An Online Network Model-Free Wide-Area Voltage Control Method Using PMUs
AACCEPTED BY IEEE TRANSACTIONS ON POWER SYSTEMS ON FEBRUARY 6, 2021 1
An Online Network Model-Free Wide-Area VoltageControl Method Using PMUs
Georgia Pierrou,
Student Member, IEEE, and Xiaozhe Wang,
Senior Member, IEEE
Abstract —This paper proposes a novel online measurement-based Wide-Area Voltage Control (WAVC) method using PhasorMeasurement Unit (PMU) data in power systems with FlexibleAC Transmission System (FACTS) devices. As opposed to previ-ous WAVC methods, the proposed WAVC does not require anymodel knowledge or the participation of all buses and considersboth active and reactive power perturbations. Specifically, theproposed WAVC method exploits the regression theorem of theOrnstein-Uhlenbeck process to estimate the sensitivity matricesthrough PMU data online, which are further used to designand apply the voltage regulation by updating the referencepoints of FACTS devices. Numerical results on the IEEE 39-Bus and IEEE 68-Bus systems demonstrate that the proposedmodel-free WAVC can provide effective voltage control in variousnetwork topologies, different combinations of voltage-controlledand voltage-uncontrolled buses, under measurement noise, andin case of missing PMUs. Particularly, the proposed WAVCalgorithm may outperform the model-based WAVC when anundetected topology change happens.
Index Terms —measurement-based estimation, Ornstein-Uhlenbeck process, phasor measurement unit (PMU), secondaryvoltage control, wide-area voltage control
I. I
NTRODUCTION
Voltage stability is crucial to ensure the normal operation ofpower systems. To avoid voltage instability/collapse, a classicvoltage control scheme, also known as primary voltage control,is typically adopted, which utilizes the conventional generatorAutomatic Voltage Regulators (AVR) to maintain voltage lev-els at generator buses in case of local perturbations. However,the growing need to automatically coordinate the differentcontrol sources led to the secondary voltage control, whichhas been effectively implemented in French and Spanish powernetworks [1], [2]. Specifically, secondary voltage control aimsto improve the overall voltage profile by controlling on-fieldreactive power resources, such as Flexible AC TransmissionSystems (FACTS) devices, at some pilot buses [3].In recent years, the increasing amount of Phasor Measure-ment Units (PMUs) provides a unique opportunity for theevolution of secondary voltage control. Wide-Area VoltageControllers (WAVC) have been proposed by utilities (e.g., [4],[5]) as a more sophisticated way to conduct the secondaryvoltage control using high-frequency, time-synchronized PMUmeasurements. Indeed, different PMU-based WAVC meth-ods can be found in the literature. In [6], a PMU-based
The authors are with the Department of Electrical and Computer En-gineering, McGill University, Montreal, QC H3A 0G4, Canada. (e-mail:[email protected], [email protected])This work is supported by the Natural Sciences and Engineering ResearchCouncil (NSERC) under Discovery Grant NSERC RGPIN-2016-04570 andthe Fonds de Recherche du Qu´ebec-Nature et technologies under Grant FRQ-NT PR-253686.
Automated Voltage Control and Automated Flow Control isdesigned based on the optimal PMU location. An adaptivemeasurement-based voltage control algorithm that exploitsmeasurements as feedback control inputs for FACTS devicesis proposed in [7]. The authors in [8] present a nonlinearconstrained optimization algorithm for voltage control thatestimates the reactive power disturbances using PMUs online.In [9], a PMU-based WAVC algorithm is developed to controla FACTS device in real-time. An extension that takes intoaccount the measurement time delay is proposed in [10].Nonetheless, the WAVC formulations of all the aforemen-tioned works strictly rely on network topology, as the sus-ceptance matrix B is needed. However, grid-wide inter-areatopology information is provided only on an hourly basis [11].Thus, accurate network topology may not always be available.Indeed, topology errors have been recorded in the literatureas a result of errors in the status of circuit breakers such asisolation switches [12] or totally unobservable cyber attacks[13]. Moreover, the aforementioned WAVC works appliedthe susceptance matrix B to describe the sensitivity relationbetween the voltage deviation and the reactive power change,i.e., ∆ Q = J QV ∆ V ≈ B ∆ V , which neglects the impacts ofreal power perturbation on voltage magnitudes and is only trueunder the assumptions of flat voltages and the participation of all buses in the voltage control. However, such assumptionsmay not hold in practical applications due to various operatingconditions, missing PMUs, and limited communication links.In this paper, we propose a novel online model-free WAVCusing PMU data, which is completely independent of modelknowledge, topology information and parameter values. Par-ticularly, we regard the power system as a stochastic dy-namic system and exploit the regression theorem of Orn-stein–Uhlenbeck process to estimate the sensitivity matricesfrom PMU data, i.e., bus voltage magnitude and angle mea-surements, in near real-time. Then the estimated sensitivitymatrices are further used to design and apply the WAVC byupdating the reference points of SVCs, or other similar typesof FACTS devices. To the best of the authors’ knowledge,this work seems to be the first attempt to develop a model-free WAVC. The main advantages of the proposed method aresummarized as follows:1) The proposed WAVC algorithm, being model-free, can beimplemented online to provide effective voltage controlin various network topologies, operating conditions anddifferent combinations of voltage-controlled and voltage-uncontrolled buses.2) The proposed WAVC method does not require the par-ticipation of all buses and thus provides more flexibility a r X i v : . [ ee ss . S Y ] F e b CCEPTED BY IEEE TRANSACTIONS ON POWER SYSTEMS ON FEBRUARY 6, 2021 2 when practical constraints (e.g., missing PMUs, limitedcommunication infrastructure) arise.3) Unlike previous works, the proposed method provideseffective voltage control considering both active andreactive power perturbations.4) Numerical studies show that the proposed WAVC remainseffective under measurement noise and missing PMUs.The proposed method may even outperform the model-based WAVC if an undetected topology change occurs.In addition, the model-free nature of the proposed methodmay offer additional flexibility in a decentralized frameworkin case of network partitioning into zones (e.g., [14], [15]), yetthis paper will focus only on a centralized implementation.Lastly, it is worth noting that, to achieve the estimation ofthe sensitivity matrices, a considerable amount of PMU datais required (300s measurements with a frequency of 60Hzin this paper), which nevertheless is still a reasonable timewindow for online application [16]. However, no networkinformation and topology are needed. Thus, it is believedthat the proposed model-free WAVC utilizing PMUs maycompliment the traditional model-based WAVC to maintainthe voltage level of power systems, when accurate networkinformation is not guaranteed.The remainder of the paper is structured as follows: SectionII introduces the stochastic dynamic power system modelfor voltage control. Section III reviews the mathematicalformulation for WAVC. In Section IV, the proposed model-free WAVC algorithm is elaborated. Section V provides adetailed numerical study that demonstrates the effectivenessof the method. Section VI gives the conclusions.II. T HE S TOCHASTIC D YNAMIC P OWER S YSTEM M ODEL
Voltage stability is a dynamic phenomenon by nature. Loaddynamics and the associated control will greatly affect thevoltage stability of a power system [17]. Also, power sys-tems are constantly experiencing small perturbations aroundthe steady-state operating point, as a result of varying loadand generation patterns. In view of this, in contrast to thestatic approach adopted by previous WAVC works, in thispaper we consider the inherent stochastic dynamics of powersystems, which will be shown to provide an interesting way ofextracting important information regarding the physical modelfrom PMU measurements.
A. Load Dynamics
Load dynamics play a major role in voltage stability, whileload buses equipped with Load Tap Changers (LTCs), FACTSetc. provide great flexibility in voltage control. Therefore, loadbuses have been commonly selected as the main agents forvoltage control (e.g., [18]). In this paper, we will design theWAVC at the load side based on the load dynamics and control.However, detailed generator dynamics are also incorporated inthe simulation study in Section V to demonstrate the efficacyof the proposed method in real-world applications.Assuming that m out of L loads are dynamic loads that canbe characterized by the dynamic load model proposed in [19]: ˙ θ k = 1 τ θ k ( P k − P sk ) (1) ˙ V k = 1 τ V k ( Q k − Q sk ) (2) P k = N (cid:88) j =1 V k V j ( − G kj cos θ kj − B kj sin θ kj ) (3) Q k = N (cid:88) j =1 V k V j ( − G kj sin θ kj + B kj cos θ kj ) (4)where k ∈ { , , ..., m } ; θ k V k are the bus voltage angle andmagnitude, respectively; τ θ k , τ V k are the time constants ofactive and reactive power recovery; P k , Q k are the active andreactive power absorptions in terms of the power injected fromthe network to the dynamic load buses; P sk , Q sk are the steady-state active and reactive power absorptions.The applied load model can adequately characterize a vari-ety of load types (e.g., thermostatic loads, induction motors,loads controlled by LTCs) in the voltage stability study, whosedifference lies in different values of the time constants τ θ k and τ V k that may range from milliseconds to several minutes.The static loads can also be naturally represented by takingthe limit τ θ k → , τ V k → [20]. Similar types of loadare also introduced in [20]–[22]. The applied model capturesthe qualitative load behavior over a wide range of voltagemagnitudes observed in the voltage stability study, wherevoltage magnitudes will respond to load power to maintainreactive power balance [23].Random load variations may be caused by the aggregatebehavior of individual users [20]. To model load fluctuations,as a common and reasonable approach [24], we add Gaussianstochastic perturbations to the steady-state power: dθ k = 1 τ θ k ( P k − P sk ) dt − τ θ k P sk σ Pk dξ Pk (5) dV k = 1 τ V k ( Q k − Q sk ) dt − τ V k Q sk σ Qk dξ Qk (6)where σ Pk , σ Qk describe the standard deviation of the stochasticload perturbation for the active and reactive power; ξ Pk , ξ Qk areWiener processes.Equations (5)-(6) can be linearized and written in thefollowing vector form: (cid:34) d θ d V (cid:35) = (cid:34) T − θ T − V (cid:35) (cid:34) ∂ P ∂ θ ∂ P ∂ V ∂ Q ∂ θ ∂ Q ∂ V (cid:35) (cid:34) θV (cid:35) dt + (cid:34) − T − θ P s Σ P − T − V Q s Σ Q (cid:35) (cid:34) d ξ P d ξ Q (cid:35) (7)where θ = (cid:104) θ , ..., θ m (cid:105) T , V = (cid:104) V , ..., V m (cid:105) T , T θ = diag (cid:104) τ θ , ..., τ θ m (cid:105) , T V = diag (cid:104) τ V , ..., τ V m (cid:105) , P = (cid:104) P , ..., P m (cid:105) T , Q = (cid:104) Q , ..., Q m (cid:105) T , P s = diag (cid:104) P s , ..., P sm (cid:105) , Q s = diag (cid:104) Q s , ..., Q sm (cid:105) , Σ P = diag (cid:104) σ P , ..., σ Pm (cid:105) , Σ Q = diag (cid:104) σ Q , ..., σ Qm (cid:105) , ξ P = (cid:104) ξ P , ..., ξ Pm (cid:105) T , ξ Q = (cid:104) ξ Q , ..., ξ Qm (cid:105) T CCEPTED BY IEEE TRANSACTIONS ON POWER SYSTEMS ON FEBRUARY 6, 2021 3
We denote x = (cid:104) θ , V (cid:105) T , A = (cid:34) T − θ ∂ P ∂ θ T − θ ∂ P ∂ V T − V ∂ Q ∂ θ T − V ∂ Q ∂ V (cid:35) , H = (cid:34) − T − θ P s Σ P − T − V Q s Σ Q (cid:35) , ξ = (cid:104) ξ P , ξ Q (cid:105) T , then(7) takes the following compact form: d x = A x dt + Hd ξ (8)Hence, the stochastic dynamic load model in ambient condi-tions can be represented as a vector Ornstein-Uhlenbeck pro-cess [25]. The system state matrix A corresponds to a scaledset of sensitivity matrices and carries significant informationof the system operating state, as will be shown in Section III.In Section IV, we will propose a purely data-driven methodto estimate the matrix A , which is further utilized to developthe WAVC algorithm. B. SVC Modeling
FACTS devices have been widely used for wide-area controlwith various control objectives. Among the most commonlyused FACTS devices for reactive power compensation andvoltage control is the Static VAR Compensator (SVC), whichis a thyristor controlled reactor-based shunt FACTS device[26]. In this paper, SVCs are assumed to be installed atsome dynamic load buses to perform the voltage control. Inother words, the dynamic load buses with SVCs are termedas voltage-controlled buses. The following dynamic model isused for the SVC [27]: ˙ V M = 1 T M ( K M V k − V M ) (9) ˙ α = 1 T ( − K D α + K T T T M ( V M − K M V k ))+ KT ( V k,ref − V M ) (10)where α is the firing angle; V M is the filtered voltage atbus k ; V k is the voltage magnitude at bus k ; V k,ref is thereference voltage at bus k ; K M , K D , K are the regulator gains; T M , T , T are the regulator time constants.The reactive power injected at the voltage-controlled bus k where the SVC is connected can be described as: Q SV C = 2 α − sin 2 α − π (2 − x L /x C ) πx L V k (11)where x L is the SVC inductive reactance and x C is the SVCcapacitive reactance.An important advantage of the SVC controller is that, exceptfor the reactive power support, it can be tuned to directlycontrol the voltage of load bus k by adjusting the set-point V k,ref . It is worth noting that similar voltage control methodsmay be developed using different types of FACTS devices thatoperate based on a reference voltage setting, such as StaticSynchronous Compensators (STATCOM), as implemented in[10], [28]. III. W IDE -A REA V OLTAGE C ONTROL
A. Mathematical Formulation
When the power system is in normal operating conditions,the linearized power flow model can be applied. At thedynamic load buses, we have: (cid:34) ∆ P ∆ Q (cid:35) = (cid:34) J P θ J P V J Qθ J QV (cid:35) (cid:34) ∆ θ ∆ V (cid:35) = J (cid:34) ∆ θ ∆ V (cid:35) (12)where ∆ P = [∆ P , ..., ∆ P m ] T is the vector of active powerchanges; ∆ Q = [∆ Q , ..., ∆ Q m ] T is the vector of reactivepower changes; ∆ θ = [∆ θ , ..., ∆ θ m ] T is the vector of busvoltage angle changes; ∆ V = [∆ V , ..., ∆ V m ] T is the vectorof bus voltage magnitude changes at the dynamic load buses; J P θ = ∂ P ∂ θ , J P V = ∂ P ∂ V , J Qθ = ∂ Q ∂ θ , J QV = ∂ Q ∂ V . Equation(12) can also be written as follows: (cid:34) ∆ θ ∆ V (cid:35) = (cid:34) S θP S θQ S V P S V Q (cid:35) (cid:34) ∆ P ∆ Q (cid:35) = S (cid:34) ∆ P ∆ Q (cid:35) (13)where S = J − . Therefore, we can obtain the followingvoltage control model: ∆ V = S V P ∆ P + S V Q ∆ Q (14)In previous WAVC literature [6]–[10], the decoupled powerflow [29] formulation ∆ Q = J QV ∆ V or ∆ V = S V Q ∆ Q isapplied by neglecting the impacts of real power perturbation onvoltage deviation. In contrast, this paper considers the impactsof both active and reactive power mismatches on voltagedeviations. B. Wide-Area Voltage Control
In this work, we consider the scenario that SVCs areavailable at some dynamic load buses that could perform thevoltage control so as to minimize the voltage deviation at theother load buses that are without the capability of voltageregulation. Therefore, (14) can be written as: (cid:34) ∆ V c ∆ V u (cid:35) = (cid:34) S V P cc S V P cu S V P uc S V P uu (cid:35) (cid:34) ∆ P c ∆ P u (cid:35) + (cid:34) S V Q cc S V Q cu S V Q uc S V Q uu (cid:35) (cid:34) ∆ Q c ∆ Q u (cid:35) (15)where the subscripts c and u denote the n c voltage-controlled load buses with SVCs installed and the n u voltage-uncontrolled load buses, respectively, while n c + n u = m . Note that the submatrices S V P cc , S V P cu , S V P uc , S V P uu , S V Q cc , S V Q cu , S V Q uc , S V Q uu are obtained by reordering therows and columns of the matrices S V P , S
V Q according tothe selection of the controlled and uncontrolled load buses.The voltage deviation is defined as ∆ V = V − V ref ,where ∆ V = [∆ V c , ∆ V u ] T , V = [ V c , V u ] T and V ref =[ V c,ref , V u,ref ] T . Particularly, for voltage-controlled buses, V c,ref denote the SVC reference voltages that can be ad-justed. For voltage-uncontrolled buses, V u,ref is constant andcorresponds to the steady-state power flow solution. Also,since SVCs are used at the voltage-controlled buses, ∆ P c is CCEPTED BY IEEE TRANSACTIONS ON POWER SYSTEMS ON FEBRUARY 6, 2021 4 assumed to be . After some algebraic manipulation of (15),the following relations can also be derived: ∆ Q c = S − V Q cc [∆ V c − S V P cu ∆ P u − S V Q cu ∆ Q u ] (16) ∆ V u = S V P uu ∆ P u + S V Q uc ∆ Q c + S V Q uu ∆ Q u (17)By substituting (16) into (17), the voltage deviation of theuncontrolled buses can be written as: ∆ V u = [ S V P uu − S V Q uc S − V Q cc S V P cu ]∆ P u + [ S V Q uu − S V Q uc S − V Q cc S V Q cu ]∆ Q u + S V Q uc S − V Q cc ∆ V c (18)Assuming that some active and reactive power perturbations ∆ P u ( t i ) and ∆ Q u ( t i ) occur at uncontrolled buses at thecurrent time step t i , the goal of the voltage controller isto minimize the voltage deviation of the uncontrolled buses ∆ V u ( t i +1 ) at the next time step t i +1 = t i + ∆ t , by adjustingthe reference points of the SVCs at the controlled busesthrough ∆ V c ( t i +1 ) .Hence, the online optimization problem can be formulatedas: min ∆ V c ( t i +1 ) || ∆ V u ( t i +1 ) || ∞ s.t. V minc ( t i +1 ) ≤ V c ( t i +1 ) ≤ V maxc ( t i +1 ) Q minc ( t i +1 ) ≤ Q c ( t i +1 ) ≤ Q maxc ( t i +1 ) (19)In previous WAVC works [6]–[10], the impacts of realpower perturbation on voltage deviation are not considered,i.e., ∆ Q = J QV ∆ V or ∆ V = S V Q ∆ Q , have been used.Besides, the approximation J QV ≈ B is applied in which B is assumed to be fully known in real-time from networktopology and line parameters. However, as mentioned in theIntroduction, accurate topology information may not alwaysbe available due to telemetry failure, bad data, undetectedtopology change, etc. In addition, the assumption J QV ≈ B is true only under the condition that the voltage magnitudes ofall buses are available i.e., ∆ Q and ∆ V in the approximation ∆ Q = B ∆ V have to include all buses. Nevertheless, suchassumption may be rarely met in real life due to missingPMUs, limited communication links, etc.In this paper, we will propose a model-free WAVC methodthat is independent of network topology and can provideeffective voltage control considering both active and reactivepower perturbations ( ∆ P u , ∆ Q u in (18)) in various workingconditions, network topologies, and in case of missing PMUs.IV. A M ODEL -F REE W IDE -A REA V OLTAGE C ONTROL M ETHOD
A. A Model-Free Method to Estimate the Sensitivity Matrices
In Section II-A, we have shown that the stochastic dynamicload modeling corresponds to a vector Ornstein-Uhlenbeckprocess (see (8)). The stationary covariance matrix C xx ofthe process is defined as follows: C xx = (cid:10) [ x ( t ) − µ x ][ x ( t ) − µ x ] T (cid:11) = (cid:34) C θθ C θV C V θ C V V (cid:35) (20)where µ x is the mean of the process. The τ -lag autocorrelationmatrix is defined as: G ( τ ) = (cid:10) [ x ( t + τ ) − µ x ][ x ( t ) − µ x ] T (cid:11) (21)Since the normal operation of a power system around its steadystate is considered, the system state matrix A is stable. By the regression theorem of the vector Ornstein-Uhlenbeck processwhich holds under ambient conditions [25], the τ -lag auto-correlation matrix satisfies the following matrix differentialequation: ddτ [ G ( τ )] = AG ( τ ) (22)where G (0) = C xx . As a result, the dynamic state matrix A can be obtained by solving (22) [30]: A = 1 τ log (cid:104) G ( τ ) C − xx (cid:105) (23)Equation (23) tactfully relates the statistical properties of PMUmeasurements with the physical properties of the dynamicpower system model and provides an intriguing way of es-timating the system state matrix from PMUs. The regressiontheorem of the vector Ornstein-Uhlenbeck process has beenleveraged in [30] to estimate the system state matrix ofclassical model of generators, which can be used in modeidentification, stability monitoring, etc. However, the relation(23) deduced from the regression theorem is, for the first time,applied to extract the sensitivity matrices for voltage control,which will be shown to be accurate and effective.In practical applications, G ( τ ) and C xx can be estimatedfrom PMU measurements. Assuming that a window size of n PMU measurements x ( t i ) = [ θ ( t i ) , V ( t i )] T , i = 1 , , ..., n are available at the dynamic load buses, the sample mean ¯ x , the sample covariance matrix ˆ C and the sample τ -lagcorrelation matrix ˆ G can be calculated as follows: ¯ x = 1 n n (cid:88) i =1 x ( t i ) (24) ˆ C = 1 n − F − ¯ x Tn )( F − ¯ x Tn ) T (25) ˆ G (∆ t ) = 1 n − F n − ¯ x Tn − )( F n − − ¯ x Tn − ) T (26)where F = (cid:104) x ( t ) , ..., x ( t n ) (cid:105) is a m × n matrix with thestates’ measurements, n is a n × vector of ones, ∆ t is thetime lag, F i : j denotes the submatrix of F from the i -th tothe j -th column. Hence, the estimated dynamic system statematrix ˆ A can be obtained as: ˆ A = 1∆ t log (cid:104) ˆ G (∆ t ) ˆ C − (cid:105) = (cid:34) ˆ T − θ ˆ J P θ ˆ T − θ ˆ J P V ˆ T − V ˆ J Qθ ˆ T − V ˆ J QV (cid:35) (27)where A is a scaled Jacobian matrix (see (8)). Particularly, thesubmatrices of ˆ A correspond to an estimation of the scaledsensitivity matrices used in the WAVC method discussed inSection III. If the time constants T θ , T V can be estimated, thesensitivity matrices needed for the WAVC can be extractedpurely from PMU measurements without any knowledge of thenetwork model, which is required by previous WAVC works. CCEPTED BY IEEE TRANSACTIONS ON POWER SYSTEMS ON FEBRUARY 6, 2021 5
B. Estimating the Load Time Constants from PMUs
By looking at the dynamic load model (1)-(2), one caneasily observe that when PMU measurements for θ , V and P , Q are available, the time constants τ θ k , τ V k at each dynamicload bus k = 1 , , ..., m can be estimated through a simplelinear regression analysis [31], because we have: ∆ θ k ∆ t = 1ˆ τ θ k ( P k − P sk ) (28) ∆ V k ∆ t = 1ˆ τ V k ( Q k − Q sk ) (29)For this purpose, we denote the angle and voltage deviationsat each load bus k between the consecutive samples i , i − as ∆ θ k ( t i ) = θ k ( t i ) − θ k ( t i − ) , ∆ V k ( t i ) = V k ( t i ) − V k ( t i − ) ,where i = 1 , , ..., p . Besides, we denote the deviation ofactive and reactive power absorption as ∆ P k ( t i ) = P k ( t i ) − ¯ P k , ∆ Q k ( t i ) = Q k ( t i ) − ¯ Q k , where ¯ P k , ¯ Q k are the samplemeans, i.e. ¯ P k = p (cid:80) pi =1 P k ( t i ) , ¯ Q k = p (cid:80) pi =1 Q k ( t i ) . Thenthe estimator for each load time constant should minimize theresiduals ˆ ε τ θk k ( t i ) = ∆ θ k ( t i )∆ t − τ θ k ∆ P k ( t i ) (30) ˆ ε τ Vk k ( t i ) = ∆ V k ( t i )∆ t − τ V k ∆ Q k ( t i ) (31)So ˆ T θ = diag (cid:104) ˆ τ θ , ..., ˆ τ θ m (cid:105) , ˆ T V = diag (cid:104) ˆ τ V , ..., ˆ τ V m (cid:105) can beobtained by solving the following minimization problems: min ˆ τ θk (cid:80) pi =1 ˆ ε τ θk k ( t i ) (32) min ˆ τ Vk (cid:80) pi =1 ˆ ε τ Vk k ( t i ) (33)Once the time constants ˆ T θ , ˆ T V are estimated, the sensitivitymatrices ˆ J P θ , ˆ J P V , ˆ J Qθ , ˆ J QV can be extracted from thescaled ones estimated in Section IV-A and further be exploitedin the design of the model-free WAVC. C. The Proposed Model-Free WAVC Algorithm
We have seen from the previous discussions that the sen-sitivity matrices ˆ J P θ , ˆ J P V , ˆ J Qθ , ˆ J QV and the time constants ˆ T θ , ˆ T V can be estimated purely from PMU data (Section IV-Aand IV-B) and can be further exploited in the design of WAVC(Section III). As such, a novel online model-free WAVCmethod can be developed to minimize the voltage deviation ofthe uncontrolled load buses without any model information.The algorithm is summarized below. Particularly, Step E1 - Step E4 are for the estimation of the sensitivity matrices, while
Step C1 - Step C4 are for the online WAVC. The structure ofthe proposed model-free WAVC is also illustrated in Fig. 1.Remarks: • In this paper, a window size of 300s with a samplingfrequency of 60 Hz is used in
Step E1 , for which a goodaccuracy is achieved. It is worth noting that, despite of arelatively large window size, the above algorithm does notassume any network model information (e.g. topology, lineparameters), but can estimate the sensitivity matrices purelyfrom PMU data.
Fig. 1. An illustration of the proposed model-free WAVC. • In case the algorithm detects changes in network topologythat may significantly affect the sensitivity matrices, new PMU
CCEPTED BY IEEE TRANSACTIONS ON POWER SYSTEMS ON FEBRUARY 6, 2021 6 data needs to be collected when the system settles to thenew steady state to reestimate the sensitivity matrices andredesign the WAVC, i.e.,
Step E1-Step E4 should be repeated.Alternatively, to achieve more accurate and frequent estimationand control, the current algorithm using one window of mea-surements can be extended to an online recursive algorithmusing a moving window, similar to the approach in [32]. • Similar to the approach adopted in [18], the proposed WAVCis triggered with a delay d after a disturbance to avoidexcessive interaction between the control and the transientresponse of voltages. d = 30 s in the simulation studies ofthis paper, whereas different values may be selected based ondifferent systems’ dynamic characteristics. • The time interval between two control actions d is selectedto be 0.2s (e.g., 10 sampling time steps), similar to the valuechosen in [10], to quickly mitigate the voltage deviation dueto disturbance. However, a larger time interval can also beselected to help avoid conflict with other control mechanisms,as described in [18]. • The proposed method assumes that PMU measurementsare available at all dynamic load buses. However, as will beshown in Section V-B, the model-free WAVC remains effectivein case of some missing measurements at either voltage-controlled or voltage-uncontrolled dynamic load buses. It isworth mentioning that it is beyond the scope of the paper toto deal with the occurrence of severe bad data events. Baddata detection methods [33], [34] and missing data recoverymethods [35], [36] can be applied to pre-process the PMU databefore applying the proposed model-free WAVC algorithm.V. N
UMERICAL R ESULTS
The IEEE 39-Bus and IEEE 68-Bus systems have beenused to validate and test the proposed model-free WAVC.The nonlinear programming solver fmincon and the interior-point algorithm [10] are used to get the optimal solution at
Step C3 of the proposed algorithm. The simulation study wasimplemented in PSAT-2.1.10 [37].To quantify the performance of the proposed model-freeWAVC, the root mean square value of the voltage deviationsat the n u uncontrolled load buses in steady state can be usedas the performance index λ [8], i.e.,: λ = (cid:114) n u || ∆ V u ss || (34)where ∆ V ss u = V ss u − V u,ref . V ss u are the new steady-state voltages of the voltage-uncontrolled buses; V u,ref arethe previous steady-state voltages of those buses before theperturbation occurs. A. Validation on the IEEE 39-Bus Test System
In this section, the validation of the proposed algorithmis conducted on the IEEE 39-Bus test system. Both loaddynamics and generator dynamics are considered in the nu-merical study. Particularly, the 4th-order generator modelsequipped with AVRs are included and 19 loads are modelledas the stochastic dynamic loads described in (5)-(6). The timeconstants T θ , T V are set to be 30s. The load model considersGaussian stochastic load variations with a mean of zero and σ Pk = σ Qk = 1 in (5)-(6). SVCs are installed at 3 voltage-controlled buses (Bus 3, 9, 20), while the rest 16 load busesare voltage-uncontrolled. Different combinations of controlledand uncontrolled buses will be discussed later.Firstly, the Estimation procedure in
Algorithm 1 is fol-lowed and the accuracy of the estimated sensitivity matrixis demonstrated. To this end, 300s PMU data are collectedfrom the 19 load buses, from which the sample covariancematrix, the sample τ -lag correlation matrix, and the scaled sen-sitivity matrices ˆ T − θ ˆ J P θ , ˆ T − θ ˆ J P V , ˆ T − V ˆ J Qθ , ˆ T − V ˆ J QV areestimated ( Step E1-Step E2 ). Moving to
Step E3-Step E4 , alltime constants in ˆ T θ , ˆ T V are well estimated from 0.1s (amongthe collected 300s) PMU data by the linear regression analysis,with acceptable estimation errors less than . Next, thesensitivity matrices ˆ J P θ , ˆ J P V , ˆ J Qθ , ˆ J QV and ˆ S V P , ˆ S V Q canbe obtained. Specifically, Fig. 2 presents a comparison betweenthe estimated and the true values of matrix J , evidentlyshowing the good accuracy of the estimation.Once the estimated sensitivity matrices are obtained, theWAVC presented in the Control part of
Algorithm 1 canbe designed and activated in case of some perturbation. Totest the effectiveness of the proposed model-free WAVC, weregard that the estimation result is obtained at t n = 0 s and a25 % load increase is applied to the active and reactive powerof all voltage-uncontrolled buses at t d = 2 s. The WAVCstarts working with a delay of d = 30 s to avoid excessiveinteraction of the controller during the transient. The controlleris executed every d = 0 . s if max | ∆ V u ( t i ) | ≥ . p.u..Fig. 3 depicts a comparison of the voltage magnitude at theuncontrolled Bus 4 for different control scenarios (no control,model-based WAVC using the true matrix J , the proposedmodel-free WAVC using the estimated matrix ˆ J ). As can beseen, the model-free WAVC is as effective as the model-based WAVC, as both can boost up the voltage when loadpower increases. In addition, the performance index λ for thedifferent scenarios of no control, model-based WAVC using thetrue matrix J and the proposed model-free WAVC assuming3 voltage-controlled buses can be found in the third row ofTable I (Case C). These results clearly show the good accuracyof the estimated sensitivity matrices and the effectiveness ofthe proposed model-free WAVC. The rest of the voltage-uncontrolled dynamic load buses exhibit similar behaviors.In terms of efficiency, using a Windows computer with aquad-core Intel i7 2.2 GHz processor and 8GB RAM, Step E1-Step E4 took 0.312775s for estimating the sensitivity matrices,while the solution of the optimization problem
Step C3 tookmuch less than the time interval between two control actions(e.g. only 0.059321s). The fast computational speed clearlydemonstrates the good feasibility of the proposed WAVC inonline implementation.In the above simulations, 3 voltage-controlled buses and16 voltage-uncontrolled buses are assumed. Nevertheless,different combinations of voltage-controlled and voltage-uncontrolled buses may affect the performance of the proposedmodel-free WAVC. To investigate this, we test the performanceof the proposed method in different combinations of controlledand uncontrolled buses. Specifically, 4 different cases aretested where the number of voltage-controlled buses varies
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Fig. 2. The estimation results for matrix J of the IEEE 39-Bus system.Fig. 3. The voltage profile at Bus 4 with 3 voltage-controlled buses (Case C- Bus 3, 9, 20). TABLE IT HE PERFORMANCE INDEX λ FOR VARIOUS COMBINATIONS OFCONTROLLED AND UNCONTROLLED BUSES
Case Voltage-controlledBuses λ no WAVC model-basedWAVC model-freeWAVC A 3 0.012737 0.0080188 0.0080193B 3, 20 0.010373 0.0048502 0.0048502C 3, 9, 20 0.01113 0.0040667 0.0040674D 3, 9, 20, 23 0.010072 0.0040306 0.0040308 from 1 (Case A - Bus 3) to 4 (Case D - Bus 3, 9, 20, 23),as shown in Table I. Note that for each case, a 25 % increasein the active and reactive power consumption is applied atthe corresponding voltage-uncontrolled buses that are differentfrom Case A to Case D, i.e., the load increasing pattern variesin different cases. It is evident from Table I that, similar to theperformance of the control using the true sensitivity matrices,the proposed model-free WAVC can also provide effectivecontrol in all cases. B. Impact of Missing PMUs and PMU locations
Although the proposed model-free WAVC assumes thatPMU measurements are available at all dynamic load buses asstated in Sections III-IV, this assumption may not always holdin practical applications. In this section, we validate the perfor-mance of the proposed methodology in case of missing PMUsat either voltage-controlled or voltage-uncontrolled buses. Itshould be noted that due to the loss of PMU measurements, Equation (15) should be modified as follows: (cid:34) ∆ V (cid:48) c ∆ V (cid:48) u (cid:35) = (cid:34) S (cid:48) V P cc S (cid:48) V P cu S (cid:48) V P uc S (cid:48) V P uu (cid:35) (cid:34) ∆ P (cid:48) c ∆ P (cid:48) u (cid:35) + (cid:34) S (cid:48) V Q cc S (cid:48) V Q cu S (cid:48) V Q uc S (cid:48) V Q uu (cid:35) (cid:34) ∆ Q (cid:48) c ∆ Q (cid:48) u (cid:35) (35)where (cid:48) is used to highlight that buses with missing PMUsare excluded from the mathematical formulation of the con-troller. Specifically, S (cid:48) V Q cc , S (cid:48) V Q cu , S (cid:48) V Q uc , S (cid:48) V Q uu are ob-tained from S V Q cc , S V Q cu , S V Q uc , S V Q uu by omitting therows and columns corresponding to the buses with missingPMUs. A similar operation applies to S V P . Thus, in caseof PMU loss, the
Estimation procedure in
Algorithm 1 canstill estimate the sensitivity matrices at the dynamic loadbuses with PMUs available. Regarding the
Control procedurein
Algorithm 1 , when PMU measurements are missed atpreviously voltage-controlled buses, those buses no longerreceive the WAVC signals or updates to their SVCs’ referencepoints. Similarly, the voltage deviation of voltage-uncontrolledbuses not equipped with PMUs is no longer considered whilesolving
Step C3 .To study the impact of missing measurements, we considersome of the PMUs of Case C to be lost. To determine theminimum number of PMUs available, as discussed in [38],PMUs should be placed in about one third of the buses toachieve full observability of the system. Therefore, we assumethat PMUs are installed at (1/3)*39=13 dynamic load buses inthe IEEE 39 bus test system. Previously, we had 19 PMUs,now we have 13 PMUs available, i.e., 6 missing PMUs.In addition to missing PMUs, the installation locations ofavailable PMUs may also affect the performance of estimationand control. Many previous works have studied the optimallocations of PMUs [39]–[41]. Particularly, we follow a sen-sitivity analysis approach as in [41]. A sensitivity indicatoris defined as the power flow rise ∆ | S ij | at each transmissionline i − j following a 25 % load power increase ∆ P at thedynamic load buses, i.e., ∆ | S ij | ∆ P . It is recommended in [41]that available PMUs should be located at the dynamic loadbuses connecting the most sensitive transmission lines, i.e.,the ones that experience the largest power flow rises. In ourcase, assuming we have only 13 PMUs available, we comparethe performance of the proposed method under two cases ofmissing PMUs: • Case C.I (best case): The 13 available PMUs are placedat the most sensitive dynamic load buses connecting thetop 13 sensitive transmission lines (Bus 3, 4, 7, 8, 12, 15,16, 18, 23, 24, 25, 26, 28), whereas 6 missing PMUs areconsidered at the least sensitives buses (Bus 1, 9, 20, 21,27, 29). • Case C.II (worst case):The 13 available PMUs are placedat the least sensitive buses (Bus 1, 8, 9, 12, 16, 20, 21,23, 24, 26, 27, 28, 29) and missing PMUs are consideredat the most sensitive buses (Bus 3, 4, 7, 15, 18, 25).Once the sensitivity matrices at the dynamic load buses withPMU data available are estimated, the same load disturbanceas in Section V-A applies. Fig. 4 presents the voltage profile
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Fig. 4. The voltage profile at Bus 15 in case of missing PMUs with 3 voltage-controlled buses (Case C - Bus 3, 9, 20).TABLE IIT
HE IMPACT OF MISSING
PMU
S AND
PMU
LOCATIONS ON THEPERFORMANCE INDEX λ Case Voltage-controlledBuses λ no WAVC no missingPMUs missingPMUsat leastsensitive missingPMUsat mostsensitive C 3, 9, 20 0.01113 0.0040674 0.0058466 0.0074339 of the uncontrolled Bus 15 for the different cases. It can beseen that a large number of missing PMUs may deterioratethe performance of voltage control compared to the casewith no missing PMUs. However, the sensitivity analysis maycontribute to the optimal placement of PMUs and thus improvethe performance of the voltage control. With missing PMUsat the least sensitive buses (Case C.I, the best case), theperformance index is only slightly affected comparing to thecase with no missing PMUs, but is much better than that inthe case where missing PMUs happen at the most sensitivebuses (Case C.II, worst case), as demonstrated in Table II. Theabove results confirm the robustness of the proposed algorithmagainst missing measurements, which further reinforces itsoverall performance and feasibility in practical applications.
C. Impact of Measurement Noise
The accuracy of measurement-based methods may be de-teriorated by measurement noise. In this section, we testthe performance of the proposed model-free WAVC againstmeasurement noise. For this purpose, measurement noise isadded to the gathered PMU data of θ , V in Step E1 of the
Estimation part as well as to the PMU data of V in Step C1 of the
Control part in
Algorithm 1 . Two different types ofnoise, representing low and high noise levels, are considered.Particularly, for the low noise level, we follow the approachin [42] to add Gaussian noise with zero mean and standarddeviation equal to 10% of the largest state changes to thePMU measurements of θ and V . On the other hand, highnoise levels correspond to Gaussian noise with fixed standarddeviation − to the measurements of θ and V [43]. Oncethe sensitivity matrices are estimated from the PMU data withadded noise, a 25 % increase in the active and reactive power ofthe voltage-uncontrolled buses is applied for Case C of SectionV-A. Table III depicts the values of the performance index λ for the two noise levels. It can be observed that both noiselevels lead to only insignificant changes to the performance TABLE IIIT
HE IMPACT OF MEASUREMENT NOISE ON THE PERFORMANCE INDEX λ Case Voltage-controlledBuses λ model-free WAVCno noise low noise high noise C 3, 9, 20 0.0040674 0.0040673 0.0045575 index λ , demonstrating the robustness of the proposed methodagainst measurement noise. D. Effectiveness Under Topology Change
One salient feature of the proposed online model-freeWAVC method is its independency of network model suchthat it can adaptively update the estimation of the sensitivitymatrices J P θ , J
P V , J Qθ , J QV and the control signals as thepower grid evolves. Particularly, if an undetected topologychange occurs in the system, the proposed method mayoutperform the model-based WAVC, the performance of whichgreatly relies on an accurate network model. To show this, weconsider that in Case E with 5 voltage-controlled buses, athree-phase fault occurs at Bus 26 that lasts for 10 cycles. Toclear the fault, the breaker between Bus 26 and Bus 27 trips,which is unfortunately undetected by the topology processor.After the topology change occurs, a transient period followsand the system eventually settles down to a new steady-state where the application of the regression theorem for theestimation of the sensitivity matrix is allowed. Once the systemreaches a new steady state after the topology change, newPMU data of 300s are used to estimate the new sensitivitymatrices in ˆ J (cid:48) in the Estimation part of
Algorithm 1 . Theupdated estimated sensitivity matrices are assumed to beavailable at t n = 0 s and they are further used to designand apply the WAVC in the Control part of
Algorithm 1 .To validate the effectiveness of the model-free WAVC afterthe topology change, a increase in the reactive powerabsorption of the voltage-uncontrolled buses is applied inthe new steady state at t d = 20 s, whereas the controller isactivated with a delay of d = 30 s, i.e. at T D = 50 s.In Fig. 5, the voltage profile at the voltage-uncontrolledBus 15 is presented. Thanks to the new estimated sensitivitymatrices, the proposed model-free WAVC is still able toeffectively minimize the voltage deviation despite the topologychange. However, the model-based WAVC is unable to performthe voltage control properly, as the sensitivity matrices are notupdated after the undetected line outage. The advantage ofthe proposed model-free WAVC is also confirmed in TableIV, where the values of the performance index λ in differentcases are presented. It can be observed that the proposedonline model-free WAVC can effectively reduce the voltagedeviations in different topologies and its performance is verysimilar to the updated model-based method, highlighting theaccuracy of the estimation after a topology change. E. Validation on the IEEE 68-Bus System
To show the feasibility of the proposed method in practicallarge-scale power systems, a numerical study on the IEEE 68-Bus test system is conducted. In this case, 35 dynamic loadsare considered with time constants of 30s. Gaussian stochastic
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Fig. 5. The voltage profile at Bus 15 in case of a line outage with 5 voltage-controlled buses (Case E - Bus 3, 9, 12, 20, 23).TABLE IVT
HE PERFORMANCE INDEX λ WHEN USING DIFFERENT SENSITIVITYMATRICES UNDER TOPOLOGY CHANGE
Case Voltage-controlledBuses λ no WAVC oldmodel-basedWAVC updatedmodel-basedWAVC updatedmodel-freeWAVC E 3, 9, 12, 20, 23 0.0089529 0.012143 0.0045857 0.0047066 load variations with a mean of zero and σ Pk = σ Qk = 1 areapplied in (5)-(6). SVCs are installed at 5 voltage-controlledbuses (Bus 20, 25, 29, 41, 42).To apply the proposed WAVC method, 300s PMU measure-ments are collected to estimate the sensitivity matrices in J . Acomparison between the elements of the true and the estimatedsensitivity matrix J QV is presented in Fig. 6, showing a goodaccuracy of the estimation.To test the performance of the proposed WAVC, a 20 % increase of the reactive power at voltage-uncontrolled busesis applied at t d = 2 s. The controller is activated with adelay d = 30 s and once max | ∆ V u ( t i ) | ≥ . p.u. atsome uncontrolled bus. Fig. 7 shows the time evolution of thevoltage magnitude at the voltage-uncontrolled Bus 21. It canbe observed that, as with the control using the true sensitivitymatrix, the proposed algorithm can effectively restore thevoltages using the estimated matrix ˆ J QV in the larger testsystem. F. Comparison to Other Methods
Although the proposed method seems to be the first purelymeasurement-based WAVC by exploiting the estimated sensi-tivity matrices, there there are previous works on estimatingsensitivity matrices from PMUs [44], [45]. In this section, wecompare the performance of the proposed estimation methodand that of the Least Squares (LS) and Total Least Squares(TLS) methods developed in [44] in estimating the sensitivitymatrices. Specifically, we compare the results of the proposedmethod and the TLS method under measurement noise.As have been shown in Section V-C (also validated inFig. 8), the impact of high measurement noise levels (i.e.,Gaussian measurement noise with standard deviation − to the measurements of θ , V ) on the performance of theproposed method is very small. On the other hand, Fig. 9shows that measurement noise can significantly deterioratethe estimation result using the TLS method. Similar resultsare observed when using the LS method. It is to note that Fig. 6. The estimation results for the sensitivity matrix J QV of the IEEE68-Bus system.Fig. 7. The voltage profile at Bus 21 with 5 voltage-controlled buses (Bus20, 25, 29, 41, 42). other observations are also acquired from the comparison:1). The estimation error using the LS and TLS methodsstrongly depends on the selected samples, as different samplesmay give very different results; 2). The TLS method maysuffer from overfitting problem [46]. Thus, it may be difficultto determine the number of samples needed to achieve asatisfactory estimation result, which in turn may hamper itsonline implementation. Fig. 8. The estimation result for matrix J of the IEEE 39-Bus system usingthe proposed method under high measurement noise levels. CCEPTED BY IEEE TRANSACTIONS ON POWER SYSTEMS ON FEBRUARY 6, 2021 10
Fig. 9. The estimation result for matrix J of the IEEE 39-Bus system usingthe TLS method [44] under high measurement noise levels. VI. C
ONCLUSIONS
Leveraging on the inherent power system dynamics, thispaper has proposed a model-free
WAVC for the first time.Specifically, the proposed method can minimize voltage de-viation by controlling the available FACTS devices in anonline environment, without knowing the topology and lineparameters of power network. The proposed WAVC methodalso provides more flexibility regarding the number and com-bination of buses participating in the voltage control schemeand unlike previous works, considers both active and reactivepower perturbations. Numerical results on the IEEE 39-Busand 68-Bus systems show that the proposed online model-freeWAVC can provide effective voltage control in various net-work topologies, different combinations of voltage-controlledand voltage-uncontrolled buses, under measurement noise, andin case of missing PMUs. In addition, the proposed model-free WAVC may outperform the model-based WAVC if anundetected topology change occurs.R
EFERENCES[1] J. P. Paul, J. Y. Leost, and J. M. Tesseron, “Survey of the secondaryvoltage control in France: Present realization and investigations,”
IEEETrans. Power Syst. , vol. 2, no. 2, pp. 505–511, 1987.[2] J. L. Sancha, J. L. Fernandez, A. Cortes, and J. T. Abarca, “Secondaryvoltage control: Analysis, solutions and simulation results for the Span-ish transmission system,”
IEEE Trans. Power Syst. , vol. 11, no. 2, pp.630–638, 1996.[3] M. D. Ilic, X. Liu, G. Leung, M. Athans, C. Vialas, and P. Pruvot,“Improved secondary and new tertiary voltage control,”
IEEE Trans.Power Syst. , vol. 10, no. 4, pp. 1851–1860, 1995.[4] C. W. Taylor, D. C. Erickson, and R. E. Wilson, “Reducing blackoutrisk by a wide-area control system (WACS): Adding a new layer ofdefense,” in
Power Syst. Comput. Conf. , Liege, Belgium, 2005.[5] M. Perron at al., “Wide-area voltage control system of flexible ACtransmission system devices to prevent voltage collapse,”
IET Gen.,Transm., Distrib. , vol. 11, no. 18, pp. 4556–4564, 2017.[6] Z. Liu and M. D. Ilic, “Toward PMU-based robust Automatic VoltageControl (AVC) and Automatic Flow Control (AFC),” in
IEEE PESGeneral Meeting , Providence, RI, USA, 2010.[7] H.-Y. Su, and C.-W. Liu, “An adaptive PMU-based secondary voltagecontrol scheme,”
IEEE Trans. Smart Grid , vol. 4, no. 3, pp. 1514–1522,2013.[8] H.-Y. Su, F.-M. Kang, and C.-W. Liu, “Transmission grid secondaryvoltage control method using PMU data,”
IEEE Trans. Smart Grid , vol.9, no. 4, pp. 2908–2917, 2018. [9] A. S. Musleh, S. M. Muyeen, A. Al-Durra, and H. M. Khalid, “PMUbased wide area voltage control of smart grid: A real time implementa-tion approach,” in
IEEE ISGT Asia , Melbourne, VIC, Australia, 2016.[10] A. S. Musleh, S. M. Muyeen, A. Al-Durra, I. Kamwa, M. A. S.Masoum, and S. Islam, “Time-delay analysis of wide-area voltagecontrol considering smart grid contingencies in a real-time environment,”
IEEE Trans. Ind. Informat. , vol. 14, no. 3, pp. 1242–1252, 2018.[11] H. Zhu and G. Giannakis, “Sparse overcomplete representations forefficient identification of power line outages,”
IEEE Trans. Power Syst. ,vol. 27, no. 4, pp. 2215-2224, 2012.[12] Y. Zhou, J. Cisneros-Saldana, and L. Xie, “False analog data injectionattack towards topology errors: formulation and feasibility analysis,” in
IEEE PES General Meeting , Portland, OR, USA, 2018.[13] J. Zhang and L. Sankar, “Physical system consequences of unobservablestate-and-topology cyber-physical attacks,”
IEEE Trans. Smart Grid , vol.7, no. 4, pp. 2016-2025, 2016.[14] Y. Guo, H. Gao, H. Xing, Q. Wu, and Z. Lin, “Decentralized coordi-nated voltage control for VSC-HVDC connected wind farms based onADMM,”
IEEE Trans. Sustain. Energy , vol. 10, no. 2, pp. 800–810,2019.[15] L. Yu, D. Czarkowski, and F. de Leon, “Optimal distributed voltageregulation for secondary networks with DGs,”
IEEE Trans. Smart Grid ,vol. 3, no. 2, pp. 959–967, 2012.[16] J. Shi, B. Foggo, X. Kong, Y. Cheng, N. Yu, and K. Yamashita, “Onlineevent detection in synchrophasor data with graph signal processing,” in
IEEE SmartGridComm , Tempe, AZ, USA, 2020.[17] T. Van Cutsem, C. Vournas,
Voltage stability of electric power systems ,Norwell, MA, USA: Kluwer, 1998, 2008.[18] A. Ashrafi and S. M. Shahrtash, “Dynamic Wide Area voltage controlstrategy based on organized multi-agent system,”
IEEE Trans. PowerSyst. , vol. 29, no. 6, pp. 2590–2600, 2014.[19] C. A. Ca˜nizares, “On bifurcations, voltage collapse and load modeling,”
IEEE Trans. Power Syst. , vol. 10, no. 1, pp. 512–522, 1995.[20] H. D. Nguyen and K. Turitsyn, “Robust stability assessment in thepresence of load dynamics uncertainty,”
IEEE Trans. Power Syst. , vol.31, no. 2, pp. 1579–1594, 2016.[21] K. T. Vu and C.-C. Liu, “Shrinking stability regions and voltage collapsein power systems,”
IEEE Trans. Circ. and Syst.-I , vol. 39, no. 4, pp.271–289, 1992.[22] T. Overbye, “Effects of load modelling on analysis of power systemvoltage stability,”
Int. Journal of Electr. Power & Energy Syst. , vol. 16,no. 5, pp. 329–338, 1994.[23] C. L. DeMarco and T. Overbye, “An energy based security measure forassessing vulnerability to voltage collapse,”
IEEE Trans. Power Syst. ,vol. 5, no. 2, pp. 419–425, 1990.[24] C. O. Nwankpa and R. M. Hassan, “A stochastic voltage collapseindicator,”
IEEE Trans. Power Syst. , vol. 8, no. 3, pp. 1187–1193, 1993.[25] C. Gardiner,
Stochastic methods: a handbook for the natural and socialsciences , 4th ed. Berlin, Germany: Springer-Verlag, 2009.[26] A. E. Gurrola, J. Segundo-Ram´ırez, and C. N´u˜nez Guti´errez, “Widearea control in electric power systems incorporating FACTS controllers:Review,” in
IEEE International Autumn Meeting on Power, Electronicsand Computing (ROPEC) , Ixtapa, Mexico, 2014.[27] E. Acha, V. G. Agelidis, O. Anaya-Lara, T. J. E. Miller,
Power ElectronicControl in Electrical Systems , 1st ed. Oxford, UK: Newnes, 2001.[28] T. T. Nguyen and V. L. Nguyen, “Application of Wide-Area network ofphasor measurements for secondary voltage control in power systemswith FACTS controllers,” in
IEEE PES General Meeting , San Francisco,CA, USA, 2005.[29] J. J. Grainger, W. D. Stevenson,
Power System Analysis , New York, NY,USA: McGraw-Hill, 1994.[30] H. Sheng and X. Wang, “Online measurement-based estimation ofdynamic system state matrix in ambient conditions,”
IEEE Trans. SmartGrid , vol. 11, no. 1, pp. 95–105, 2020.[31] R. A. Johnson, I. Miller, J. E. Freund,
Miller & Freund’s Probabilityand Statistics for Engineers , 9th ed. Boston, MA, USA: Pearson, 2017.[32] G. Pierrou and X. Wang, “Online PMU-based method for estimatingdynamic load parameters in ambient conditions,” in
IEEE PES GeneralMeeting , Montreal, QC, Canada, 2020.[33] V. Kekatos and G. B. Giannakis, “Distributed robust power system stateestimation,”
IEEE Trans. Power Syst. , vol. 28, no. 2, pp. 1617–1626,2013.[34] M. B. Do Coutto Filho, J. C. Stacchini de Souza, and M. A. RibeiroGuimaraens, “Enhanced bad data processing by phasor-aided stateestimation,”
IEEE Trans. Power Syst. , vol. 29, no. 5, pp. 2200–2209,2014.
CCEPTED BY IEEE TRANSACTIONS ON POWER SYSTEMS ON FEBRUARY 6, 2021 11 [35] D. Osipov and J. H. Chow, “PMU missing data recovery using tensordecomposition,”
IEEE Trans. Power Syst. , vol. 35, no. 6, pp. 4554-4563,2020.[36] Y. Hao, M. Wang, J. H. Chow, E. Farantatos, and M. Patel, “Modellessdata quality improvement of streaming synchrophasor measurements byexploiting the Low-Rank Hankel structure,”
IEEE Trans. on Power Syst. ,vol. 33, no. 6, pp. 6966–6977, 2018.[37] F. Milano, “An open source power system analysis toolbox,”
IEEE Trans.Power Syst. , vol. 20, no. 3, pp. 1199–1206, 2005.[38] J. De La Ree, V. Centeno, J. S. Thorp, and A. G. Phadke, “Synchronizedphasor measurement applications in power systems,”
IEEE Trans. SmartGrid , vol. 1, no. 1, pp. 20–27, 2010.[39] R. F. Nuqui and A. G. Phadke, “Phasor measurement unit placementtechniques for complete and incomplete observability,”
IEEE Trans.Power Deliv. , vol. 20, no. 4, pp. 2381–2388, 2005.[40] S. Chakrabarti and E. Kyriakides, ”Optimal placement of phasor mea-surement units for power system observability,”
IEEE Trans. Power Syst. ,vol. 23, no. 3, pp. 1433–1440, 2008.[41] E. Makram, Z. Zhao, and A. Girgis, “An improved model in optimalPMU placement considering sensitivity analysis,” in
IEEE PES PowerSyst. Conf. and Expo. , Phoenix, AZ, USA, 2011.[42] N. Zhou, D. Meng, and S. Lu, “Estimation of the dynamic states ofsynchronous machines using an extended particle filter,”
IEEE Trans.Power Syst. , vol. 28, no. 4, pp. 4152–4161, 2013.[43] M. Brown, M. Biswal, S. Brahma, S. J. Ranade, and H. Cao, “Char-acterizing and quantifying noise in PMU data,” in
IEEE PES GeneralMeeting , Boston, MA, USA, 2016.[44] Y. C. Chen, J. Wang, A. D. Dom´ınguez-Garc´ıa, and P. W. Sauer,“Measurement-based estimation of the power flow Jacobian matrix,”
IEEE Trans. Smart Grid , vol. 7, no. 5, pp. 2507–2515, 2016.[45] P. Li, H. Su, C. Wang, Z. Liu, and J. Wu, “PMU-based estimation ofvoltage-to-power sensitivity for distribution networks considering thesparsity of Jacobian matrix,” in
IEEE Access , vol. 6, pp. 31307–31316,2018.[46] C. E. Davila, “Total least squares system identification and frequencyestimation for overdetermined model orders,” in
IEEE Int. Conf. onAcoustics, Speech, and Signal Processing , Minneapolis, MN, USA,1993.
Georgia Pierrou (S’19) received a Diploma in Elec-trical and Computer Engineering from the NationalTechnical University of Athens, Athens, Greece in2017. Since 2017, she has been pursuing the Ph.D.degree in Electrical Engineering with the ElectricEnergy Systems Laboratory at McGill University,Montreal, Canada. Her research interests includepower system dynamics, control and uncertaintyquantification.