Modelling and control of a hybrid power filter to compensate harmonic distortion under unbalanced operation
2017; Institution of Engineering and Technology; Volume: 10; Issue: 7 Linguagem: Inglês
10.1049/iet-pel.2016.0263
ISSN1755-4543
AutoresA. Fernández, G. Escobar, Pánfilo R. Martínez-Rodríguez, José M. Sosa, Daniel U. Campos‐Delgado, Manuel J. López-Sánchez,
Tópico(s)Multilevel Inverters and Converters
ResumoIET Power ElectronicsVolume 10, Issue 7 p. 782-791 Research ArticleFree Access Modelling and control of a hybrid power filter to compensate harmonic distortion under unbalanced operation Andres A. Valdez-Fernandez, Corresponding Author Andres A. Valdez-Fernandez andres.valdez@ieee.org School of Sciences, UASLP, SLP 78290, MexicoSearch for more papers by this authorGerardo Escobar, Gerardo Escobar School of Engineering, UADY, Merida, Yucatan, 97310 MexicoSearch for more papers by this authorPanfilo R. Martinez-Rodriguez, Panfilo R. Martinez-Rodriguez School of Sciences, UASLP, SLP 78290, MexicoSearch for more papers by this authorJose M. Sosa, Jose M. Sosa orcid.org/0000-0002-5153-6185 Laboratory of Electricity and Power Electronics, ITESI, Irapuato-Silao, GTO, 36821 MexicoSearch for more papers by this authorDaniel U. Campos-Delgado, Daniel U. Campos-Delgado School of Sciences, UASLP, SLP 78290, MexicoSearch for more papers by this authorManuel J. Lopez-Sanchez, Manuel J. Lopez-Sanchez School of Engineering, UADY, Merida, Yucatan, 97310 MexicoSearch for more papers by this author Andres A. Valdez-Fernandez, Corresponding Author Andres A. Valdez-Fernandez andres.valdez@ieee.org School of Sciences, UASLP, SLP 78290, MexicoSearch for more papers by this authorGerardo Escobar, Gerardo Escobar School of Engineering, UADY, Merida, Yucatan, 97310 MexicoSearch for more papers by this authorPanfilo R. Martinez-Rodriguez, Panfilo R. Martinez-Rodriguez School of Sciences, UASLP, SLP 78290, MexicoSearch for more papers by this authorJose M. Sosa, Jose M. Sosa orcid.org/0000-0002-5153-6185 Laboratory of Electricity and Power Electronics, ITESI, Irapuato-Silao, GTO, 36821 MexicoSearch for more papers by this authorDaniel U. Campos-Delgado, Daniel U. Campos-Delgado School of Sciences, UASLP, SLP 78290, MexicoSearch for more papers by this authorManuel J. Lopez-Sanchez, Manuel J. Lopez-Sanchez School of Engineering, UADY, Merida, Yucatan, 97310 MexicoSearch for more papers by this author First published: 01 June 2017 https://doi.org/10.1049/iet-pel.2016.0263Citations: 6AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study presents a model-based controller for a hybrid power filter (HPF) to reduce the current harmonic distortion in a three-phase system for a general operation case considering unbalanced and distorted source voltages and load currents. The HPF comprises an active power filter (APF) grid connected by means of an LC passive filter. This topology is aimed to reduce the current rating handled by the APF, representing a cost effective solution. The controller design is split into two stages, a first one to control the fundamental component of the system dynamics, and to regulate the DC voltage; and a second stage to compensate the harmonic distortion caused by a non-linear load. For this, the harmonic compensation scheme forces the injection of the required harmonic current to the point of common coupling. The proposed scheme keeps the very familiar proportional–resonant control structure. The compensation of the load reactive current is left to the passive filter. Experimental and simulation results are presented to exhibit the benefits of the proposed control solution. 1 Introduction Hybrid power filter (HPF) topologies [1-7] have emerged as a composition (series or parallel connection) of passive power filter (PPF) and active power filter (APF) to combine the advantages of both approaches. In fact, HPFs are less sensitive to parameter variations than simple PPFs [1]. Normally, in an HPF the APF is of lower rated power [2]. This is perhaps the most remarkable advantage, as it represents a significative reduction of cost and switching losses [3]. An interesting HPF topology was proposed in [4]. This HPF combines the use of a parallel PPF and a small rated series APF. In this topology, the series APF enhances the parallel PPF characteristics by providing harmonic isolation. That is, the series APF imposes high resistance to harmonic frequencies, while it offers zero resistance at the fundamental frequency [5]. In this way, the HPF achieves the so-called active impedance. Another interesting HPF is reported in [6], which comprises a small-rated APF in cascade connection with a PPF. The resultant HPF is connected in parallel to a NL load. In this case, the PPF suppresses the harmonic currents, whereas the APF improves the filtering characteristics of the PPF. As a result, the current rating of this APF is much smaller than in a conventional implementation. However, due to the series connection between APF and PPF, all reactive powers absorbed by the PPF circulates through the switching devices of the voltage source inverter (VSI) in the APF. To alleviate this issue, an improved HPF topology is proposed in [7]. In this topology, the APF is connected between the inductor and the capacitor of the PPF, which considerably reduces the reactive power handled by the APF, and thus, its rated current is reduced as well, even in the case of a predominantly inductive load. In this particular HPF topology, the voltage drop on the PPF capacitor reduces the inverter voltage rating, while the PPF inductor deviates the harmonic current at which the PPF has been tuned. This results in a lower stress in both voltage and current of the APF [8]. Different control strategies have been recently reported in [9-12] for this particular HPF topology. In [9] the authors present a controller comprising stages, namely, a current harmonics compensation stage and a regulation stage. The design, however, considers a balanced voltage source, hence, a good performance is guaranteed only under balanced operation conditions. In [10], a proportional–integral (PI)-based control scheme is proposed. The design considers a small-signal model of this HPF topology. This solution proposes the insertion, in the HPF topology, of an extra resistor for damping purposes, which leads to unnecessary power losses. In [11] a bidirectional control principle is proposed to solve the problem of harmonic over-current in the HPF. The authors achieve grid-side current harmonics reduction with immunity to passive filter parameters deviation. However, the algorithm requires voltage measurements of PPF passive elements, that is, six additional voltage sensors are needed for a three-phase HPF system. This paper addresses the modelling and control design process of the HPF topology proposed in [7]. Based on the mathematical model of the HPF, a controller is suggested to compensate harmonic distortion in the general case of distorted and unbalanced source voltage and load current. It is our belief that the success of a controller design strongly depends on the knowledge of the complete mathematical model that describes the overall system dynamics. In this case, the HPF involves the dynamics of both PPF and APF. The contents of this paper are based on the conference paper [12]. The present updated version includes a slight modification of the proposed controller, experimental evidence, and an updated revision of the state-of-the-art, among other modifications. The paper preserves some material presented in the conference paper to make it self-contained. The proposed model-based controller comprises proper damping terms to guarantee stability; an explicit mechanism composed by a bank of resonant regulators to achieve harmonics compensation [13]; and an adaptive algorithm to cope with parametric uncertainties. The damping terms in the proposed scheme avoid the unnecessary power losses of the control algorithm proposed in [10], while the adaptive algorithm avoids the unnecessary use of extra voltage sensors of the controller proposed in [11]. Positive and negative sequences are explicitly considered along the modelling and controller design to guarantee a proper operation under unbalanced conditions. Hence, the novelty of this paper, as compared to previous works [9-11], is the proposal of a model-based controller to compensate current harmonic distortion at the point of common coupling (PCC) in the general case of distorted and unbalanced load current, under a distorted and unbalanced source voltage operation. That is, our proposal considers simultaneous conditions of distortion in both voltage and current that have not been reported so far. Besides, the proposed controller does not require the insertion of any passive element, as in [10], which avoids unnecessary power losses. In addition, the proposed scheme does not require voltage measurements of PPF passive elements, as in [11], which avoids the use of six additional voltage sensors. The proposed controller involves two stages, namely, regulation stage and harmonics compensation stage. In the regulation stage, the DC-link capacitor voltage in the APF is maintained at a constant reference level [14]. For this, the APF only takes active power from the source, while letting the reactive current to be handled by the PPF, as in [6]. The second stage exclusively deals with the compensation of the harmonic distortion in the grid-side current. Simulation and experimental tests were carried out in a 2 kVA system operating under unbalanced and distorted conditions to assess the performance of the proposed model-based controller. 2 System description The power system under study is depicted in Fig. 1. In this system, a three-phase AC power supply feeds a NL unbalanced load, while an HPF is connected in parallel to such a load at the PCC. Fig. 1 shows an example of an array to form such a load, which comprises a three-phase NL load, and a single-phase NL load connected between two phases to create the unbalance. This array is considered here as an illustration only, and to experimentally test the proposed controller, as it is shown later. However, the proposed controller is, by no means, limited to this particular array of NL loads. Fig. 1Open in figure viewerPowerPoint Scheme of the power system under study including the HPF topology and an example of a non-linear unbalanced load The HPF topology considered in this work has been reported in [7], which comprises a PPF and an APF as shown in Fig. 1. The PPF consists of an inductor and a capacitor in series connection. The capacitor is directly connected to the PCC, while each phase of the APF is attached to the points of connection between inductors and capacitors of the PPF. These are indicated as points of APF connection in Fig. 1. The APF comprises a three-phase VSI, a three-phase inductive filter , and a DC-link capacitor C. 2.1 Mathematical model of the HPF The mathematical model of the HPF system in Fig. 1 is represented by (1) (2) (3) (4)where is the vector of the injected voltages referred to point '0', and is considered from now on as the control input; is the vector of grid-side currents and is the vector of hybrid injected currents; , , and are the vectors of APF currents, PPF inductor currents and load currents, respectively; is the vector of grid voltages at the PCC; is the vector of filter capacitor voltages; is the DC-link capacitor voltage at the APF; and are the capacitor and inductance of the PPF; is the APF inductance; C is the equivalent DC-link capacitance. For security reasons, a large resistor is usually connected in parallel to the DC-link capacitor of the active filter. This resistor discharges the capacitor whenever the system is turned off. In practice, this resistor, together with the switching and conduction losses are lumped in the circuit model as an unknown constant resistive element R. The fundamental frequency of the grid voltage source is represented by ; matrix B represents an interconnection matrix, which is given by In this model, each vector in three-phase coordinates has the form . Here and in what follows, vectors and matrices are denoted with bold face characters. Remark 1.In this work, the averaged model is considered instead of the exact switched model. For this, the switching sequence vector , , generated with a suitable modulation scheme, has been replaced in the model equations by the corresponding duty ratios vector , . This is based on the assumption that the switching frequency is high enough as compared to the HPF system bandwidth. Transformation from original three-phase coordinates (123-coordinates) to fixed-frame coordinates (-coordinates) is performed by means of the following Clarke's transformation: (5)where , and denotes the pseudo-inverse of T. Application of transformation (5) to model (1)–(4) yields the following system dynamics in terms of the (fixed-frame) -coordinates (6) (7) (8) (9) (10)where the property has been used. Notice that and , where , u, , , , , , and are all vectors of the form . Here and in what follows, subindices are omitted to simplify the notation. 2.2 Main assumptions The design of the proposed controller considers the following assumptions: A1.System parameters , , , and C are considered unknown positive constants. A2.The fundamental frequency is considered a known constant. A3.The APF current , the grid-side current , the load current , and the grid voltage are all considered unbalanced periodical signals [Notice that this controller is limited to compensate periodic signals only. Non-periodic, random and non-characteristic harmonics are out of the scope of this paper and are left for future research.]. They can be described as the sum of a fundamental component and higher order harmonics of the fundamental frequency . To deal with the harmonics distortion issue in the controller design, all vectors of signals involved in this application are expressed as the sum of two terms, namely, a fundamental component term, and a higher-order harmonic components term, i.e. (11)where x represents either of , u, , , , , , and . Moreover, to consider unbalanced operation conditions in the controller design, the above two terms are expressed, in their turn, as the sum of their symmetrical components, i.e. (12) (13)where and represent the positive and negative sequences of the fundamental component term, respectively; while and represent the positive and negative sequences of the harmonic components term, respectively. A4.(decoupling assumption): Dynamics (6)–(8) evolve much faster than capacitor voltage dynamics (9). This assumption facilitates the control design, as it can be split in the design of a fast (inner) tracking loop and a slow (outer) regulation loop. 2.3 Control objectives The control design is divided into two stages, namely, a first stage dealing with the fundamental component of the system dynamics only, and a second stage aimed to the design of a harmonics compensation loop, exclusively. The first stage, in its turn, comprises two loops, namely, a fast fundamental current tracking loop and a slow DC voltage regulation loop. The design of these two loops in the first stage appeals to the decoupling Assumption A4. Roughly speaking, the main objective of the first stage is to keep charged the DC-link capacitor of the APF by taking pure active power from the grid. The DC-link capacitor must maintain a given voltage level to allow a proper operation of the harmonic compensation stage. In what follows, all variables involved in the first stage, are identified with a subscript f, as they represent fundamental components of signals. Besides, all state variables involved in the second stage are identified with a subscript h, as they are related with harmonic components. The objectives to be fulfilled by a proper design of the loops forming the controller scheme can now be stated as follows: First stage (inner loop) – fundamental current tracking. Design a control loop to force the fundamental component of the APF current to track a reference . This objective can be expressed as (14) (15)where represents the positive sequence of the fundamental component of the source voltage , , and represents the active power reference. The scalar is calculated in the DC voltage regulation loop to be described later. Notice that is proportional to the positive sequence of . As a consequence, a balanced synchronised current is produced on the grid-side. Therefore, achieving this objective is equivalent to extract pure active power from the voltage source. First stage (outer loop) – DC voltage regulation. Design a control loop to regulate (in average) the capacitor voltage towards a constant reference . This objective can be expressed as (16) The accomplishment of this control objective guarantees that enough energy is stored in the capacitor for a proper operation. This is a necessary condition to achieve, in its turn, the harmonic compensation objective to be described later. The outcome of this loop is the variable used in the previous loop. As above mentioned, in this first stage, the DC-link capacitor of the APF is regulated to a certain voltage level by taking pure active power from the grid. iii. Second stage – harmonics compensation loop. The aim of this control loop is to reject the harmonic distortion present in the grid-side currents vector , which is caused by the non-linear load. This objective can be expressed as (17) 3 Controller design As above presented, the control design is divided into two main stages. The first stage comprises two loops, namely, an inner fundamental current tracking loop and an outer DC voltage regulation loop. The design in the first stage (first) assumes that the APF operates as a simple rectifier; (second) it deals only with the fundamental components of variables; and (third) it appeals to the decoupling Assumption A4. The second stage comprises a harmonics compensation loop to deal with the harmonic components of variables only. For control design purposes, it is convenient to introduce the following definition: (18)where represents the new control input variable, which replaces the control vector . Notice that, after this definition, the measured disturbance is cancelled out of (6). Hence, the simplified model can be rewritten as follows: (19) (20) (21) (22) 3.1 First stage (inner loop) – fundamental current tracking The aim of this loop is to find an expression for the fundamental component of the control input, referred as , to accomplish the current tracking objective (14). That is, to find a that forces the fundamental component of the grid current to track its reference defined in (15). As above mentioned, it is assumed that the APF operates as a rectifier in this first stage. Therefore, none of the load current components is compensated at this point. The design of this loop considers only subsystem (19)–(21). Moreover, the design deals only with the fundamental component of system dynamics (19)–(21). These dynamics, referred here as the fundamental dynamics, are given by (23) (24) (25)The system dynamics (23)–(25) are expressed in terms of the increments (or tracking errors), which are defined as , and . This yields the following dynamical system in terms of the increments: (26) (27) (28)where is a periodic disturbance according to A3. The variables and are the references of and , respectively; and both satisfy the following dynamics: (29)Notice that variables and are bounded as far as is bounded. Based on the structure of subsystem (26), the following controller is proposed: (30)where is a design parameter aimed to add the required damping, and is a term introduced to reject the periodical disturbance . Following a similar approach as in [13], an expression for the harmonic compensation term is obtained, which consists of the following harmonic oscillator: (31)where is a design parameter representing the gain of the fundamental oscillator (31). In fact, this control term is in agreement with the internal model principle [15]. Therefore, it allows perfect tracking of the fundamental component towards its purely sinusoidal reference defined in (15). 3.2 First stage (outer loop) – DC voltage regulation The DC voltage regulation loop is aimed to drive (in average) the DC-link capacitor voltage towards a desired constant reference . Out of this loop, an expression for the active power reference is obtained. Recall that is used to reconstruct the current reference according to (15). The design of the regulation loop involves the capacitor dynamics (22), and the fundamental dynamics (23)–(25) above described. Moreover, based on the decoupling Assumption A4, it is assumed that the fundamental current tracking objective in (14) has been achieved after a relatively short time. That is, the dynamics (23)–(25) have reached the equilibrium point located at . Moreover, , and from (30), . After evaluation of (22), at this equilibrium point, and neglecting all harmonic components, the following subsystem is obtained: (32)where with , and . Moreover, the property has been used, and it is assumed that . Notice that system (32) is a first-order linear time-invariant (LTI) system, where is the output and represents the control input. In addition, this system is perturbed by an unknown constant signal , while the control input is affected by the negative constant . Under these circumstances, the following PI controller represents an effective solution: (33)where and are the proportional and integral gains, respectively. This controller guarantees that (in average) as , while remains bounded. 3.3 Second stage – harmonics compensation loop To facilitate the design of this loop, it is proposed here to express subsystem (19)–(21) in terms of the grid-side currents . Moreover, only the harmonic part of all terms in this subsystem must be considered. The resulting system is referred, here and in what follows, as the harmonics dynamics. These are described by (34) (35)where and . The subscript h indicates that these variables represent the harmonic contents only. Notice that (34) and (35) is a controllable second-order system perturbed by both the harmonic distortion and its time derivative . However, this system does not fulfil the matching condition [15]. Notice that these disturbances appear in both rows of system (34) and (35), while the control input appears only in the first row. Roughly speaking, the controller cannot simultaneously bring to zero both states and . Therefore, the priority in this control loop consists in regulating towards zero, while a small amount of harmonic distortion is allowed in . For this, and to facilitate the design of the control signal , it is convenient to introduce the following transformation: (36)where is a periodic bounded signal defined by . Direct application of transformation (36) to harmonics dynamics (34) and (35) yields the following system: (37) (38)where is a periodical disturbance according to A3. In fact, has the same harmonic content as . Notice that, after transformation (36), the disturbance term, now represented by , has been swapped to the first row. The matching condition is now fulfilled, as the control input appears in the first row as well. Based on the structure of system (37) and (38), the following controller is proposed: (39)where is a design parameter. Notice that the above controller includes a damping term to reinforce the stability and a harmonic compensation term represented by to be designed later. Following a similar approach as in [13], an expression for the harmonic compensation term can be derived as follows. This term consists of the following bank of harmonic oscillators (or resonant filters): (40)where is a design parameter representing the gain of the kth harmonic oscillator (). Notice that, in this harmonic compensation mechanism, the kth oscillator is tuned at the kth harmonic under concern. Out of this, perfect tracking is achieved at that precise kth harmonic, which follows from the internal model principle [15]. 3.4 Estimation of fundamental, harmonic and positive-sequence components The implementation of the proposed controller, represented by expressions (15), (30), (31), (39) and (40), requires the knowledge of both the fundamental component and the harmonic components , as well as the positive sequence of the fundamental component . However, none of them is available from measurements. Therefore, it is proposed here to reconstruct these missing components by means of a fundamental component estimator similar to the one proposed in [16]. The aim of this estimator is to reconstruct, out of a measured signal vector x, its fundamental component at frequency . The measured signal x represents, in the present work, either , or . The estimator proposed in [16] is given by (41) (42)where represents the estimate of the fundamental component of x; is an auxiliary vector with the same characteristics (amplitude and angular frequency) as , except for a phase difference of 90° ahead; and is a positive gain that defines the convergence speed. Based on (41) and (42), the fundamental components and can be reconstructed according to the following more familiar expressions: (43)whereas the harmonic components can be recovered out of the estimated and the measured as follows: (44)According to [16], the positive sequence component can be reconstructed as (45)Expressions (41) and (42) can be used to express the estimator (45) in the form of a transfer function. In particular, the estimator to reconstruct the positive sequence component out of is given by (46) Remark 2.Notice that, the above estimators expressions assume that the frequency is a known constant. However, in case of frequency variations an adaptive version such as the one presented in [16] can be used instead. 3.5 Summary of the proposed controller Summarising, the final expression for the proposed controller comprises a fundamental current tracking loop composed by (30) and (31), a DC voltage regulation loop given by (33) and a harmonics compensation loop composed by (39) and (40). As above mentioned, the implementation of these control expressions requires the knowledge of , and , which are reconstructed by means of (43), (44) and (46). Moreover, the overall control signal is the result of the sum of its fundamental component and its harmonics component components obtained in (30) and (39), respectively, i.e. . Out of this, the original control input (the injected voltage) can be obtained as . Finally, according to (10), the vector of duty ratios can be obtained as . A block diagram of the overall proposed controller is shown in Fig. 2. Fig. 2Open in figure viewerPowerPoint Block diagram of the overall proposed controller 4 Experimental and simulation results The proposed controller has been experimentally tested in an HPF laboratory prototype as shown in Fig. 1. The source voltage is obtained from a Chroma 61705 programmable AC power supply, where a fundamental frequency of has been selected. This power supply emulates an unbalanced operation in the voltage vector , whose fundamental components are described by (47)which represents an unbalance factor of around 6.5%. Moreover, harmonic distortion in the grid voltage is introduced by means of an auxiliary circuitry as the Chroma 61705 power supply is unable to provide this function. This auxiliary circuitry, not shown here to avoid confusion consists of a relatively small resistor connected between the power supply and the PCC, together with a highly non-linear three-phase load connected to the PCC. The voltage drop in the small resistor contains harmonic components, which are added to the unbalanced voltage coming from the Chroma 61705 power supply. As a result, an unbalanced and distorted voltage signal is produced at the PCC, which is represented by in Fig. 1. Table 1 describes the harmonic components of the resulting unbalanced and distorted grid voltage at PCC , which were obtained with a three-phase power quality analyser Fluke 434. These harmonic components are given as a percentage of the corresponding fundamental component (of three phases) and until the harmonic 15th. The corresponding THDs are described in Table 2. Table 1. Harmonic factors of grid voltage and load current (as % of the corresponding fundamental component) Harmonic 3rd [0.6, 2.4, 2.1] [11.8, 11.1, 4.8] 5th [1.5, 1.3, 1.4] [16.3, 12.7, 18.7] 7th [1.8, 0.9, 1.1] [10.4, 8.3, 4.0] 9th [0.5, 0.5, 0.6] [3.3, 3.5, 2.5] 11th [0.2, 0.4, 0.3] [3.1, 3.9, 5.0] 13th [0.4, 0.2, 0.3] [3.4, 2.6, 1.2] 15th [0.2, 0.2, 0.2] [1.9, 1.6, 2.0] Table 2. Total harmonic distortion of grid voltage and load current Signal Phase 1 % Phase 2 % Phase 3 % Three-phase THD % vS123 2.5 3.0 2.9 2.7909 iL123 23.5 19.9 20.7 21.5519 As shown in Fig. 1, the non-linear load used for experimental tests consists of a three-phase non-controlled diode rectifier feeding a resisti
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