This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT 1 Dynamic Phasor Estimates Under the Bellman’s Principle of Optimality: The Taylor-LQG-Fourier Filters Luis Fernando Sánchez-Gómez and José Antonio de la O Serna, Senior Member, IEEE Abstract —Recently Taylor K –Kalman–Fourier filters were pro- posed for estimating dynamic phasors to provide instantaneous estimates and drastically reduce the total vector error by a factor of 10. However, they exhibit resonant frequencies at the edges of the pass band, and high-interharmonic gains. In this paper, the optimal linear quadratic (LQ) control is applied to design feedback filters referred to as Taylor K -LQG-Fourier filters. This method reduces the interharmonic gains and the resonant frequency at the passband edges of the Taylor K –Kalman–Fourier filter. The estimates from oscillating signals obtained through this optimal technique are quasi-instantaneous, and provide estimates of the instantaneous frequency, and its rate of change, preserving its synchrony with the signal for control applications. The effectiveness of the proposed algorithm is verified through simulations and experimental results. Index Terms—Kalman filter, linear-quadratic control, linear- quadratic-Gaussian controller, optimal control, oscillating phasor, phasor estimate. I. Introduction M ONITORING and controlling power systems, or any other system whose behavior is sinusoidal, depend on accurate and instantaneous phasor estimates, specially when signals fluctuate. Better phasor estimates under dynamic con- ditions are obtained with signal models that assume fluctuating amplitude and phase, instead of the traditional signal model with constant parameters. Nowadays, the phasor estimation problem has attracted a lot of attention, due to the proliferation of electronic devices designed to operate under oscillations conditions, like phasor measurement units. This interest was expressed through the recently reviewed synchrophasor stan- dard [1]. Phasor estimates under oscillating conditions were recently proposed in several studies. The Taylor K –Kalman–Fourier (T K –K–F) filter [2] was developed to solve the delay of the Taylor K -Fourier Transform (TFT) [3]. The TFT is more appropriate under dynamic conditions than the discrete Fourier transform; however, both produce phasor estimates systemati- Manuscript received December 4, 2012; revised June 11, 2013; accepted June 13, 2013. This paper was recommended by Associate Editor C. Muscas. The authors are with State University of Nuevo León, UANL, Mon- terrey, Nuevo León 66450, Mexico (e-mail: fersago.00@gmail.com; jde- lao@ieee.org). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2013.2270316 cally delayed, while the T K –K–F filter provides instantaneous estimates that can be truly synchronized; opening the way to control applications. Another important advantage of these filters is that they estimate not only the phasor, but also its first derivatives. This becomes important because now in addition to the phasor estimates during oscillations, or abrupt changes, it is also necessary to estimate the system frequency, and its rate of change. The main idea of this paper is to show the control action of the linear-quadratic controller based on the Bellman’s principle of optimality [4], [5] using the Taylor K –Kalman (T K –K) filters as observers to generate the state vectors from the available signal samples. This optimal method leads us to the Taylor K -LQG filter, and the Taylor K -LQG-Fourier filters with new feedback gains able to modify the trajectory of the phasor estimates obtained through the Kalman filter, and improve its frequency response. This paper was motivated by the research of the opti- mum optimorum method for phasor estimation under dynamic conditions. We want to know which one among the least mean squares, Kalman filtering, or the linear quadratic (LQ) algorithm is the best one for that purpose. All of them claim to be optimal, but difficulties emerge in their implementation as those shown in this paper. The LQ algorithm cannot operate over the available signal samples, its implementation requires an observer to generate the states of the signal. In this paper, we used the Kalman algorithm for this purpose. In this way a hybrid method was found. Of course, many other strategies are possible, that is why we share the findings of this paper to stimulate the research in this interesting field. The main contribution of this paper is to provide a new class of filters, developed from the modified Linear-Quadratic- Gaussian (LQG) classical algorithm, and demonstrate that such filters are able to improve the magnitude response of the T K –K and T K –K–F filters, by reducing their high-resonance frequencies in the subharmonic bands. The LQG method with a flexible signal model estimates phasors of oscillation signals with enough accuracy and provides quasi-instantaneous phasor estimates. This paper is organized as follows: Section II describes the state-space representation of the signal model. Section III describes the overview of the LQG controller and the changes applied in the implementation presented in this paper. Then, in Section IV the controller equations LQG and their respective 0018-9456/$31.00 c  2013 IEEE