A Data-Driven Energy Storage System-Based Algorithm for Monitoring the Small-Signal Stability of Power Grids with Volatile Wind Power
AA Data-Driven Energy Storage System-BasedAlgorithm for Monitoring the Small-Signal Stabilityof Power Grids with Volatile Wind Power
Ilias Zenelis, Georgia Pierrou, and Xiaozhe Wang
Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2K6, [email protected], [email protected], [email protected]
Abstract —In this paper, we propose a data-driven energystorage system (ESS)-based method to enhance the online small-signal stability monitoring of power networks with high penetra-tion of intermittent wind power. To accurately estimate inter-areamodes that are closely related to the system’s inherent stabilitycharacteristics, a novel algorithm that leverages on recent ad-vances in wide-area measurement systems (WAMSs) and ESStechnologies is developed. It is shown that the proposed approachcan smooth the wind power fluctuations in near real-time usinga small additional ESS capacity and thus significantly enhancethe monitoring of small-signal stability. Dynamic Monte Carlosimulations on the IEEE 68-bus system are used to illustrate theeffectiveness of the proposed algorithm in smoothing wind powerand estimating the inter-area mode statistical properties.
Index Terms —Data-driven methods, energy storage systems,small-signal stability monitoring, wind power
I. I
NTRODUCTION
Wind power is a rapidly evolving renewable energy technol-ogy worldwide because of its cleanness, abundance and cost-effectiveness. However, the volatile and stochastic nature ofwind poses many challenges to the secure operation of moderngrids. Concerning small-signal stability, the intermittency ofwind energy may not only lead to instability [1] but alsoresult in great obstacles regarding the power system stabilitymonitoring and assessment [2]. Particularly, poorly-dampedinter-area oscillations may become undetectable if small-signalstability monitoring is deteriorated. Such a phenomenon cancause major power outages [3]. Recent works have proposedprobabilistic approaches to study and quantify the impact ofwind power on small-signal stability analysis, which maynevertheless require large computational effort [4], [5].The recent advances in energy storage systems (ESSs) haveprovided power engineers with an effective means to minimizethe unwanted impacts of wind energy on power networks bysmoothing wind power variations [6]. However, even withESSs, additional challenges may arise as power grids are trans-forming into large-scale networks with increasing complexity,due to the continuous integration of power electronics-baseddevices, the transmission system expansion, etc. In fact, [7] hasshown that the conventional model-based methods for small-signal stability monitoring may fail when the power system
This work is supported by the Fonds de Recherche du Qu´ebec - Natureet technologies under Grant FRQ-NT PR-253686 and the Natural Sciencesand Engineering Research Council (NSERC) under Discovery Grant NSERCRGPIN-2016-04570. experiences unexpected disturbances or undetected topologychanges. In this direction, measurement-based methods havebeen proposed in the recent literature to monitor and controlsmall-signal stability considering load uncertainties [7]–[10].These strategies mainly rely on the enormous growth of wide-area measurement systems (WAMSs) and phasor measurementunits (PMUs) over the last − years [11]. Despite ofproviding advancements, these works do not consider windstochasticity.To address these challenges, in this paper, a novel data-driven ESS-based algorithm for monitoring the small-signalstability of power grids with volatile wind power is proposed.Our method exploits two of the key emerging technologies–WAMS and ESS–that have already been massively installedin most power networks [12], to accurately monitor the inter-area modes of systems with random renewables in near real-time. Particularly, we install ESSs at the wind generator side tosmooth out the random wind power fluctuations. The assump-tion of having ESSs at wind generator side may be optimisticnow, but is commonly made in the literature due to the fastrate of ESS deployment in power grids [6], [12], [13]. Wethen apply an online data-driven mode identification approachto estimate the dynamic system state matrix and the inter-areamode characteristics. Unlike probabilistic stability assessmentapproaches, the proposed method enables small-signal stabilityassessment online (within a minutes window). In addition, itwill be shown that the ESS capacity required for wind powersmoothing can be determined based on the statistical propertiesof wind farm power output, whereas its size is not significantwhen compared with the large scale of power grids. To the bestof authors’ knowledge, this work represents the first attemptto enhance the data-driven monitoring of small-signal stabilityconsidering the stochastic nature of renewable energy sources.II. S TOCHASTIC M ODEL FOR P OWER G RID D YNAMICS
Inter-area modes lie in the low-frequency portion of theelectromechanical mode spectrum (i.e. . − Hz) [14]. Thus,fast generator dynamics can be neglected and aggregatedsynchronous machines can be represented by the classicalmodel [14]. By numbering generator buses as i = 1 , ..., n : ˙ δ i = ω i (1) M i ˙ ω i = P m i − P e i − D i ω i (2) a r X i v : . [ ee ss . S Y ] F e b e i = E i n (cid:88) j =1 E j | Y i,j | cos( δ i − δ j − φ i,j ) (3)where δ i is the rotor angle, ω i the rotor speed deviation fromsynchronous speed, M i the inertia coefficient, D i the dampingcoefficient, P m i the mechanical power input, P e i the electricalpower output, E i the transient emf magnitude, and | Y i,j | ∠ φ i,j the (i,j) th entry of the Kron-reduced admittance matrix Y . A. Stochastic Load Model
Generator dynamics prevail over load dynamics in the studyof inter-area modes. Therefore, we model loads as constantimpedances to simplify the computations and obtain the gen-erator electromechanical dynamics from the network dynamics[14]. Considering a steady-state grid operation, we assume thatinter-area modes are excited by Gaussian load fluctuations thattranslate into variations of the diagonal elements of Y [15]: Y i,i (cid:48) = | Y i,i | (1 + σ i ξ i ) ∠ φ i,i , i = 1 , ..., n (4)where ξ i are mutually independent standard Gaussian randomvariables, σ i is the standard deviation of load variations and Y i,i ∠ φ i,i = G i,i + jB i,i is the (i,i) th element of Y . Substituting(4) into (3), i.e. replacing | Y i,i | with | Y i,i | (1 + σ i ξ i ) gives P e i (cid:48) = P e i + E i G i,i σ i ξ i (5) B. Stochastic Wind Speed Model
In this work, wind farms associated to a wind speed modelare integrated into the power network. Due to its intermittency,wind speed adds stochastic perturbations to the grid that canbe described by various continuous probability distributions,such as the Weibull distribution, the beta distribution, etc [16].Therefore, wind speed can be statistically represented by ageneric vector stochastic process v w = [ v w , ..., v w m ] T where m is the number of wind farms installed in the grid. In thesimulation study of this paper, wind speed is modeled as aWeibull distributed stochastic process by a set of stochasticdifferential algebraic equations, following [17]–[19].The power captured by a variable speed wind farm is P w j = n g j ρ c p j A r j v w j , j = 1 , ..., m (6)where n g j is the number of wind turbines that composethe wind farm, ρ is the air density, c p j is the performancecoefficient, and A r j is the turbine rotor swept area. Hence,wind farm power output is also a stochastic process. Windpower dynamics are closely coupled to the voltage phasors ofthe buses where wind farms are installed [20], which subse-quently affect the electrical power output P e i of synchronousgenerators (see (3)). That being said, P e i , i = 1 , ..., n is afunction of v w . Thus, (5) can be re-written as: P e i (cid:48) = P e i ( v w ) + E i G i,i σ i ξ i (7) C. Stochastic Dynamic Power System Model
Substituting (7) to (2) , we obtain the power system dynamicmodel operating around steady state under the influence ofsmall random load fluctuations and wind speed perturbations: ˙ δ i = ω i (8) M i ˙ ω i = P m i − P e i ( v w ) − D i ω i − E i G i,i σ i ξ i (9)Linearization around the stationary point ( δ , ω ) yields: (cid:20) ∆ ˙ δ ∆ ˙ ω (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ˙ x = (cid:20) n × n I n × n − M − ∂ P e ( v w ) ∂ δ − M − D (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) A (cid:20) ∆ δ ∆ ω (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) x + (cid:20) n × n − M − E G Σ (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) B ξ ξ (10)where δ = [ δ , ..., δ n ] T , ω = [ ω , ..., ω n ] T , ∆ δ = δ − δ , ∆ ω = ω − ω , ξ = [ ξ , ..., ξ n ] T , P e ( v w ) =[ P e ( v w ) , ..., P e n ( v w )] T , G = diag ([ G , , ..., G n,n ]) , M = diag ([ M , ..., M n ]) , D = diag ([ D , ..., D n ]) , E = diag ([ E , ..., E n ]) , and Σ = diag ([ σ , ..., σ n ]) . B ξ expressesthe effect of load variations. A is the state matrix whoseeigenproperties provide all the modal information includingfrequencies, damping ratios, etc. Thus, the accurate knowledgeof A plays a crucial role in identifying inter-area modes andperforming online small-signal stability monitoring. Tradition-ally, the calculation of A requires the knowledge of ∂ P e ( v w ) ∂ δ , M and D . However, it may be hard to obtain the exact valuesof M and D in large-scale power grids [9]. Moreover, thecomputation of the Jacobian ∂ P e ( v w ) ∂ δ requires informationabout the network model and its parameters (e.g. Y and E ),which may be unknown or corrupted in practice [8]. D. Data-Driven Inter-Area Mode Estimation
To overcome the aforementioned challenges, the purelydata-driven strategy [7] can be exploited to estimate A , andthus the inter-area mode properties, from PMU data. Forsimplicity, we assume that all generator terminal buses areequipped with PMUs that provide measurements of real-timephasors of voltages and currents. However, the method canalso handle cases of missing PMUs [7]. Rotor angles δ andspeed deviations ω can be estimated from synchrophasordata around steady state [21]. Hence, the state vector x =[∆ δ , ∆ ω ] T is obtained. The stationary covariance matrix C xx and the τ -lag time correlation matrix G xx ( τ ) of x satisfy: C xx = E ([ x ( t ) − ¯ x ] , [ x ( t ) − ¯ x ] T ) (11) G xx ( τ ) = E ([ x ( t + τ ) − ¯ x ] , [ x ( t ) − ¯ x ] T ) (12)According to the regression theorem for the Ornstein-Uhlenbeck process A can be estimated purely from the statis-tics of state variables that can be obtained from PMU data: A = 1 τ log (cid:2) G xx ( τ ) C xx − (cid:3) (13)which bypasses the knowledge of network topology andgenerator parameters. After estimating A purely from fieldmeasurements, we extract the inter-area mode informationby modal analysis. Each inter-area mode is associated to aneigenpair λ i ± = r i ± jh i , i ∈ { , ..., n } of A and has its ownfrequency f i = h i π and damping ratio ζ i = − r i | λ i | .When wind speed fluctuations are trivial (i.e. the varianceof v w is negligible), P w j remains approximately constant andthe normally distributed load randomness prevails over windrandomness. Since A is almost fixed, x can be termed as avector Ornstein-Uhlenbeck process in steady-state operation.Consequently, the data-driven method (13) is expected toield accurate inter-area mode estimation results that greatlyenhance small-signal stability monitoring. On the other hand,if wind speed variations become more significant, the non-Gaussian distributed wind randomness dominates load un-certainty. Meanwhile, P w j might be highly fluctuating. As aresult, A is no longer constant such that the data-driven method[7] based on the property of the vector Ornstein-Uhlenbeckprocess may fail to provide accurate estimation for inter-areamodes. Nonetheless, the recent advancement of ESSs can beused to smooth out the random wind power, thus improvingthe inter-area mode estimation accuracy. It will be shown inthe next section that by installing a small extra capacity tothe existing ESS infrastructure, only a negligible additionaleffort is required to achieve an accurate real-time small-signalstability monitoring of power grids with volatile wind power. E. Energy Storage System (ESS) for Wind Power Smoothing
The randomness of wind power results in active powerimbalance on the wind generator side: P im j ( t k ) = P w j ( t k ) − P ref j , k = 1 , ..., N (14)where P w j ( t k ) is the actual power output, P ref j is thereference (rated) power output, and P im j ( t k ) is the initialpower imbalance of wind farm j ∈ { , ..., m } at time instant t k . To smooth the wind power fluctuations, we assume thatevery wind farm is equipped with an ESS [13]. Inspired bythe single-bus multitimescale method introduced in [22], wemodel ESS discrete dynamics as follows: S j ( t k +1 ) = S j ( t k ) + η c j C j ( t k ) − η d j D j ( t k ) (15)where S j ( t k ) ≤ S max j , C j ( t k ) ≤ C max j , D j ( t k ) ≤ D max j are the stored, charging and discharging power of ESS j ∈{ , ..., m } at time instant t k , respectively; η c j is the charg-ing efficiency (ratio of charged to input power); η d j is thedischarging efficiency (ratio of output to discharged power). S max j , C max j and D max j denote the ESS power capacityfor smoothing purposes, the maximum ESS charging powerand the maximum ESS discharging power, respectively, while S j ( t ) is known. If P im j ( t k ) ≥ , there is a surplus of energyat wind farm j and the ESS is charged with ≤ C j ( t k ) ≤ min { P im j ( t k ) , C max j } and D j ( t k ) = 0 . If P im j ( t k ) ≤ ,there is a deficit of energy at wind farm j and the ESS isdischarged with ≤ D j ( t k ) ≤ min {− P im j ( t k ) , D max j } and C j ( t k ) = 0 . The goal of ESS control (15) is to minimize theexpected average magnitude of the residual power imbalance P res j ( t k ) = P im j ( t k ) − C j ( t k ) + D j ( t k ) (16)i.e. the wind power imbalance after ESS operation. In otherwords, µ P resj = E ( N (cid:80) Nk =1 | P res j ( t k ) | ) needs to be as closeto zero as possible so as to give a smoothed wind farmpower output. It can be proved that a greedy policy (i.e.charging/discharging sequence) π o ∗ = { ( C j ∗ ( t k ) , D j ∗ ( t k )) : k = 1 , ..., N } solves the aforementioned minimization prob-lem optimally [22]. Particularly, if P im j ( t k ) ≥ then: Calculate
Std( P imj ( t k )) , j = 1 , ..., m Step 1
Std( P imj ( t k )) > γ p ? Step 2
ESS j is OFFESS j is ON S max j = α × Std( P imj ( t k )) Step 3
Set C max j = S maxj / η cj and D max j = η d j S max j Step 4
Apply optimal policy (17)–(18) to smooth P imj ( t k ) Step 5
Obtain x = [∆ δ , ∆ ω ] T from PMU data Step 6
Estimate state matrix A by (13) Step 7
Perform modal analysis on A and obtain theinter-area mode properties f i , ζ i , i ∈ { , ..., n } Step 8
Yes NoFig. 1. Flowchart of the proposed ESS algorithm. C j ∗ ( t k ) = C max j , if C max j ≤ min { P im j ( t k ) , S maxj − S j ( t k ) η cj } P im j ( t k ) , if P im j ( t k ) < min { S maxj − S j ( t k ) η cj , C max j } S maxj − S j ( t k ) η cj , otherwise (17)In contrast, if P im j ( t k ) ≤ then: D j ∗ ( t k ) = D max j , if max { P im j ( t k ) , − η d j S j ( t k ) } < − D max j − P im j ( t k ) , if max {− η d j S j ( t k ) , − D max j } ≤ P im j ( t k ) η d j S j ( t k ) , otherwise (18)This ESS policy allows the accurate online small-signalstability monitoring [7] by smoothing the wind farm poweroutput variations. Note that the considered generic ESS couldbe pumped hydro storage, battery energy storage, etc [23].Based on the above, we propose a data-driven ESS-basedalgorithm for monitoring the small-signal stability of powersystems with intermittent wind power (see Fig. 1). Particularly: • In Step 1 , the standard deviation
Std( P imj ( t k )) can beeither calculated using the CDF of v w j and (6) or directlyobtained from wind power data or a probabilistic model [22]. • In Step 2 , γ p is selected as a threshold for the acceptabledeviations in the varying wind farm power output. In thispaper, γ p = 0 . is chosen, the suitability of which is confirmedby the numerical experiments (see Section III). • In Step 3 , the value of α is determined based on the relevantESS literature [22] and our simulation experience. To reducethe expected average magnitude of the initial power imbalance,i.e. µ P imj = E ( N (cid:80) Nk =1 | P im j ( t k ) | ) , by , S max j is set toe only × Std( P imj ( t k )) , i.e. α = 7 . The resulting S max j is only 1 per unit (p.u.). • In Step 4 , the efficiencies η c j and η d j typically lie in [50% , [22] and are considered as pre-known parameters.III. N UMERICAL R ESULTS
In this section, the effectiveness of the proposed data-driven ESS strategy in enhancing online small-signal stabilitymonitoring is validated. Simulations are conducted on theIEEE -bus system; see Fig. 2. Detailed modal analysisreported in the literature [9] reveals the presence of three inter-area modes with typical frequencies f = 0 . Hz (mode ), f = 0 . Hz (mode ) and f = 0 . Hz (mode ).The classical model has been used to represent synchronousmachines. Loads are modeled by constant impedances experi-encing Gaussian variations with σ i = 20 [9]. The rest of thepower system is represented according to [24]. Wind poweris integrated into the grid through the widely used doubly-fedinduction generator (DFIG) [20]. The stochasticity of windspeed v w is modeled by the Weibull distribution with shapeparameter k v w = 1 and scale parameter λ v w = 0 . obtainedfrom real-life applications [25]. Wind power fluctuations aresmoothed using ESS with η c j = η d j = 70% and S max j = η c j C max j = D max j /η d j = 100 MW ( p.u.), j = 1 , ..., m .Time-domain simulations are implemented in PSAT toolbox[26], while ESS operates every / s [22]. A. Validation of the Data-Driven ESS Algorithm for Enhanc-ing Inter-Area Mode Identification
In this case study, we install
MW DFIG-based windfarms to the zero-injection buses { , , , } , reaching awind penetration level of . Buses , , , correspondto j = 1 , , , , respectively (see (14)). Next, we conduct Monte Carlo time-domain simulations (i.e. differentwind speed realizations v w ) to perform the probabilistic small-signal stability monitoring of the grid. Note that the proposedalgorithm can work using a single scenario, yet Monte Carlosimulation results are more rigorous from a statistical sense.Since Std( P im j ( t k )) > γ p = 0 . p.u., ∀ j , ESS is activated(ON). Table I presents a comparison of the expected averagemagnitude of the power imbalance when the ESS is OFF(i.e. µ P imj ) and ON (i.e. µ P resj ). It can be observed thatthe ESS algorithm achieves a decrease of almost inthe expected average magnitude of the power imbalance eventhough S max j , j = 1 , ..., is very small given the systemscale. This result is consistent with the findings of relevantworks [22]. Furthermore, the per-unit value of µ P resj ( ≈ . p.u. = 5 MW) approaches zero and is less than of thereference wind farm power output ( MW = 5 p.u.).Next, we compute the statistical properties of A , f i and ζ i , i = 1 , , . To this end, we use s PMU data witha sampling frequency of Hz. Simulation results showthat wind stochasticity mostly affects the estimation of mode . Therefore, a comparison of the mean true and estimatedfrequency and damping ratio of inter-area mode with and Fig. 2. 68-bus, 16-generator, 5-area benchmark system [24].TABLE IW
IND P OWER I MBALANCE W ITH AND W ITHOUT
ESSBus j µ P imj (p.u.) µ P resj (p.u.) Decrease (%)
19 1 0 .
142 0 .
044 69 . .
138 0 .
046 66 . .
121 0 .
040 66 . .
132 0 .
041 68 . TABLE IIM
EAN T RUE AND E STIMATED P ROPERTIES OF I NTER -A REA M ODE ESS E ( f t ) (Hz) E ( f e ) (Hz) Err. (%) E ( ζ t ) (%) E ( ζ e ) (%) Err.(%)OFF .
753 0 .
760 0 .
930 1 .
756 1 .
378 21 . ON .
754 0 .
758 0 .
531 1 .
754 1 .
660 5 . Note: “ t ” stands for “True”, “ e ” for “Estimated”, and “Err” for “Error”. Fig. 3. Initial power imbalance P im ( t k ) (displayed in blue) and residualpower imbalance P res ( t k ) (displayed in red) for the wind farm of bus . without ESS is presented in Table II. Notice that, E ( f ) is ac-curately estimated when ESS is OFF despite wind uncertainty.On the other hand, the estimation of E ( ζ ) without ESS ispoor, thus deteriorating the small-signal stability monitoring.These results demonstrate that the proposed data-driven ESSalgorithm can reduce significantly the modal estimation errorwhile requiring only a very small ESS capacity. Similarconclusions can be drawn for the other inter-area modes. B. The Algorithm Operation under High Wind Penetration
In this case study, the effectiveness of the proposed ESSalgorithm is validated under higher wind power penetrationlevels. Particularly, we extend the base-case scenario of sectionIII-A by installing additional DFIG-based wind farms of
MW on top of the existing ones in buses { , , , } .Thus, three different wind penetration levels are considered:(1) Buses { , , , } – penetration (base case)(2) Buses { , , , , , } – penetration(3) Buses { , , , , , , , } – penetrationwhile { , , , } → j = { , , , } , respectively. Tovisualise the effectiveness of the proposed ESS algorithm, Fig. ABLE IIIM
EAN A BSOLUTE P ERCENTAGE I NTER -A REA M ODE E STIMATION E RROR
Wind Penetration ESS
MAP E f (%) MAP E ζ (%) OFF .
316 9 . ON .
914 2 . OFF .
653 9 . ON .
214 1 . OFF .
043 16 . ON .
260 3 . ( j = 4 ), for a single realizationof v w . With the exception of a few remaining power im-balance spikes, P res ( t k ) becomes zero, thus resulting in anapproximately constant wind farm power output. Importantly, S max j = 100 MW is sufficient to almost eliminate wind powervariations despite the penetration increase.Next, the accuracy of small-signal stability monitoring isassessed. Again, modal analysis is carried out based on
Monte Carlo simulations. Table III summarizes the comparisonbetween the mean absolute percentage error (MAPE) in theestimation of the inter-area mode frequencies (
M AP E f ) anddamping ratios ( M AP E ζ ) when ESS is OFF and the corre-sponding MAPE when ESS is ON. For illustration purposes,the average of all three modes is computed. As can be seen,frequency errors are trivial irrespective of the ESS applica-tion. Nonetheless, damping ratios, which are the main focusof small-signal stability monitoring, exhibit large estimationerrors when ESS is deactivated, especially for higher windpenetration levels (e.g. ). As a result, a poorly-dampedinter-area mode can be potentially identified as well-damped.Clearly, our method promotes the accurate estimation of ζ i bysmoothing the unwanted wind power fluctuations.IV. C ONCLUSION AND P ERSPECTIVES
This paper proposed a novel data-driven ESS algorithmfor small-signal stability monitoring of power systems withstochastic wind power penetration. Our method can accu-rately estimate the inter-area mode properties by smoothingthe wind power variations, thus enhancing the small-signalstability assessment in near real-time. Numerical simulationsdemonstrate that the proposed technique achieves the inter-area mode identification and smoothing goals using only asmall ESS capacity. Future endeavors will focus on developingmethodologies for optimal ESS placement in wind farms andon implementing the proposed data-driven ESS in practice.R
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