The electric power transmission grid has been progressively developed for over a century, from initial design of local dc networks in low-voltage levels to three-phase high voltage ac networks, and finally to modern bulk interconnected networks with various voltage levels and multiple complex electrical components. The development of human society and economic needs is the major driving force the revolution of transmission grids stage-by-stage with the aid of innovative technologies. The current power industry is being modernized and tends to deal with the challenges more proactively by using the state-of-the-art technologies in the areas of sensing, communications, control, computing, and information technology. The shift in the development of transmission grids to be more intelligent has been summarized as “smart grid” [see Fig.1].
In a smart transmission network, flexible and reliable transmission capabilities can be facilitated by the advanced Flexible AC Transmission Systems (FACTS), high-voltage dc (HVDC) devices, and other power electronics-based devices. The FACTS devices are optimally placed in the transmission network to provide a flexible control of the transmission network and increase power transfer levels without new transmission lines. These devices also improve the dynamic performance and stability of the transmission network. Through the utilization of FACTS technologies, advanced power flow control, etc., the future smart transmission grids should be able to maximally relieve transmission congestions, and fully support deregulation and enable competitive power markets. In addition, with the increasing penetration of large-scale renewable/alternative energy resources, the future smart transmission grids would be able to enable full integration of these renewable energy resources(Wira et al., 2010, Sauter & Lobashov 2011, Varaiya et al., 2011).
Smart substations would provide advanced power electronics and control interfaces for renewable energy and demand response resources so that they can be integrated into the power grid on a large scale at the distribution level. By incorporating micro-grids, the substation can deliver quality power to customers in a manner that the power supply degrades gracefully after a major commercial outage, as opposed to a catastrophic loss of power, allowing more of the installations to continue operations. Smart substations should have the capability to operate in the islanding mode taking into account the transmission capability, load demand, and stability limit, and provide mechanisms for seamlessly transitioning to islanding operation. Coordinated and self-healing are the two key characteristics of the next generation control functions. These applications require precise tracking of the utility’s phase-angle information, for high performance local or remote control, sensing and fault diagnosis purposes(Froehlich et al., 2011, Han et al., 2009).
On the other hand, the proliferation of nonlinear loads causes significant power quality contamination for the electric distribution systems. For instance, high voltage direct transmission (HVDC), electric arc furnaces (EAFs), variable speed ac drives which adopts six-pulse power converters as the first power conversion stage, these devices cause a large amount of characteristic harmonics and a low power factor, which deteriorate power quality of the electrical distribution systems. The increasing restrictive regulations on power quality problems have stimulated the fast development of power quality mitigation devices, which are connected to the grid to improve the energy transmission efficiency of the transmission lines and the quality of the voltage waveforms at the common coupling points (PCCs) for the customers. These devices are known as flexible AC transmission systems (FACTS) (Fig.2), which are based on the grid-connected converters and real-time digital signal processing techniques. Much work has been conducted in the past decades on the FACTS technologies and many FACTS devices have been practically implemented for the high voltage transmission grid, such as static synchronous compensators (STATCOMs), thyristor controlled series compensators (TCSCs) and unified power flow controllers (UPFCs) (Fig.3), etc(Cirrincione et al., 2008, Jarventausta et al, 2010).
The stable and smooth operation of the FACTS equipments is highly dependent on how these power converters are synchronized with the grid. The need for improvements in the existing grid synchronization approaches also stems from rapid proliferation of distributed generation (DG) units in electric networks. A converter-interfaced DG unit, e.g., a photovoltaic (PV) unit (Fig.4), a wind generator unit (Fig.5) and a micro-turbine-generator unit, under both grid-connected and micro-grid (islanding) scenarios requires accurate converter synchronization under polluted and/or variable-frequency environment to guarantee stable operation of these grid-connected converters(Jarventausta et al., 2010).
Besides, an active power filter (APF) (Fig.6) or dynamic voltage restorer (DVR) (Fig.7) rectifier also requires a reference signal which is properly synchronized to the grid. Interfacing power electronic converters to the utility grid, particularly at the medium and high voltages, necessitates proper synchronization for the purpose of operation and control of the grid-connected converters. However, the controller signals used for synchronization are often corrupted by harmonics, voltage sags or swells, commutation notches, noise, phase-angle jump and frequency deviations(Abdeslam et al., 2007, Cirrincione et al., 2008).
Therefore, a desired synchronization method must detect the phase angle of the fundamental component of utility voltages as fast as possible while adequately eliminating the impacts of corrupting sources on the signal. Besides, the synchronization process should be updated not only at the signal zero-crossing, but continuously over the fundamental period of the signal(Chang et al., 2009, Chang et al., 2010). This chapter aims to present the harmonic estimation and grid-synchronization method using the adaptive linear neural network (ADALINE) (Figs.8 and 9). The mathematical derivation of these algorithms, the parameter design guidelines, and digital simulation results would be provided. Besides, their practical application for the grid-connected converters in smart grid would also be presented in this chapter.
2. Mathematical model of the adaptive linear neural network (ADALINE)
The adaptive linear neural network (ADALINE) was used to estimate the time-varying magnitudes and phases of the fundamental and harmonics from a distorted waveform. The mathematical formulation of ADALINE is briefly reviewed. Consider an arbitrary signal
The square error on the pattern
where the weights are assumed to be fixed at
Notably, matrix R is real and symmetric, and
which is a linear function of weights. The optimal set of weights, , can be obtained by setting , which yields:
The solution of the Eq. (10) is called Weiner solution or the Weiner filter:
The Weiner solution corresponds to the point in weight space that represents the minimum mean-square error
where the learning rate
From Eq. (14), it can be found that the long-term average of approaches hence can be used as unbiased estimate of .If the input data set is finite (deterministic), then the gradient can be computed accurately by collecting the different gradients over all training patterns
When mean-square error
Thus the fundamental component of the measured signal
Obviously, the dimension of the weight vector
3. Synchronization for grid-connected converters using ADALINE technique
This Section formulates the generalized methodology for the phase-locked loop (PLL) design and synthesis by using adaptive linear neural network (ADALINE) technique. The mathematical derivation, the stability analysis and the detailed description of the proposed ADALINE-PLL are outlined consecutively herein. In subsection 3.1, the optimal control parameters selection of the proposed ADALINE-PLL is discussed in terms of the continuous domain and the discrete domain analysis. Furthermore, the time-domain simulation results of the proposed ADALINE-PLL under different control parameters are also presented for verification.
3.1. Mathematical formulation of the ADALINE-PLL
This section presents the grid synchronization technique using the ADALINE algorithm. Firstly, the formulation of the ADALINE problem by using single-phase representation is outlined as follows. An arbitrary grid voltage can be represented as:
where and are the initial phase angle of the fundamental and
where and represent the estimated phase angle of the fundamental grid voltage and the estimation error, respectively, obtained from the ADALINE-PLL. Therefore, the phase angle of the
From Eq. (21), it can be deduced that the original signal denoted by Eq. (18) can be regenerated by adjusting the coefficients , (
where is the estimated output of the grid voltage by using the LMS-based linear optimal filter methodology. The vector
Equation (23) can be rewritten as:
Notably, the salient difference between the ADALINE algorithm and the ADALINE-PLL algorithm is that, the frequency and phase angle signals utilized in the ADALINE weights updating process were assumed to be constant. However, in case of the ADALINE-PLL, the frequency and phase angle of fundamental component grid voltage is recursively updated by the loop filter (LF) and voltage controlled oscillator (VCO) of the PLL. In other words, the weights updating procedure of the ADALINE is utilized as the phase detector (PD) for the PLL, which generate the error signal to drive the loop filter (LF) and voltage controlled oscillator (VCO), according to the initial definition of PLL The graphical interpretation of the proposed ADALINE-PLL is illustrated in Fig.9. In order to better illustrate the working principle of the proposed ADALINE-PLL, the weights updating law and stability conditions are discussed in detail as follows.
In the discrete domain, the weight vector of the ADALINE should be changed in a minimum manner, subject to the constraint imposed on the updated filter output. Let denote the old weight vector of the ADALINE filter at the
For each (
Hence the weights adaption process is achieved by solving the optimization problem, as indicated by Eqs. (26)-(27). The cost function at the
where denotes the real-valued Lagrange multiplier. The term denotes the squared Euclidean norm of the weight change .The cost function is a quadratic function of the weight vector , as shown by expanding Eq. (28) into:
The optimum weight vector can be found by minimizing the cost function . Differentiate the cost function with respect to , we get:
By setting Eq.(30) equal to zero, the optimum value for , corresponding to the stationary point of the cost function , can be derived as:
Hence, the output of the ADALINE as denoted by Eq. (22) can be rewritten as:
Then, the Lagrange multiplier can be obtained as:
In order to ensure stable operation of the weight vector updating process, a positive real scaling factor
The aforementioned weights updating scheme, in essence, belongs to the well-known least mean square (LMS) algorithm, which may introduce convergence problem in case of small input vector since the squared norm appears in the denominator, as indicated by Eq. (36). To solve this problem, Eq. (36) can be modified as (Chang 2009):
where is a sufficiently small real number and . The weight adaptation law represented in Eq.(37) is adopted and practically implemented herein.
3.2. Stability analysis of the ADALINE
The selection of the step-size parameter
From Eqs.(37)-(39), the incremental in the estimation error can be derived as:
As stated above, the underlying idea of the ADALINE design is to minimize the incremental change in the weight vector from the
Taking the squared Euclidean norms of both sides of Eq. (41), rearranging terms, and then taking the expectations on both sides of equation, we get:
where denotes the undisturbed error signal defined by
From Eq.(43), it shows that the mean-square deviation decrease exponentially with the increase of iterations, hence the ADALINE is therefore stable in the mean-square error sense (i.e., the convergence process is monotonic), provided is satisfied, which corresponding to the following condition:
Considering the limited rate of variation in parameters for the practical grid-connected converter applications, if faster adaptation for the weight vector than the parameter variation of the input signal is ensured, it can be shown that this inequality can always be satisfied. It should be noted that the selection of the step-size parameter has a significant effect on the frequency characteristics of the ADALINE-PLL, which would be discussed in the forthcoming subsection. Here we first describe the proposed ADALINE-PLL and its implementation in Matlab/Simulink www.mathworks.com
3.3. Description of the proposed ADALINE-PLL
Figs.10-11 show the single-phase and three-phase version of the proposed ADALINE-PLL. The following discussion is mainly focused on the single-phase version of the ADALINE-PLL, but the similar analysis can be easily extended to the three-phase version. For the sake of brevity, only the fundamental component, fifth and seventh order harmonics are considered in the grid voltages, hence the estimation blocks corresponding to these three components are considered in the single-phase ADALINE-PLL. One may extend the order of the ADALINE-PLL by incorporating higher order harmonic blocks in the algorithm according to the particular applications. Fig.10(a) shows the top layer representation of the single-phase ADALINE-PLL, it can be observed that the estimation error, phase angle of the fundamental component in grid voltage, the learning rate are utilized as the input signals to the subsystems, namely, the fundamental frequency block, the fifth order harmonic block and the seventh order harmonic block.
Figs.10(b)-(d) shows the three subsystems for individual harmonic component estimation, namely, the fundamental component, the fifth and the seventh order harmonic components. Once again, the weights of the fundamental frequency component are denoted as and , hence the phase estimation error denoted by can be regulated to zero by using a properly designed closed-loop control system, which resembles that of the existing grid synchronization schemes. As shown in Fig.10(b), the per unit representation of the weight is utilized as the input signal for the loop filter (LF) of the PLL, which can be simply derived as:
The derived signal , is then used as input for the phase tracking algorithm. However, by incorporating the adaptive linear optimal filter methodology, the proposed ADALINE-PLL exhibits noticeable advantages compared to the existing grid synchronization algorithms in terms of response speed, accuracy and robustness.
Fig. 11 shows the corresponding three-phase version of the proposed ADALINE-PLL, which has a similar architecture with that of the single-phase version. One of the salient features of the three-phase ADALINE-PLL algorithm is that the Clark’s transformation and Park’s transformation are utilized consecutively to derive the
3.4. Parameter selection of the ADALINE-PLL
In this section, the parameter design of the single-phase version ADALINE-PLL is discussed by using continuous domain (
3.4.1. Continuous-domain (s-domain) analysis
Assuming the phase angle of the fundamental grid voltage detected by the closed-loop ADALINE-PLL is denoted by , which is an integral of the estimated angular frequency . In the steady state, the estimated angular frequency can be considered to be constant, hence the phase angle can be approximated as . Therefore, the block diagram of the ADALINE-PLL indicated by Fig.10 can be simplified as Fig.12, provided that the estimated angular frequency is within its neighborhood, i.e., (and represent the lower and upper boundaries which defines the lock range of the PLL). Referring to the fundamental frequency block in Fig.12, the estimated fundamental component in time domain can be represented as:
where represents the estimation error of the ADALINE, represents the estimated fundamental component of grid voltage, represents the operator of integration and asterisk denotes convolution. Applying Laplace transform to Eq. (46), rearranging terms, we get:
where , ,corresponds to the Laplace transform of , and , respectively. In Eq. (47), is represented as:
Similarly, for the
For the present case, the fundamental component, fifth and seventh order harmonics are considered, hence the error transfer function from the input to , can be represented as:
Similarly, the transfer function from the input to the estimated fundamental component , is:
Fig. 13 shows the bode-plot of the ADALINE when only the fundamental frequency block is considered. The frequency response of the ADALINE under the variations of the center frequency and the integration gain are shown in Fig.13(a) and Fig.13(b), respectively. Fig.13(a) shows the open-loop frequency response of the ADALINE with the variation of center frequency, it is interesting to notice that this characteristic provides the flexible frequency tracking capability, compared to the adaptive linear neural network (ADALINE) algorithm since the frequency response of ADALINE cannot adapt to the frequency variation in the input signal. It can be observed from Fig.13(b) that the integration gain, i.e., the learning rate (
Fig.14 shows the frequency response of the ADALINE when the fundamental component, fifth and seventh harmonic components are considered. Fig.14(a) shows the bode-plot from the input signal
It should be noted that the frequency domain analysis is based on the quasi-steady state model of the ADALINE, which serves the purpose of phase detection (PD) for the PLL. The estimated phase error signal is then utilized as the input for the loop filter (LF), which is selected as the standard proportional-integral (PI) regulator for the present case. Here the linearized model for the phase estimation can be described as Fig.15(a). It is interesting to observe that the derived linearized model for the phase estimation resembles that of the existing PLL algorithms. The closed-loop transfer function of the linearized model indicated by Fig.15(a) can be represented as:
where , denote the Laplace transform of the estimated phase angle and the actual phase angle respectively. To achieve a good trade-off between the filter performance and system stability, the proportional-integral (PI) type filter is utilized for the loop filter (LF), which can be given as:
The open loop transfer function of Fig.15 (a) can be derived as:
The root locus for the PLL modeled in the
3.4.2. Discrete-domain (z-domain) analysis
In the discrete domain, Eq. (50) can be rewritten as:
where , and
Assuming that , ,
The root locus for the ADALINE modeled in the
The ADALINE subsystem is assumed to be stable in the following discrete domain analysis, which implies that the phase detection is achieved. The
For the second order loop using the PI type filter,
The root locus for the PLL modeled in the Jitter—The time variation of a characteristic of a periodic signal in electronics and telecommunications, often in relation to a reference clock source.
Jitter—The time variation of a characteristic of a periodic signal in electronics and telecommunications, often in relation to a reference clock source.
3.5. Time-domain simulation results of the ADALINE-PLL
Figs.18-19 show the time-domain simulation results of the single phase version of the proposed ADALINE-PLL under different control parameters. The grid voltage is assumed to contain 0.1 p.u. 5th order harmonic and 0.1 p.u. 7th order harmonic components and a transient voltage sag occurs at t=0.05s to test the dynamic response of the ADALINE-PLL. Fig.18 shows the performance of the single-phase ADALINE-PLL with the variation of learning rate (
Fig.19 shows the performance of the ADALINE-PLL with the variation of regulator gains when the learning rate is predefined. It can be observed that the dynamic response of the ADALINE-PLL is mainly determined by the proportional gain
4. Performance comparison with the existing PLL algorithms
This section presents the performance comparison among the existing PLL algorithms and the proposed ADALINE-PLL. Firstly, a brief introduction of the enhanced PLL (EPLL) and the
4.1.The enhanced phase-locked loop (EPLL)
In recent literature, the enhanced PLL (EPLL) system was proposed (Karimi-Ghartemani et al, 2004). The major improvement introduced by the EPLL is in the PD mechanism, which is replaced by a new strategy allowing more flexibility and provides more information such as amplitude and phase angle. The mechanism of this EPLL is based on estimating in-phase and quadrature-phase amplitudes of the desired signal, hence, has potential application in communication systems which employ quadrature modulation techniques.
The Matlab/Simulink diagram of this EPLL is shown in Fig.20. It can be observed that there are three gains, denoted as
Parkphase-locked loop ( Park-PLL)
It was reported that the PD is always asymptotically stable around the equilibrium condition . As for the selection of time constants, if
After the cutoff frequency of the low pass filters is selected, the compensator gains, namely,
4.3. The performance evalution among the EPLL, the
Park-PLL, and the ADALINE-PLL
Fig.22 shows the simulation results corresponding to the estimated frequency in grid voltage and the phase estimation error when the grid is subjected to 0.7 per unit (p.u.) voltage sag. Here the existing grid synchronization schemes, namely, the enhance PLL (EPLL) and the
Fig.23 shows the simulation results corresponding to the estimated frequency in grid voltage and the phase estimation error when the grid is contaminated by harmonics. The 0.3 per unit (p.u.) 5th order harmonic and 0.3 per unit (p.u.) 7th order harmonic components are added to the grid voltage at t=0.05s with a duration of 0.15s to test the immunity of the various grid synchronization schemes. The
Fig.24 shows the simulation results corresponding to the estimated frequency in grid voltage and the phase estimation error when the grid voltage is contaminated by random noise. The random noise of power density 10e-5 per unit (p.u.) is added to the grid voltage at t=0.05s with a duration of 0.15s to test the immunity of the various grid synchronization schemes. Similar to the case of a sudden applying harmonics, the park-PLL and EPLL show noticeable oscillations in the estimated frequency when the noise is added to the grid voltage. Besides, the park-PLL shows longer settling time when the grid voltage recovers to the normal condition. The EPLL shows the highest estimation error in grid frequency with amplitude of about 5 Hz, and the park-PLL shows the estimation error of about 2Hz when the noise is imposed. However, the proposed ADALINE-PLL shows the lowest frequency oscillation (0.2Hz) and highest estimation accuracy in the estimated frequency compared to the other grid synchronization schemes. Moreover, the phase estimation error of the park-PLL and the EPLL is remarkable during transients, and the park-PLL is found to have a large settling time when the grid voltage recovers. Besides, it shows that the park-PLL has the maximum phase estimation error of about 3 degrees, and the phase estimation error of EPLL is less than 2 degrees. However, the proposed ADALINE-PLL shows negligible estimation error compared to the other algorithms, with amplitude of less than 0.5 degree. The estimated frequency and the phase estimation error in Fig.24 indicate that the proposed ADALINE-PLL shows better robustness when grid voltage is contaminated by random noise.
The electrical power systems are under a transition to the smart grid owing to the advancement of modern control, communication technologies and the requirement of real-time marketing. In the smart grid, the power converters are indispensable components which connect the renewable energy resources and the FACTS devices, power quality conditioning devices to the grid. Hence the accurate grid-synchronization of these power converters to the grid is crucial to ensure their stable operation. This book chapter aims to provide a systematic approach for the adaptive linear neural network (ADALINE) algorithm for the real-time harmonic estimation and phase synchronization for the grid-connected converters, which are the fundamental building blocks for the smart grid infrastructure.
The mathematical derivation of the ADALINE algorithm and the ADALINE-PLL scheme is presented, followed by the stability analysis, the continuous domain and the discrete domain models, and the guidelines for parameter selection of the ADALINE-PLL algorithm. The performance of the ADALINE-PLL is further validated by performance comparison with the existing park-PLL and EPLL algorithms. It can be expected that the presented ADALINE-based algorithms can find wide application in the grid-connected converters for smart grid applications.
This work is financially supported by the Fundamental Research Funds for the Central Universities of China under grant No.ZYGX2011J093.
- Jitter—The time variation of a characteristic of a periodic signal in electronics and telecommunications, often in relation to a reference clock source.