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Local Sequential Ensemble Kalman Filter for Simultaneously Tracking States and Parameters

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Local Sequential Ensemble Kalman Filter for Simultaneously Tracking States and Parameters Local Sequential Ensemble Kalman Filter for Simultaneously Tracking States and Parameters Ning Zhou, Zhenyu Huang, Yulan Li Greg Welch Energy and Environment Directorate The Department of Computer Science Pacific Northwest National Laboratory The University of North Carolina at Chapel Hill Richland, WA 99352, USA Chapel Hill, NC 27599, USA E-mail: {ning.zhou, zhenyu.huang, yulan.li}@pnnl.gov E-mail: [email protected] Abstract— Accurate information about dynamic states and simulation clearly reveals the inadequacy of the model. Low parameters is important for efficient control and operation of a confidence in the accuracy of a model usually leads to power system. To improve the estimation accuracy of states and conservative operation and reduces asset utilization. To parameters, this paper applies a local sequential ensemble improve model accuracy, many studies involving staged tests Kalman filter (EnKF) method to simultaneously estimate have been carried out to identify parameters [5,6]. However, dynamic states and parameters using phasor-measurement-unit most staged calibration methods require injecting probing (PMU) data. Based on simulation studies using multi-machine signals and maneuvering real/reactive power, and therefore systems, the proposed method performed favorably in tracking cannot capture parameter changes during normal operations. It both states and parameters in real time. would be of significant value if some parameters of dynamic models can be calibrated on line using event data. I. INTRODUCTION The real-time states of electromechanical dynamics reflect Electromechanical dynamic models are used widely to the current status of a power system and can be used to study problems involving transient and small signal stability in coordinate controllers in a wide area. Traditionally, estimating power systems. Accurate information about states (e.g., rotor dynamic states (e.g., rotor speeds, angles) over a wide area speeds, angles) and parameters is essential for efficient control was not possible because data from the Supervisory Control and operation of a power system. States are the minimum set and Data Acquisition (SCADA) system are normally sampled of variables that can determine the current status of a dynamic once every 2 to 4 seconds. This sampling rate is too low for system [1]. Parameters are the coefficients that relate the the data to reveal electromechanical dynamic responses. In input, output, and state variables of a model. A dynamic addition, SCADA data are not well synchronized. As a result, model with accurate parameters and states can faithfully most controllers (e.g., power system stabilizers) for reveal system responses. Therefore, the model can be used to controlling the electromechanical dynamics only use local enhance stability control and establish accurate operation states to achieve local objectives. Compatibility among limits, which in turn improve the reliability and efficiency of controllers is only studied in planning models during offline the power system. studies. Without global objectives and systematic coordination Traditionally, component-based approaches are used to over wide areas, the influence of local controllers at the grid build a dynamic model. The model parameters are derived level may not always be desirable. For example, the wide area from equipment manuals or staged tests [2]. A dynamic model small signal problem has been associated with, “… the can be used to answer ‘what-if’ questions regarding transient introduction of high gain, and low time constant automatic and small signal stability problems, provide dynamic security voltage regulators” [7]. assessments for power system operations[3], and guide The parameters and states of dynamic models can be controller design. Yet, because of the large number of estimated in real time using measurement data from phasor components in a typical power grid, efforts to build and measurement units (PMU). These data typically have a maintain an accurate, comprehensive model are not trivial. sampling rate of 30 or 60 samples per second, are well This often results in a model that does not adequately reflect synchronized with the Global Positioning System clock, and true system behaviors. For example, the initial model for can continuously capture the dynamic responses of a power simulating the U.S. Western Interconnection breakup on system under normal and abnormal conditions. Prior work by August 10, 1996 could not replicate the oscillations that were [8] revealed the benefits and potential of using PMU data with measured during the event [4]. The discrepancy between the extended Kalman filter (EKF) for online parameter responses measured during a real event and responses from a identification and state estimation. Ghahremani et al. This paper is based on work sponsored by the U.S. Department of Energy (DOE) through the Applied Mathematics Program in the Office of Advanced Scientific Computing Research. Pacific Northwest National Laboratory is operated by Battelle for DOE under Contract DE-AC05- 76RL01830. extended EKF to simultaneously estimate the generator states B. Generalized Dynamic Model and unknown inputs [9], as well as using the unscented To generalize the problem formulation, non-linear Kalman filter for estimating states using a single machine differential algebraic equations are used to describe power infinite bus system [10]. However, the required sampling rate system dynamics as expressed in (3). of data in those studies are 2,000 samples per second and 10,000 samples per second, which is higher than what can be ⎧ x = fc (x,α) + wc provided by a commercial PMU. ⎪ ⎨α = 0 + ε c (3) ⎪ To overcome the difficulty, this paper proposes a local ⎩z = hc (x,α) + vc sequential ensemble Kalman filter (EnKF) method to nx where vector x ∈R represents state variables; vector simultaneously estimate dynamic states and parameters using R nα represents the parameters to be identified (for a PMU data. The paper is organized as follows. Section II α ∈ classical generator; the parameters are H, KD, ); formulates the state and parameter tracking problem using X dʹ classical generator models and then generalizes the problem vector z ∈R m represents available measurements (e.g., PMU ~ ~ for other dynamic components. Section III provides an measurements V and I at generator buses); the function h(*) overview of the basic EnKF. Section IV presents a local is the measurement function; the symbol “c” indicates the sequential EnKF method. Section V introduces the inflation continuous form of the model; and the random variables method for the proposed EnKF. Section VI evaluates the nx , nα , and m represent process, parameter, wc ∈ R ε c ∈ R vc ∈ R performance of the proposed EnKF using simulation models. and measurement noise, respectively. For a classical generator In Section VII, conclusions and future work are discussed. model, the states are δ and Δω as shown in (1) and (2). II. PROBLEM FORMULATION Assuming a sampling interval of Δt seconds, the discrete This section introduces the problem of tracking states and form of the model corresponding to (3) can be written as (4). parameters using PMU data. First, a classical generator model ⎧ xk = f (xk−1,α k−1) + wk−1 is used as an example. The problem then is generalized using a ⎪ ⎨α k = α k−1 + ε k−1 (4) non-linear dynamic model. ⎪ ⎩ zk = h(xk ,α k ) + vk A. Classical Generator Model where subscript “k” denotes variables and quantities at time kΔt. The state transition function f(*) in (4) can be A classical generator model can be described by (1) and implemented through numerical integration of fc(*) in (3). The (2) [2]: process, parameter, and measurement noise are assumed to be white noise with a mean value of zero and a covariance matrix 0 0 ⎡ δ ⎤ ⎡ ω0 ⎤⎡ δ ⎤ ⎡ ⎤ of Q and R as defined by (5) and (6): = K + (T −T ) (1) ⎢ ⎥ ⎢0 D ⎥⎢ ⎥ ⎢ 1 ⎥ m e ⎣Δω⎦ ⎣ − 2H ⎦⎣Δω⎦ ⎣ 2H ⎦ ⎛⎡wk ⎤ T T ⎞ EV E⎜ w ε ⎟ = Q (5) T P sin( ) ⎜⎢ ⎥[ k k ]⎟ e = e = δ −θ (2) ⎝⎣ε k ⎦ ⎠ X dʹ T where δ is the rotor angle and Δω is the rotor speed deviation. E(vk vk ) = R (6) The symbol Tm is the mechanical torque, Te is the electric air- The generalized state and parameter tracking problem then gap torque, H is the inertia constant, and KD is the damping can then be stated as follows. factor. The symbol ω0 is the rated value of the rotor speed. The generator is connected to a terminal bus through a Using the PMU measurement sequence of z1, z2, …, zk, and transient reactance X ʹ . The symbol E is the internal voltage model structure (4), (5), and (6), estimate the states of xk and d the parameters of α . magnitude of the generator. Assume that PMU measurements k of voltage and current phasors are available at the terminal bus Note that unknown parameters α propagate over time in a of the generator. The state and parameter tracking problem for way that is similar to the states x in (4). By treating parameters a classical generator model then can be stated as follows. as special states, and given sufficient measurements such that ~ the system is observable, the state and parameter tracking Using PMU measurements of voltages (V =V∠θ ) and ~ problem can be solved using a Kalman filter. currents ( I ) at the generator buses, estimate the states of δ, Δω, and model parameters of H, K , . D X dʹ III. REVIEW OF AN ENSEMBLE KALMAN FILTER Note that (1) and (2) are models only for a dynamic In this paper, EnKF is chosen for tracking states and component (in this case, a generator) in a power grid. All the parameters because it has been successfully applied in component models are coupled through the grid network oceanographic studies (under the term “data assimilation”) to model to form the full system model [2]. All the other track states of high-order nonlinear dynamic systems [11].
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