Asymptotic rejection of sinusoidal disturbances based voltage balance control in back-to-back power converters

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust. Nonlinear Control 2014; 24:2974–2994 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.3036 Asymptotic rejection of sinusoidal disturbances based voltage balance control in back-to-back power converters Francisco Umbr´ıa∗,†, Francisco Gordillo, Francisco Salas, Javier Aracil and Fabio G´omez-Estern Department of Systems Engineering and Automation, University of Seville, 41092 Seville, Spain SUMMARY This paper addresses the imbalance problem of the dc-link capacitor voltages in the three-level diodeclamped back-to-back power converter. In order to cope with it, a mathematical analysis of the capacitor voltage difference dynamics, based on a continuous model of the converter, is first carried out. It leads to an approximated model which contains explicitly several sinusoidal functions of time. In view of this result, the voltage imbalance phenomenon can be addressed as an output regulation problem, considering the sinusoidal functions as exogenous disturbances. Thus, a novel approach to deal with the mentioned problem in the back- to-back converter is presented. Then, the particular features of the disturbances are used to design several controllers. They all follow an asymptotic disturbance rejection approach. In this way, the estimations of the disturbances are used to apply a control law that cancels them while regulating the capacitor voltage balance as well. Finally, the performance of the proposed control laws is evaluated, presenting the simulation results obtained when the different controllers are implemented. Copyright c 2014 John Wiley & Sons, Ltd. Received ... KEY WORDS: back-to-back power converter; diode-clamped converter (DCC); voltage balancing; disturbance rejection; internal model principle; state observer; adaptive algorithm 1. INTRODUCTION In the field of power conversion, back-to-back converter topology plays a fundamental role, that has found widespread applications such as train traction systems, automotive applications, energy generation and conversion,among others, in various industrial sectors [1]. In particular, regenerative conveyorsand variable speed drives [2–4], for conversion from a three-phase source to a three-phase load with an arbitrary frequency and amplitude, and, especially, wind power generators [5,6], due to the recently increasing influence of the renewable energies [7,8], have been under research and development over the last years. In respect of the back-to-back configuration, it consists of a connection of two power converters. The left-hand side converter generally works as a rectifier for the grid interface, and the right-hand side one as an inverter for the load, as shown in Fig. 1. Both converters are able to switch their operation functions and to connect asynchronous systems, allowing an indirect ac-ac conversion [9,10], also referred to as ac-dc-ac conversion, including energy storage elements in the dc-link. These elements provide a decoupling of both converter stages for control purposes [11], as well as energy buffering when the instantaneouspowersin the rectifier andinverterare unbalanced.Besides, the back-to-back converter presents other remarkable features such as good dynamic response, the ∗Correspondence to: Francisco Umbr´ıa, Departamento de Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Espa˜na. †E-mail: fumbri[email protected] Copyright c 2014 John Wiley & Sons, Ltd. Prepared using rncauth.cls [Version: 2010/03/27 v2.00] 2F. UMBR´ IA ET AL. Figure 1. Three-phase three-level diode-clamped back-to-back power converter used as a grid interface for applications such as wind power generators or variable speed drives. same hardware is required for both converters, high power factor and bidirectional power flow. In other words, they absorb energy from the grid or transfer it to the grid, meaning that the activepower flow direction can be reversed at any instant due to the four-quadrant capability of the setup. Among the different back-to-back configurations, the setup based on the three-level diodeclamped converter (DCC), also known as neutral-point-clamped converter [12], is one of the most attractive converter topologies [13,14]. It adds the typical advantages of the multilevel topology, compared with the conventional two-level converter topology, to those provided by the back-to- back connection. In this way, it leads to an increase of the output voltage magnitude, lower switching frequency, reduction of the total harmonic distortion, better output voltage spectrum, robustness and higher efficiency [15]. Nevertheless, there exists the issue of the voltage imbalance of the dc-link capacitors, which is a relevant and well-known drawback of the DCC. Concerning the difficulty of balancing of the dc-link capacitor voltages, multilevel DCCs with a large number of levels present some theoretical and practical limits related to the modulation index and the power factor of the load [16], so a proper voltage balancing is not possible in all the operating conditions [17]. Focusing on the back-to-back converter, although the particular connection of the rectifier and the inverter might yield a natural compensation of the voltage unbalance, this configuration is only able to achieve an acceptable voltage balancing including a suitable controller or modulation strategy [18–20]. Thus, the most popular method to mitigate the voltage imbalance is the use of modulation strategies based on the exploitation of the redundant switching states of the converter [21,22]. That is, those distinct positions of the converter switches that generate the same output voltage but they cause different effects on the capacitor voltages. This paper addresses the voltage imbalance phenomenon in the three-level diode-clamped back- to-back converter. In order to cope with it, an analysis of the dc-link capacitor voltage difference dynamics is carried out in this paper. The analysis is based on a continuous model of the converter and, considering a singular perturbation approach [23], yields a particular model for the voltage difference dynamics that contains explicitly several sinusoidal functions of time. This important fact is the key of the work presented here since it leads to relate the voltage imbalance problem to a problem of periodic disturbance cancellation and regulation of the output of the system. In this way, this new problem statement, which has been widely studied in the literature and applied to many other industrial system, is the first contribution of the current paper. The mathematical analysis derived enables the use of classical control theory to design the controller. Thereby, it provides an alternative approach to deal with the problem, compared with the strategies usually considered and previously mentioned [18–22]. One of the most important problems in control theory is that of controlling a fixed plan in order to retain closed-loop stability and output regulation in the presence of disturbances produced by Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 ASYMPTOTIC REJECTION OF SINUSOIDAL DISTURBANCES IN BACK-TO-BACK CONVERTERS 3 Figure 2. Three-phase three-level diode-clamped back-to-back power converter. some external generator. The problem has been first treated for linear systems [24–26], showing that any regulator which solves the matter at hand incorporates in the feedback a reduplicated model of the dynamic structure of the disturbance which must be tracked and/or rejected. This property is commonly known as the internal model principle [25,26]. Incorporating the earlier results established in the linear setting, the theory was then extended to the nonlinear case. The most classical formulation of the problem for this case was presented by Isidori and Byrnes in [27]. Still today, significant effort is being made in this research field [28–30]. Regarding the topic of identifying and asymptotically rejecting periodic disturbances, huge progresses have been reported [31–33]. Both the case when the disturbance frequency is known as well as the case when it is an unknown parameter have been studied. Even the varying frequency disturbance scenario, typical in applications such as rolling machines and magnetic tape drives, has been also discussed [34,35]. Thus, methods related to adaptive control [36,37], repetitive control [38,39], backstepping [40], among others, have been used providing solid results. Based on the analysis of the converter dynamics presented in the first part of this paper, some control schemes are designed to achieve the voltage balancing of the dc-link capacitors of the back- to-back converter. In this way, the design of the controllers applying well-known techniques to reject sinusoidal disturbances is the second contribution of this work. The authors have worked previously with control techniques based on disturbance rejection in other converter topologies [41,42], where the circuit studied was the neutral-point-clamped rectifier. In particular, in [41] an analysis of the voltage difference dynamics was also formulated but leading to an expression which contains only one sinusoidalfunction. In the current paper, the ideas presented in [41,42] are adopted and extended to the multiple frequency approach of the back-to-back topology. The remainder of this paper is organized as follows. In Section2, the convertermodelis described, presenting the control objectives. Afterwards, the dynamics of the capacitor voltage difference are analyzed in detail in Section 3. Considering the previous analysis, Section 4states the problem of disturbance rejection in the converter topology of this paper and proposes several control methods to guarantee the correct behavior of the capacitor voltages. Then, the admissible implementationsof the proposed control laws are discussed in Section 5. The simulation results obtained are presented in Section 6. Finally, some conclusions are drawn in Section 7. 2. MODEL OF THE BACK-TO-BACK CONVERTER AND CONTROL OBJECTIVES Fig. 2illustrates a schematic diagram of the three-phase three-level diode-clamped back-to-back power converter, which is the setup considered in this paper. The dc-link is composed of capacitors C1and C2, both of identical capacitance C, being their respective voltages vc1and vc2. As mentioned before, both left-hand side and right-hand side converters are able to switch their operation modes. However, in the following, the left-hand side converter is named rectifier, and the right-hand side one is named inverter, in order to distinguish one from the other, as well as to simplify the notation. Both converters are connected to a corresponding ac system through the inductors Lrand Li. The whole system represents a general back-to-back converter. In other words, it is not only focused on a particular application of this configuration, but it includes the all purpose Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 4F. UMBR´ IA ET AL. Table I. Variables of the system in αβγ coordinates. Variable Description iαβ,r= [ iα,riβ,r]TRectifier phase current vector iαβ,i= [ iα,iiβ,i]TInverter phase current vector vαβ,r= [ vα,rvβ,r]TRectifier phase voltage vector vαβ,i= [ vα,ivβ,i]TInverter phase voltage vector δαβγ,r= [ δα,rδβ,rδγ,r]TRectifier control vector δαβγ,i= [ δα,iδβ,iδγ,i]TInverter control vector interconnection of two ac power systems with different frequencies, where the instantaneous power flows from one ac system to another. The variables of the system in abc coordinates are, on one hand, the phase currents and phase voltages of the rectifier, denoted by ia,r,ib,r,ic,rand va,r,vb,r,vc,r, respectively. On the other hand, considering the inverter, the phase currents ia,i,ib,i,ic,i, and the phase voltages, given in this case by va,i,vb,iand vc,i. In addition, the gating signals that regulate the positions of the switches of both converters are defined by δ∗ a,r,δ∗ b,rand δ∗ c,rin the rectifier, and by δ∗ a,i,δ∗ b,iand δ∗ c,iin the inverter. Among the large variety of converter models such as, e.g., time-varying and time-invariant averaged models, detailed switching models and small signal models [43], the continuous model of the back-to-back converter presented in [44] is adopted in this paper. The model is based on a direct average of the characteristics and waveforms associated with each of the components of the converters, providing a model valid all over the range of values of the control inputs. Thereby, the model obtained describes, in αβγ orthogonal coordinates, using the Clarke-Concordia Transform, the dynamics of the phase currents and of the total dc-link voltage [44], including the dc-link capacitor voltage difference dynamics, as well. Particularly, these last dynamics are given by Cdvd dt =δT αβ,riαβ,r √3δγ,r+δ2 α,r−δ2 β,r 2√6−δα,rδβ,r √6iαβ,r (1) −δT αβ,iiαβ,i √3δγ,i−δ2 α,i−δ2 β,i 2√6−δα,iδβ,i √6iαβ,i, where vdis the difference between the upper and lower dc-link capacitor voltages, that is, vd=vc1−vc2 2. (2) Notice that the system variables have been transformed into αβγ coordinates, yielding the vectors summarized in Table I. It is assumed that the gating signals can be replaced by their averaged values in a switching period, defining in this way the control vectors δαβγ,rand δαβγ,i. They are the control inputs of the model and are carried out as an indirect control through a modulator. The components αβ of these control vectors are also represented together in (1) by δαβ,r=δα,rδβ,iT(3) δαβ,i=δα,iδβ,iT. (4) Moreover, (1) includes the current vectors iαβ,rand iαβ,i, meaning that the capacitor voltage difference dynamics are coupled with the current ones. 2.1. Control objectives As pointed out in the introduction, this paper addresses the imbalance problem of the voltages of the dc-link capacitors. The voltage difference vdshould be controlled to avoid the presence of unbalance Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 ASYMPTOTIC REJECTION OF SINUSOIDAL DISTURBANCES IN BACK-TO-BACK CONVERTERS 5 in the system, so it should be kept close to zero vd≃0, (5) providing closed-loop stability as well. It is worth stressing that it is not aimed for vd(t)→0 as t→ ∞ for all possible initial conditions of the closed-loop state variables. Consequently, the requirements of the classical output regulation problem [24,27] are smoothed. Besides, at this point of the paper, the disturbances in the system have not been considered yet. The analysis and control methods presented in the next sections have been worked out to keep balanced the dc-link capacitor voltages. Nevertheless, other requirements of the system such as, e.g., desired instantaneous active power, total dc-link voltage and power factor at both ac sides should be also fulfilled. Hence, it is necessary to design some other controllers to satisfy these other aims. Thus, the other variables to control are the rectifier instantaneous active and reactive powers denoted by prand qr, and in the inverter by piand qi, being their references described, respectively, by p∗ r, q∗ r,p∗ iand q∗ i, and the total dc-link voltage defined as vdc =vc1+vc2. (6) In the following section it is assumed that there exist certain controllers to deal with the regulation of the instantaneous powers as well as with the total dc-link voltage. Specifically, the control vectors (3) and (4) are used by the power controller to this end, remaining control inputs δγ,rand δγ,ias the degrees of freedom for coping with the voltage imbalance problem. 3. ANALYSIS OF THE DC-LINK CAPACITOR VOLTAGE DIFFERENCE DYNAMICS Since current and dc-link capacitor voltage difference dynamics in the back-to-back converter are tightly coupled, the design of a controller to balance the capacitor voltages is not an easy task. This section is devoted to study in detail the voltage difference equation in order to provide a better understanding of its behavior, as well as some remarkable results that are the basis of the proposed control methods in this paper. The analysis, worked out from a mathematical point of view, is carried out considering some essential assumptions related to a time-scale approach, that are stated next. 3.1. Time-scale assumptions Assumption 1 The instantaneouspower dynamics are faster than the dc-link capacitor voltage difference dynamics. Under the assumption of different velocities of the power and voltage dynamics [45], and in order to study (1), it leads to suppose that the instantaneous powers have been regulated around their references. Therefore, applying δαβ,rand δαβ,iin an appropriate control scheme, the instantaneous powers at both rectifier and inverter sides are approximated by pr≃p∗ r(7) qr≃q∗ r(8) pi≃p∗ i(9) qi≃q∗ i. (10) Assumption 2 The total dc-link voltage and the rectifier instantaneous active power reference dynamics are slower than the dc-link capacitor voltage difference dynamics. Concerning this second assumption, which is related to the design of the parameters of the voltage balancing methods, it leads to assume that the total dc-link voltage as well as the rectifier instantaneous active power reference are treated as constants in the analysis of (1). Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 6F. UMBR´ IA ET AL. Regardingthe rectifier power reference p∗ r, it is assigned to control the total dc-link voltage, whose dynamics considering Assumption 1are given by Cd dt v2 dc 2≃p∗ r−p∗ i. (11) It is worth mentioning that, in the back-to-back topology, the rectifier and inverter are usually controlled individually, so one of them should be devoted to regulate the total dc-link voltage regardless of the operating state of the another. Hence, whereas the power references q∗ r,p∗ iand q∗ iare fixed parameters, the dynamics of the rectifier power reference p∗ rare defined to drive the total dc-link voltage towards its reference v∗ dc. 3.2. Final expression of the voltage difference dynamics Considering the time-scale assumptions and following the procedure to analyze (1) given in Appendix A, it results in the final expression of the dynamics of the voltage difference between the dc-link capacitors defined by Cdvd dt =krδγ,r−kiδγ,i+φr+φi. (12) Due to Assumption 2,krand kiare the constants described by kr=2p∗ r √3vdc (13) ki=2p∗ i √3vdc . (14) and φrand φiare the sinusoidal functions given by φr=µ1,rsin (3ωrt+ 3θr+ arctan (µ2,r)) (15) φi=µ1,isin (3ωit+ 3θi+ arctan(µ2,i)) . (16) Parameters µ1,r,µ2,r,µ1,iand µ2,iare constant values defined in Appendix A. Notice that the frequencies of the sinusoidal functions correspond with three times the frequencies of the rectifier and inverter side phase voltages (77)-(80), denoted by ωrand ωi, respectively. Summarizing, this section has simplified the initial highly complicated expression of the dc-link capacitor voltage difference dynamics (1), leading to a simpler equation (12), under some particular conditions.Besides, (12) is a linear function of control inputs δγ,rand δγ,i, containing also explicitly two additional terms which take the form of sinusoidal functions. The special features of φrand φi, which are regarded in the following as external disturbances, are used to propose several methods to design the voltage balance controllers. 4. PROBLEM STATEMENT AND PROPOSED VOLTAGE BALANCING METHODS Due to the appearance of the disturbances in the capacitor voltage difference dynamics, it leads to relate the voltage imbalance problem to a problem of periodic disturbance rejection. In this way, the purpose of the controllers to design is now twofold. The first goal, as pointed out in Section 2.1, is to regulate the dc-link capacitor voltage difference vdwhich should remain close to its reference given by v∗ d, providing closed-loop stability. This reference is set to zero to keep balanced the capacitor voltages. The second goal is the asymptotic cancellation of the sinusoidal disturbances, that is, the asymptotic cancellation of the disturbances φrand φidefined in the previous section. In this way, to achieve a proper output regulation, suitable models of both disturbances should be included in the controller, as stated in [25,26]. Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 ASYMPTOTIC REJECTION OF SINUSOIDAL DISTURBANCES IN BACK-TO-BACK CONVERTERS 7 Figure 3. Schematic block diagram of the system (dc-link capacitor voltage difference dynamics) when the controller as well as the state observer are both implemented. The balancing methods herepresented are based on an asymptoticdisturbance rejection approach. The proposed controllers estimate asymptotically the sinusoidal disturbances, having in common the idea of applying a control law to cancel them while regulating the capacitor voltage balance as well. The methods use some control concepts widely discussed in [25,26,47–49], and also proposed for the neutral-point-clamped rectifier in [41,42], extending them to the multiple frequency approach of the back-to-back topology of the current paper. In order to apply the control action, whereas vectors δαβ,rand δαβ,iare used to control the instantaneous powers, the remaining control inputs δγ,rand δγ,iare still free parameters, so they both can be used for the balancing of the capacitor voltages. However, in this section, a control signal denoted by uγis defined containing both δγ,rand δγ,ias follows uγ=krδγ,r−kiδγ,i. (17) As a result, the final equation of the voltage difference dynamics given by (12) is expressed as Cdvd dt =uγ+φr+φi. (18) The control term uγis only introduced in the controller design stage to simplify the notation. The implementation of the definitive control law is discussed in the following section. Notice that (18) takes the form of the system considered in [27] for the linear setting, defining the disturbance win that reference as the sum of φrand φi. Fig. 3depicts a schematic block diagram of (18) when the controller is included in the system. The block denoted by P(s)is the transfer function defined in the Laplace domain by P(s) = 1 Cs. (19) 4.1. State space estimation provided by a Luenberger observer The first proposed control method includes the asymptotic estimations of (15) and (16) provided by a state space observer, concretely by a Luenberger observer. Taking into account that the frequencies of the sinusoidal disturbances φrand φiare 3ωrand 3ωi, respectively, (18) is expressed as the augmented system Sdescribed by S:                      ˙xd=1 Cxφ1 r+1 Cxφ1 i+1 Cuγ ˙xφ1 r=xφ2 r ˙xφ2 r=−(3ωr)2xφ1 r ˙xφ1 i=xφ2 i ˙xφ2 i=−(3ωi)2xφ1 i. (20) Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 8F. UMBR´ IA ET AL. The capacitor voltage difference is defined by the variable xd. The sinusoidal disturbances φrand φiby xφ1 rand xφ1 i, and their time derivatives by xφ2 rand xφ2 i, respectively. The control input of the system is uγ. Considering that the output of the system is the variable xd, the state space representation of system Sis given by S:(˙x=Ax +Buγ y=Cx,(21) where x∈R5is the state vector of the system, and it is defined by x=xdxφ1 rxφ2 rxφ1 ixφ2 iT, (22) y∈Ris the output, and uγ∈Ris the control input. The state matrix A, and the input and output matrices, Band C, respectively, are given by A=             01 C01 C0 0 0 1 0 0 0−(3ωr)2000 0 0 0 0 1 000−(3ωi)20             ,B=             1 C 0 0 0 0             ,C=           1 0 0 0 0           T . (23) Since the observability matrix of system Sis full rank, the system is observable, and it is possible to reconstruct the system state from its output measurements via a state space observer. Thereby, to provide an estimate of x, the Luenberger observer So, assuming that the dc-link capacitor voltage difference is measurable, is designed as follows So:(˙ ˆx=Aˆx+Buγ+L(y−ˆy) ˆy=Cˆx.(24) The estimated state vector ˙ ˆx∈R5is given by ˆx=ˆxdˆxφ1 rˆxφ2 rˆxφ1 iˆxφ2 iT, (25) the estimated output by ˆy∈R, and the observer gain matrix is described by L=p1p2p3p4p5T. (26) The rest of the variables and parameters of (24) have been already defined in (21). The values of the elements of Lare chosen in such a way that the poles of the observer Soconverge faster than the poles of system S. Finally, once the observer is implemented, the controller applies the estimates of φrand φithat the observer provides, represented by the state variables ˆxφ1 rand ˆxφ1 i, respectively. Thus, the control law is defined by uγ=k(v∗ d−vd)−ˆxφ1 r−ˆxφ1 i, (27) where kis a design constant parameter, and v∗ dis the capacitor voltage difference reference, which is set to zero. Notice that the estimates ˆxφ1 rand ˆxφ1 iare applied to asymptotically cancel the terms φrand φiin (18), and they are implemented in parallel with a proportional controller to ensure that vdtends exponentially towards its reference. Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 ASYMPTOTIC REJECTION OF SINUSOIDAL DISTURBANCES IN BACK-TO-BACK CONVERTERS 9 4.2. Internal model principle based disturbance estimation The second approach is based on the internal model principle of control theory [25,26], which states that the regulation of a system is achieved when the control law encapsulates, either implicity or explicitly, some representation of the process to be controlled. In this way, the feedback path should incorporate suitably reduplicated models of the dynamic structure of the disturbances. Focusing on the capacitor voltage difference dynamics, some transfer functions are used to approximate the behavior of the sinusoidal disturbances. In this manner, the models of φrand φi consist of ˆ φrand ˆ φi, respectively, which are defined in the Laplace domain as ˆ φr(s) = −gr s s2+ (3ωr)2(v∗ d(s)−vd(s)) (28) ˆ φi(s) = −gi s s2+ (3ωi)2(v∗ d(s)−vd(s)) , (29) where grand giare positive constants, and v∗ dis the capacitor voltage difference reference. Notice that the models of both disturbances have a pair of imaginary poles at the same frequency of the disturbances φrand φi. The disturbance approximations are applied to achieve the asymptotical cancellation of the disturbances and are constructed in parallel with a proportional controller in a feedback control system. Therefore, it leads to the following control law uγ=k(v∗ d−vd)−ˆ φr−ˆ φi, (30) where kis the gain of the proportional controller. 4.3. Disturbance estimation applying an adaptive algorithm Under the third control approach, an adaptive algorithm is introduced to asymptotically estimate the disturbances, in order to apply an adaptive feedforward cancellation at the input of the system of both φrand φi, considering that their expressions are known. The control method starts by proposing equivalent expressions for φrand φiapplying, among others, the trigonometric identity pa2+b2sin α+ arctanb a =asin (α) + bcos (α). (31) Thereby, (15) and (16) are expressed as φr=η1,rsin (3ωrt) + η2,rcos (3ωrt)(32) φi=η1,isin (3ωit) + η2,icos (3ωit). (33) The constants η1,r,η1,i,η2,rand η2,iare defined as follows η1,r= cos (3θr)µ1,r q1 + µ2 2,r−sin (3θr)µ1,rµ2,r q1 + µ2 2,r (34) η2,r= cos (3θr)µ1,rµ2,r q1 + µ2 2,r + sin (3θr)µ1,r q1 + µ2 2,r (35) η1,i= cos (3θi)µ1,i q1 + µ2 2,i−sin (3θi)µ1,iµ2,i q1 + µ2 2,i (36) η2,i= cos (3θi)µ1,iµ2,i q1 + µ2 2,i + sin (3θi)µ1,i q1 + µ2 2,i . (37) Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 16 F. UMBR´ IA ET AL. 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 650 700 750 800 Time (s) vdc ,v∗ dc (V) Figure 5. Evolution of the total dc-link voltage (solid) and its reference (dashed). 0.8 1 1.2 1.4 1.6 1.8 2 9600 9700 9800 9900 10000 10100 Time (s) pr,p∗ r(W) Figure 6. Behavior of the rectifier instantaneous active power pr(solid) and its reference p∗ r(dashed). that their frequencies are unknown shows the worst voltage balancing, since it contains a higher number of parameters to estimate (the frequencies are also estimated parameters applying this control method). However, the voltage difference still remains close to zero. Nevertheless, if the frequencies ωrand ωrchange their nominal values (but not much because the terms P3ωr R,P3ωr I, P3ωi Rand P3ωi Iof (60) and (61) have been estimated and set considering the nominal values of the disturbance frequencies), this controller provides a better estimation of the disturbances than the one provided by the other voltage balance controllers, which have been designed for specific values of ωrand ωi(Table II). Focusing on the steady state behavior, the voltage difference presents a very small ripple, which is lower than 0.5V and is not relevant. Hence, the variable vdis properly regulated to zero with all the proposed control methods, so a good voltage balancing is ensured. Concerning the behavior of the total dc-link voltage in the simulation carried out implementing the controller of Section 4.1, note that, when the voltage reference step occurs, the variable vdc tends also correctly towards its new reference (Fig. 5). As mentioned before, this variable is regulated applying the instantaneous powers of the rectifier. Concretely, via the instantaneous active power reference p∗ r, whose evolution is illustrated in Fig. 6. The power variable pris also depicted in the figure to illustrate the usefulness of the instantaneous power controller. Notice that its behavior is practically identical to the one of p∗ r, and it is difficult to distinguish prfrom p∗ r. Moving on to the control method that estimates unknown frequency sinusoidal disturbances (Section 4.4), Fig. 7illustrates the evolution of the frequency estimations of the rectifier and inverter phase voltages. They are denoted by, respectively, the variables ρ2,rand ρ2,i. The initial states of both variables have been set to ρ2,r(0) = 941 rad/s and ρ2,i(0) = 1130 rad/s. As stated in Section 3.2, the frequencies of the sinusoidal disturbances φrand φiare the triple frequencies of those of the rectifier and inverter phase voltages, respectively. Since the values of the phase voltage frequencies in the simulations are ωr= 2π·50 rad/s and ωi= 2π·60 rad/s, notice that both ρ2,rand ρ2,itend properly to their specific references (ρ∗ 2,r= 3ωr≃942.48 rad/s and ρ∗ 2,i= 3ωi≃1130.97 rad/s), even when the total voltage reference step occurs. Finally, Fig. 8shows the amplitude frequency spectrums of the dc-link capacitor voltage difference. On one hand, when the observer-based disturbance estimation controller is implemented to regulate the voltage balance. On the other hand, when a traditional PI controller is introduced in the system with the purpose of achieving the same control aim. Notice that both spectrums present the highest frequency content at the frequencies of the disturbances (3fr= 150 Hz, 3fi= 180 Hz). Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 ASYMPTOTIC REJECTION OF SINUSOIDAL DISTURBANCES IN BACK-TO-BACK CONVERTERS 17 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 938 940 942 944 946 Time (s) ρ2,r,ρ∗ 2,r(rad/s) (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1110 1120 1130 1140 Time (s) ρ2,i,ρ∗ 2,i(rad/s) (b) Figure 7. Frequency estimations provided by the controller with estimation of unknown frequency disturbances applying an adaptive algorithm. From top to bottom: (a) Rectifier frequency estimation variable ρ2,r(solid) and its reference value ρ∗ 2,r(dashed). (b) Inverter frequency estimation variable ρ2,i(solid) and its reference value ρ∗ 2,i(dashed). Nevertheless, the magnitude obtained applying the first control method is much smaller than that provided considering the PI controller. Therefore, the proposed control methods in this work present the important benefit of a smaller frequency content of the dc-link capacitor voltage difference, in contrast with the use of other voltage balancing strategies (the spectrum features of vdconsidering the remaining control methods presented in Section 4are analogous to the one obtained applying the observer). 7. CONCLUSIONS In this work, several control methods designed to achieve the balancing of the dc-link capacitor voltages in three-phase three-level diode-clamped back-to-back power converters have been presented. Under some time-scale assumptions, the dynamics of the capacitor voltage difference are analyzed, resulting in a expression that contains several complex terms, which are approximated by sinusoidal functions of time. The significance of this fact is highly relevant, since it is the main idea behind the design of the proposed controllers, which are based on the estimations of these sinusoidal functions, considering them as external disturbances of the system. In this way, the paper relates the capacitor voltage imbalance problem to the problem of regulating the output of the system, under the presence of periodic disturbances that should be canceled. The simulations presented in Section 6have proved the usefulness of the voltage balance controllers. It is worth stressing that the control laws are not difficult to implement. Due to the disturbance frequencies, which are three times the frequencies of the rectifier and inverter side phase voltages, the voltage balancing methods have been designed and introduced in the system taking into account the values of the frequencies used in the simulations (see Table II). If other nominal values of the frequencies are considered, the controllers should be redesigned to cancel the disturbances, since the disturbance frequencies are modified and present other values. Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 18 F. UMBR´ IA ET AL. 0 50 100 150 200 250 300 0 0.1 0.2 0.3 Frequency (Hz) |Vd(f)| (a) 0 50 100 150 200 250 300 0 0.1 0.2 0.3 Frequency (Hz) |Vd(f)| (b) Figure 8. From top to bottom, single-sided amplitude frequency spectrums of the dc-link capacitor voltage difference vd, applying the control methods: (a) Controller that estimates the disturbances via a Luenberger observer. (b) Traditional PI controller. A. APPENDIX Mathematical analysis of the capacitor voltage difference dynamics of the converter Focusing on the capacitor voltage difference dynamics defined by (1), notice that they are expressed as a function of the components of control vectors δαβγ,rand δαβγ,i, and of the phase current vectors iαβ,rand iαβ,i. Since Assumption 1is related to the power dynamics, it is more suitable for the analysis to express (1) in terms of power instead of current. Thereby, considering (7)-(10), the phase currents are approximated by iα,r≃p∗ r |vαβ,r|2vα,r−q∗ r |vαβ,r|2vβ,r(73) iβ,r≃p∗ r |vαβ,r|2vβ,r+q∗ r |vαβ,r|2vα,r(74) iα,i≃p∗ i |vαβ,i|2vα,i−q∗ i |vαβ,i|2vβ,i(75) iβ,i≃p∗ i |vαβ,i|2vβ,i+q∗ i |vαβ,i|2vα,i, (76) defining this well-known change of variables in three-phase circuits. It is important to mention that the phase voltages in αβ coordinates are expressed as vα,r=|vαβ,r|cos (ωrt+θr)(77) vβ,r=|vαβ,r|sin (ωrt+θr)(78) vα,i=|vαβ,i|cos (ωit+θi)(79) Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 ASYMPTOTIC REJECTION OF SINUSOIDAL DISTURBANCES IN BACK-TO-BACK CONVERTERS 19 vβ,i=|vαβ,i|sin (ωit+θi), (80) where ωrand ωiare the frequencies of the phase voltages in the rectifier and inverter sides of the system, respectively. Their phases are denoted by θrand θi. In the same way, concerning the control vectors δαβ,rand δαβ,i(devoted to regulate the instantaneous powers), their expressions are obtained introducing (73)-(76) in the phase current dynamics [44], assuming that the capacitor voltage imbalance is small, i.e., vd≃0. Consequently, when the instantaneous powers are located around their desired values, they are described by δαβ,r≃2 vdc λ1,rvαβ,r−2 vdc λ2,rJ vαβ,r(81) δαβ,i≃2 vdc λ1,ivαβ,i+2 vdc λ2,iJ vαβ,i. (82) Constant parameters λ1,r,λ2,r,λ1,i,λ2,ias well as the matrix Jare introduced to simplify the notation. They are defined as follows λ1,r= 1 + Lrωrq∗ r |vαβ,r|2(83) λ2,r=Lrωrp∗ r |vαβ,r|2(84) λ1,i= 1 −Liωiq∗ i |vαβ,i|2(85) λ2,i=Liωip∗ i |vαβ,i|2(86) J=0−1 1 0 . (87) Moving on to the dynamics under analysis, introducing the change of variables (73)-(76) as well as the control vectors (81) and (82) in (1), the variation over the time of the voltage difference results in the complex expression Cdvd dt =2p∗ r √3vdc δr γ−2p∗ i √3vdc δi γ +2p∗ rvα,r−2q∗ rvβ,r √6|vαβ,r|2v2 dc (λ1,rvα,r+λ2,rvβ,r)2−(λ1,rvβ,r−λ2,rvα,r)2 −4p∗ rvβ,r+ 4 q∗ rvα,r √6|vαβ,r|2v2 dc (λ1,rvα,r+λ2,rvβ,r) (λ1,rvβ,r−λ2,rvα,r)(88) +2q∗ ivβ,i−2p∗ ivα,i √6|vαβ,i|2v2 dc (λ1,ivα,i−λ2,ivβ,i)2−(λ1,ivβ,i+λ2,ivα,i)2 +4p∗ ivi β+ 4 q∗ ivα,i √6|vαβ,i|2v2 dc (λ1,ivα,i−λ2,ivβ,i) (λ1,ivβ,i+λ2,ivα,i). Nonetheless, according to the phase voltage definitions (77)-(80), and considering some trigonometric identities and the constants µ1,r,µ2,r,µ1,iand µ2,igiven by µ1,r= 2 |vαβ,r| √6v2 dc λ2 1,r+λ2 2,rpp∗2 r+q∗2 r(89) Copyright c 2014 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control (2014) Prepared using rncauth.cls DOI: 10.1002/rnc.3036 20 F. UMBR´ IA ET AL. µ2,r=λ2 1,r−λ2 2,rp∗ r+ 2 λ1,rλ2,rq∗ r −λ2 1,r−λ2 2,rq∗ r+ 2 λ1,rλ2,rp∗ r (90) µ1,i= 2 |vαβ,i| √6v2 dc λ2 1,i+λ2 2,iqp∗2 i+q∗2 i(91) µ2,i=−λ2 1,i−λ2 2,ip∗ i+ 2 λ1,iλ2,iq∗ i λ2 1,i−λ2 2,iq∗ i+ 2 λ1,iλ2,ip∗ i , (92) from (88) the reduced expression defined by (12) is derived. Hence, (12) is the final expression of the analysis worked out. ACKNOWLEDGEMENTS This work has been funded under grants MICINN-FEDER DPI2009-09661 and Junta de Andaluc´ıa P07- TIC-02991. REFERENCES 1. Kouro S, Malinowski M, Gopakumar K, Pou J, Franquelo LG, Wu B, Rodr´ıguez J, P´erez MA, Le´on JI. 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