecontrollers. Once engineered to behave passively, the systemwas guaranteed to be stable for an arbitrary interconnectionof converters. The robustness of such passivity-based control

laws was analyzed in [61].There is, therefore, the need for simple but reliable stability

certificates that can be routinely used for a wide class of micro-grid configurations. In an ideal operating paradigm, a standardscould be developed for typical microgrid components that willallow stable operation under arbitrary interconnections. Sucha system would provide a significant step towards realizingthe so-called “plug-and-play” operation of a microgrid [62].

C. Wide Area Inference

The IEEE/CIGRE Joint Task Force on Stability Terms andDefinitions defines transient stability as “the ability of thepower system to maintain synchronism when subjected to asevere disturbance, such as a short circuit on a transmissionline” [1]. Transient stability is therefore an essential com-ponent of what is known as Dynamic Security Assessment(DSA) [63]. An overview of the modern transient stabilitytechniques which are available in the literature are cataloguedin [64]. Despite years of research on the topic, most EPS oper-ators still depend on copious time domain integration studies inorder to ensure the transient stability of their system [65]. Dueto the highly interconnected nature of the massive US powergrid, such studies are a considerable computational burden.Additionally, the dynamical model is prone to having mistakesdue to the time varying nature of the system.

With the increased observability provided by PMUs throughWide Area Monitoring Systems (WAMS), system operatorsare beginning to have complete real time observability of theconnections (i.e. tie lines) between their respective “internal”regions and the “external” regions to which they are tied. Withthis complete observability comes the opportunity to infer theunderlying input/output dynamics of these neighboring sys-tems. Thus, creating a dynamical equivalent of these externalsystems is appearing increasingly realistic. Using advancedsystem identification (SysID) techniques, high sample ratePMU data can be used to construct an equivalent black boxdynamical model. Once this inferred model is constructed, itcan potentially be used to replace or update the erroneousanalytical models of the external systems used in the transientstability solver. If this black box model is updated online,the transient stability solver will always have access to themost up-to-date external system models [66]. Such analysis isfurther motivated by the fact that in stability simulations, localsystem effects are of primary concern while external systemscan often times be reduced [67]. Most modern EPS simulators,such as PSS/E and DYNRED, have external system reductiontechniques built into the software package [66], [68].

Since PMU data can only reconstruct dynamics of up to ∼8Hz, the dynamics of interest are primarily electromechanicaldynamics. There is a fairly vast literature on the estimation ofelectromechanical modes in EPSs [69]–[77]. One of the mostpopular methods, known as Prony analysis, was first appliedto power systems in the Pacific Northwest in 1990 [78]. Themethod uses so-called transient “ring-down” data in orderto linearly approximate the frequency, phase, amplitude, anddamping of the different system modes. Prony-type methodsare primarily applicable when system transient response is

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strong. New methods seek to perform modal identificationfrom ambient data; many of the methods are outlined in [79].Authors in [76] collect 50 minutes of ambient data followedby transient data associated with the controlled application ofa 1400 MW brake resistor. They show that Prony analysis ofthe transient data and solutions to the Wiener-Hopf Equationsfor linear prediction analysis of the ambient data are both ableto predict the four dominant system modes and their respectivedamping ratios.

Methods for explicitly building the input/output transferfunction matrices associated with wide area systems fromPMU data are less developed. Authors in [69] propose ag-gregation techniques based on coherency methods in orderto construct simple dynamic equivalents of wide areas; pa-rameters of these equivalent reductions are then inferred.Using a simple autoregessive with exogenous input (ARX)model (where parameter values are chosen based on a linearleast squares solution), [67] develops process for computing adynamic equivalent with simulated or measured data. Voltagemagnitude and frequency are chosen as inputs, with activeand reactive power flows across key tie-lines are chosen asoutputs. Each tie line is considered independently. SimpleMIMO transfer function approaches are developed in [68],[80], where model order is arbitrarily selected and polesand zeros are iteratively selected based on some unspecifiednumerical procedure. In each of these papers [67], [68], [80],measurement noise and process noise (i.e. load noise) arenot considered, and dynamics are inferred after a transientswitching event rather than from ambient data.

For online oscillation damping control tuning, [81] uses alower order autoregressive moving average exogenous (AR-MAX) model for MIMO system identification. In [73], transferfunction identification is performed on PMU data, but themodes of the system are assumed known apriori, and “arx.m”from MALTAB is used to perform the system identification.Accordingly, there exists a need in the EPS community forthe further development of methods which are capable ofperforming online SysID of external wide area systems fromambient data in numerically expedient ways; such methods arecurrently lacking.

D. Distribution System State Estimation (DSSE)

While the transmission grid is vitally important for trans-porting electrical power long distances, the distribution net-works facilitate the final step of power delivery to homesand businesses. Distributed energy resources (DERs), suchas Telsa powerwalls and residential rooftop photovoltaic sys-tems, automated sensing equipment equipped with telemetrycapabilities, such as µ-PMUs and smart meters, and loadswhich are capable of reactively responding to real-time pricingsignals, are all majorly disrupting the standard operation ofdistribution networks [82]. Accordingly, distribution systemoperation and control are receiving much more attention inthe EPS research community than they have in the past.

In order to properly operate and control these systems,knowledge of the nework “state” is vitally important in-formation. In the seminal state estimation works by Fred

Schweppe [83]–[85], the state of an EPS is defined as “thevector of the voltage magnitudes and angles at all networkbuses.” Furthermore, state estimation is defined as “a data pro-cessing algorithm for converting redundant meter readings andother available information into an estimate of the static-statevector.” The many potential advantages associated with DSSEare discussed in [86]. Despite these advantages, very few util-ities have implemented real time DSSE in their systems [87].This fact alone suggests that more research must be completedbefore utilities choose to invest in and adopt DSSE as acommon tool for system management. Although transmissionsystem state estimation (TSSE) is common practice amongsystem operators (often being run every two minutes), TSSEmethodology must be properly adapted for use in distributionnetworks. The primary differences (challenges and benefits)associated with DSSE are as follows: large “R/X” ratios(resistance cannot be neglected); unbalanced operation; dis-jointed phase extensions; highly time varying loads; disparatemeasurements (µ-PMU, SCADA, AMI, etc); primarily radialin topology; a higher degree of network topology uncertainty;and a high degree of unobservability.

A famous solution to the three-phase load flow problemon distribution circuits, known as forward-backward sweepmethod, is first proposed in [88], and a recent review ofstate-of-the-art DSSE techniques is given in [87]. Researchgaps are also identified: DSSE methodologies for wide areamonitoring, data synergy techniques for incorporating hetero-geneous data types, measurement collection and coordination,and the integration of TSSE and DSSE solvers. Authorsin [89] provided side-by-side comparisons of primary DSSEframeworks: weighted least squares (WLS), weighted leastabsolute value (WLAV), and the Schweppe Huber GeneralisedM (SHGM) estimator.

One of the more famous DSSE methods, formulated withrectangular branch current flows in a system with few mea-surements and many pseudo-measurements, is given in [90].The effect of including PMUs is considered in [91]. A similar(but linearized) formulation, developed for “smart distributionsystems”, is proposed in [86]. In this paper, measurementvariances are carefully constructed using the so-called deltamethod. Stochastic optimization methods are leveraged in [92]in order to minimize meter investment costs while takingDSSE uncertainty into consideration. Meter placement is alsoconsidered in [93], where the authors seek to minimize the busvoltage variance at buses without measurement equipment.

Many other novel DSSE techniques have been proposedin the literature in recent years. The effects of direct smartmeter measurement integration are characterized in [94]. Alinear Bayesian state estimator is compared to the typicalweighted least squares formulation in [95]. By leveraging loadforecasting, [96] also incorporates a Bayesian state estimatorin a linearized network with limited sensing. State estimationvia Kalman filtering, combined with load control, is proposedin [97]. In order to leverage the increasing observabilityprovided by smart meters in rigorous yet numerically efficientand practical ways, there exists a need in the EPS communityfor new DSSE solution methods.

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II. PROPOSED RESEARCH OBJECTIVES

The following subsections outline the proposed technicalcontributions of this thesis. For research which has not alreadybeen completed, explanations for how the research will beconducted are also included.

A. Forced OscillationsIn order to exploit the characteristically narrow bandwidth

of FOs, we have proposed the use of frequency domain meth-ods in order to tackle the problem of locating the sources ofFOs. Via an equivalent circuit representation, we have shownthat FOs in the time domain show up like current injections inthe frequency domain. This is shown schematically in Fig. 1,where generators are converted into their equivalent frequencyresponse functions (FRFs). These FRFs relate AC terminalvoltage and current perturbations and can therefore be thoughtof as admittances.

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Fig. 1. Equivalent circuit representation (right) of a power system.

The plot in Fig. 2 shows the spectrum a generator which isproducing a FO at 2 Hz. At that frequency, there is significantdeviation between the measured current and the predictedcurrent, well beyond what the measurement noise Σ2 couldexplain. Thus, FO sources can be located by determining thelocations of these equivalent current injections. The full resultsof this analysis are presented in our IEEE Transactions onPower Systems journal paper [31].

0 2 5 10Frequnecy (Hz)

10-3

100

Squar

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tral

Mag

nitude ...~I! Y ~V

...2

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Fig. 2. Equivalent circuit representation.

These methods are extended to consider situations where thegenerator parameters (time constants, controller gains, etc.)which are used to construct the FRFs are uncertain. In thissituation, we propose leveraging the spectral content outsideof the FO frequency band to “tune” the generator parameters.To formulate the problem mathematically, we propose a proba-bilistic framework via Bayesian analysis. Assuming Gaussianmeasurement noise and Gaussian parameter uncertainty, weconstruct a prior function which conveys certainty aboutparameter value and a likelihood function which quantifiesthe probability of the observed data given the stated model.We then maximize the posterior distribution (so-called MAP)via numerical optimization in a two-stage process to find themost likely set of generator parameters and current injections

Fig. 3. DEF energy flows for a system with lossy (left) and lossless (right).

which explain the observed data. Results from these methods,along with further implementation details, may be found inour IEEE Transactions on Power Systems journal paper [98].

Using passivity theory, the physical characteristics of ourproposed FRFs (and the full equivalent circuit representationof an EPS) can be leveraged to investigate the celebratedenergy-flow based DEF FO source location method of [30]. Inour PESGM conference paper [44], we show that the quadraticenergy flow which is traced by the DEF method via a timedomain integral is positive definite in a lossless “classical” [99]power system due to the underlying passivity of the FRFs inthe network. The elements are only passive, however, afteran identified “passivity transformation” has been applied tothe FRFs. We go on to prove that classical generators arepassive devices, constant power loads are lossless, and constantimpedance terms are indefinite.

In a follow-up IEEE Transactions on Power Systems jour-nal submission (currently under its second round of re-views) [100], we propose an expsion of this passivity frame-work in order to offer a rigorous justification of the DEFat the system level. Using a set of defined basis matrices,we first prove that there exists no passivity transformationwhich can simultaneously render all components of a classicalpower system passive. With many other intermediate results,we ultimately go on to develop an algorithm which is capableof analytically predicting whether or not the DEF method willperform successfully in an arbitrary network. The particularalgorithm is simulation-free and does not depend on theoriginating characteristics of a FO source; only the topologicallocation of the source is necessary. For example, the circuiton the left in Fig. 3 shows perturbative “energy” flows (as theresult of simulation) for when a FO is applied at bus 31. Dueto resistive components in the network, all generator busesappear as energy sinks; this unsuccessful DEF performanceis predicted by our algorithm. The figure on the right showsenergy flows for when the resistive components have beenentirely removed, thus making the source readily identifiable;this successful DEF performance is also predicted by ouralgorithm. Full details are found in [100].

B. Decentralized Stability Criteria for Microgrids

To further build upon the passivity framework developed inthe previous works, we have proposed a decentralized stability

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Fig. 4. Eigenvalue plots for applied transformation M1 (left) and M2 (right).

criterion for microgrid power systems1. As is commonlyknown, if the transfer functions of the components in a systemare all passive, then the arbitrary interconnection of those samecomponents is guaranteed to be stable. The same result istrue, though, if there is some transformation matrix whichrenders all system matrices passive. In our published ACCconference paper [101], we further generalize the result toshow that if we can, at each narrow frequency band, findsome common matrix which renders all FRFs passive, thenthe system is stable at each narrow frequency band. Thus, byconsidering each frequency band independently, the overallstability of the system can be assessed. For example, theeigenvalues associated with a droop controlled inverter andits interconnecting line are shown in Fig. 4. Passivity amongboth components is guaranteed for ω/Ω0 > 0.15 for someapplied transformation matrix M1, while passivity amongboth components is guaranteed for ω/Ω0 < 1 for someother applied transformation matrix M2. We are thus ableto guarantee stability for arbitrary interconnection of thesecomponents.

We plan to expand these results by studying a varietyof common microgrid components, including doubly fedinduction generators (DFIGs), grid following inverters, andsynchronous generators with fast voltage regulation. By con-sidering the family of eigenvalues across the frequency domainassociated with these components’ transfer functions, we planto use semidefinite programming techniques in order to deter-mine the passivity transformation matrices which will renderthe system components passive. Typically, there are particularmodes of instability which are of greatest concern to powersystem operators. Therefore, we will identify these modes anddevelop methods which are capable of ensuring stability inthese specific regions of the frequency domain.

Since analytical models are not always available, methodsfor measuring the FRFs of various microgrid components willalso be leveraged (such as those developed in the followingsubsection). Thus, determining the matrices which rendersystem components passive will be posed as a type of inverseproblem. Through publication in the IEEE Transactions onPower Systems journal, we plan to disseminate this research tomicrogrid operators and researchers. Using our novel measureof “passivity margins”, operators will be given a better senseof how far controller gains can be pushed until the system canno longer be rendered passive in a certain frequency band.

1While our proposed methodology is general and can be applied to anyEPS, microgrids are where the approach will be most practical.

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Fig. 5. Internal and external subsystems of a test EPS.

Fig. 6. In/Out: Magnitude and Phase (80% of correlation)

C. Wide Area Inference

In order to accurately infer wide area system dynamics, we(MIT and Stefano Grivet-Talocia’s lab at Politecnico di Torino)have experimented with the use of frequency domain vectorfitting (FDVF) [102]. The FDVF procedure uses observedfrequency domain data to reconstruct the underlying linearmodel which produced the data via an iterative selection ofpoles. Using FDVF, our goal is to reconstruct the MIMOtransfer function associated an external EPS. To perform initialtests, we constructed various configurations of the experimen-tal test system given in Fig. 5. In this system, the externalsystem contains two third order synchronous generators withvoltage control and a “ZIP” load. The internal system containsa generator, an induction motor, a ZIP load, and droop-controlled inverter circuit. Initially, we have excited the systemby applying load switching noise to the loads in the internalsystem. Reconstruction of the the external system’s frequencyresponse function, as compared with the actual frequencyresponse function, can be seen in Fig. 6.

Our primary objective is to develop a practical FDVF-basedalgorithm which can be used by system operators to inferthe equivalent dynamics at the boundaries of their systems.Many considerations, such as measurement noise, multiplesources of excitation, and optimal input/output signal choices,have been considered and will be experimented with. Inher-ently, this is an underdetermined inverse problem, becausethere are more unknown polynomial coefficients than thedata can necessarily reconstruct. Therefore, in this work, wewill develop methods which are capable of constructing thesubspace of the unobservable nullspace. This will give systemoperators an understanding of the quality of the inferencesolution. To overcome the ill-posed nature of the problem,a Bayesian framework will be formulated which will lightlyregularize the solution. Since the inference is being performedonline, previously known solutions can be used to regularizethe results. Other known system characteristics will also beexploited.

In order to implement and test our methods, we plan touse a variety of small, medium, and large scale test networksto perform simulation-based testing. Once our algorithms

7

have matured, we will move on to real PMU data (althoughvalidating the results will be challenging). At the conclusionof this project, we plan to publish our results in the IEEETransaction on Power Systems journal. The results will bedisseminated among utility and system operators who mayhave interest in the inference algorithm.

D. Distribution System State Estimation

While smart meter data is becoming a common measure-ment to exploit in the DSSE problem, the literature usesit exclusively in the context of the nonlinear power flowproblem. By leveraging smart meter voltage, current injection,and power factor angle measurements, we are exploring theconstruction of a novel state estimation framework whichuses this smart meter data to construct a series of linearconstraints on the power flow problem. These constraintswill be truly linear, rather than something imposed from alinearized system. Once constrained, the lower dimensionalpower flow problem will have a highly restricted solutionspace, allowing for definite computational benefits.

Using the prior work [95] of our collaborator (Luca Schen-ato’s lab at the University of Padova) as a starting point, wewill seek to construct an optimal Bayesian estimator usingthe reduced power flow formulation described above. Usingunscented transformation techniques via proper sigma pointselection and distribution variances parameterized by the timepassed since the measurement was taken, we will carefullyoptimize our posterior distribution such that we arrive at themost likely (probabilistically speaking) state of the network.

In order to make our approach general enough for practicalapplications, it will be developed for use in realistic unbal-anced three phase networks with disjoint phase extensions. Indeveloping and testing our DSSE, we will leverage small scaletest cases for proof-of-concept expositions, and large scale(8500 node test feeder circuit) test cases for numerical testing.The work will be published through the IEEE Transactions onPower Delivery journal.

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