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Load estimator-based hybrid controller design for two-interleaved boost converter dedicated to renewable energy and automotive applications

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This paper is devoted to the development of a hybrid controller for a two-interleaved boost converter dedicated to renewable energy and automotive applications. The control requirements, resumed in fast transient and low input current ripple, are formulated as a problem of fast stabilization of a predefined optimal limit cycle, and solved using hybrid automaton formalism. In addition, a real time estimation of the load is developed using an algebraic approach for online adjustment of the hybrid controller. Mathematical proofs are provided with simulations to illustrate the effectiveness and the robustness of the proposed controller despite different disturbances. Furthermore, a fuel cell system supplying a resistive load through a two-interleaved boost converter is also highlighted.

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Load estimator-based hybrid controller design for two-interleaved boost converter dedicated to renewable energy and automotive applications

  1. 1. Research Article Load estimator-based hybrid controller design for two-interleaved boost converter dedicated to renewable energy and automotive applications Mohamed Bougrine a,b , Mohammed Benmiloud a , Atallah Benalia a , Emmanuel Delaleau c , Mohamed Benbouzid b,d,n a University of Laghouat, LACoSERE Lab, Laghouat, Algeria b University of Brest, FRE CNRS 3744 IRDL, Brest, France c Ecole Nationale d'Ingénieurs de Brest, Mechatronics Department, Plouzané, France d Shanghai Maritime University, Shanghai, China a r t i c l e i n f o Article history: Received 8 June 2016 Received in revised form 14 August 2016 Accepted 6 September 2016 Available online 16 September 2016 This paper was recommended for publica- tion by Dr. Jeff Pieper Keywords: Interleaved boost converter Hybrid dynamical system Optimal limit cycle Stabilization Adaptive control Fuel cell source a b s t r a c t This paper is devoted to the development of a hybrid controller for a two-interleaved boost converter dedicated to renewable energy and automotive applications. The control requirements, resumed in fast transient and low input current ripple, are formulated as a problem of fast stabilization of a predefined optimal limit cycle, and solved using hybrid automaton formalism. In addition, a real time estimation of the load is developed using an algebraic approach for online adjustment of the hybrid controller. Mathematical proofs are provided with simulations to illustrate the effectiveness and the robustness of the proposed controller despite different disturbances. Furthermore, a fuel cell system supplying a resistive load through a two-interleaved boost converter is also highlighted. & 2016 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Climate change in the last century, mainly due to the increasing carbon dioxide (CO2) released through human activities as shown in Fig. 1, brings to light serious issues like global temperature and sea level augmentation, ocean warming and acidification, shrink- ing ice sheets, etc. These facts have pushed the scientific com- munity for renewable and clean energy solutions to supply various technological applications via switched converters. Due to the power source and load constraints, the converter structures have to meet some practical challenges such as relia- bility, high power density, high efficiency, and low current/voltage ripples. Parallel connection of switched converters or specifically the interleaving approach meets the above requirements with better power scalability characteristics compared to the classical ones [2,3]. Fig. 2a depicts the topology of a DC–DC two-interleaved boost converter, the two-phase boost topology was proposed as an alternative to the classical single phase boost converter [4], and the two-phase buck topology as an alternative to the classical buck converter [5]. These interleaved topologies are widely used in varieties of applications and systems that incorporate solar panels or fuel cell sources, as reflected in the literature [6–11]. The interleaving technique is also investigated for the microprocessors power supply to achieve better computing performance using the topology in Fig. 2b, which is known as multiphase/multi-channel synchronous/interleaved buck converter [12–15]. The interleaved bidirectional topology, obtained by changing each diode with a controlled switching device in the interleaved boost converter, is also explored in systems with rechargeable energy storage ele- ments like batteries or supercapacitors [16–20]. Research studies have discussed these topologies from different practical and fundamental points of view. The authors in [21] have discussed the number of phases that can be used to obtain a tra- deoff among some indexes such as: the switching losses, the inductor volume, the input current ripples and the switches cost. Power management in fuel cell hybrid vehicles involving battery/ Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2016.09.001 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding author at: University of Brest, FRE CNRS 3744 IRDL, Brest, France. E-mail addresses: m.bougrine@lagh-univ.dz (M. Bougrine), med.benmiloud@lagh-univ.dz (M. Benmiloud), a.benalia@lagh-univ.dz (A. Benalia), delaleau@enib.fr (E. Delaleau), mohamed.benbouzid@univ-brest.fr (M. Benbouzid). ISA Transactions 66 (2017) 425–436
  2. 2. supercapacitor as an auxiliary regenerative source is thoroughly explored in [16,22]. A maximum power point tracking (MPPT) approach in systems with solar module or a fuel cell source associated to interleaved boost converter is performed in [23,24]. Other works deal with the energetic efficiency improvement by considering variants of the interleaved topology such as soft switching and resonant techniques, or coupled inductors [25,26]. Furthermore, the control of interleaved converters has been dealt using several control strategies. Linear control techniques based on the average model are effective only around a specific operation points due to the neglected nonlinearities of the con- verter model and the power source [27]. In [14,27], a sliding mode and adaptive sliding mode controllers are designed and defended to be robust ones. Model predictive control is developed for the interleaved boost topology in [28]. Regardless of the fact that several control techniques have been proposed, the transient and steady-state control problem of the interleaved converter still pose challenging issues. More precisely, the transient control deals with a minimal response time and number of commutations require- ments, which is beneficial for a fast MPPT achievement in a solar panel based system for example. In addition, in electric vehicles, a fast tracking of the supercapacitor reference current, may also require an interleaved bidirectional converter to guarantee the instantaneous peak power demands or to recover a maximum energy through regenerative braking [29]. Moreover, information processing in computers is directly related to the response time of the microprocessors power supply in fast current variations tracking [12]. The existing controllers may tackle the problem of a fast transient with different approaches. Nevertheless, the steady- state behavior, represented by states ripples, is not directly con- trolled. Indeed, ripples amplitude and harmonic content of the current is one of the various phenomena influencing fuel cell lifespan as well as battery lifetime [11,22]. Besides, a small input current ripple is advantageous for MPPT to operate around a maximum power point without too much fluctuation [30]. Motivated by the above facts, we aim to address the control problem of the transient and steady-state of the two-interleaved boost converter. For this purpose, an exact instantaneous model of the converter will be investigated in the next section using the theory of hybrid dynamical systems. An adaptive transient and steady-state controllers will be developed in Section 3 to solve the problem of fast stabilization of a predefined optimal limit cycle. Simulation results will be discussed in Section 4 and followed by conclusions and future works that outline the main contributions of the present paper. 2. Hybrid model of 2IBC and control formulation We consider the 2IBC presented in Fig. 2a, supplying a resistive load (R) and operating in continuous conduction mode (CCM). Phase k consists of a non-ideal inductor ðLk; rkÞ, a diode ðDkÞ, and a controlled semiconductor device ðswkÞ by a binary input signal uk Af0; 1g. The instantaneous model of the 2IBC is given by: diLk dt ¼ 1 Lk ðvi ÀrkiLk ÀukvoÞ; k ¼ 1; 2 dvo dt ¼ 1 Co X2 k ¼ 1 ukiLk À vo R ! ; 8 >>>>< >>>>: ð1Þ where iLk is the inductor current of phase k. The voltages vi and vo are the input and output voltages respectively. Á : f0; 1g↦f0; 1g is the Not function. The input signal ðu1; u2Þ combinations offer four different con- verter operations without redundancy, known as discrete modes in hybrid systems vocabulary. From (1), each discrete mode can be represented by the following affine differential equation: _x ¼ AqðtÞxþBqðtÞ ¼ f qðtÞðxÞ ð2Þ with x ¼ ½iL1 iL2 voŠT AX is the continuous state vector defined in a physical operating region X DR3 . qðtÞ : Rþ ↦Q is the switching signal, with Q ¼ fq1; q2; q3; q4g is the set of discrete modes. The state matrices Aqi AR3Â3 and Bqi AR3Â1 are given by: Aqi ¼ À r1 L1 0 Àu1 L1 0 Àr2 L2 Àu2 L2 u1 Co u2 Co À 1 RCo 0 B B B @ 1 C C C A ; Bqi ¼ vi L1 vi L2 0 0 B @ 1 C A For each discrete mode, the corresponding values of the inputs are given in Table 1. The control design consists in the adequate state feedback switching low design q(x) that orchestrates the switching among the discrete modes to meet the control requirements. In closed loop, the hybrid automaton H of the 2IBC, operating in CCM, can be 1980 1985 1990 1995 2000 2005 2010 2015 330 350 370 390 410 Years CO2 (ppm) Seasonal Monthly Fig. 1. Recent global monthly and seasonal mean CO2 in parts per million (ppm) over marine surface sites (October 2015) [1]. Fig. 2. Topology of interleaved converter. M. Bougrine et al. / ISA Transactions 66 (2017) 425–436426
  3. 3. represented by the following 6-tuple: H ¼ ðQ; X; Sc; T; G; InitÞ ð3Þ where Sc : ðQ Â XÞ↦R3 is the application that assigns to every discrete mode a continuous dynamic given by (2). T ¼ fTij; i; jAf1; …; 4gg represents a set of all possible transitions between discrete modes, G : T-2X associates to each transition a continuous set where the transition is valid (called a guard condition), and InitDX Â Q gives the initial states. Based on (3) the switching design corresponds to the free elements definition of the hybrid automaton ðT; G; InitÞ in order to solve the stabilization problem discussed in the next subsection. In the remainder of the paper, we will consider r1 ¼ r2 ¼ r and L1 ¼ L2 ¼ L. The control of interleaved boost converters addresses the reg- ulation of the output voltage vo around the desired voltage Vref by controlling the input current Iin around an average reference generated by an outer loop as follows: Iref ðtÞ ¼ INom ref þ Z t 0 K sgnðVref ÀvoðτÞÞ Vref ÀvoðτÞ α dτ ð4Þ where K is a sufficiently large positive real number, αAð0; 1Þ is a small coefficient to ensure a finite time settling, INom ref ¼ V2 ref Rvi is the nominal value of Iref in static steady-state out of model uncer- tainties. To guarantee the balancing of the converter, the input current must be shared between the two phases. Hence for the 2IBC, the phase currents must be regulated around the average reference Irefk ¼ 0:5Iref . On the other hand, it is clear from the hybrid nature of the converter that the regulation of the phase currents can be obtained in a cyclic manner. This means that in steady-state, the converter trajectory converges to a limit cycle. The optimal limit cycle is characterized by low input ripple which corresponds to the following condition on the inputs: 1. The input signals have the same duty cycle d. 2. The phase shift between the two phases is equal to π. The duty cycle d depends on the required output voltage as fol- lows: d ¼ 1À vi ÀrIref Vref ð5Þ Fig. 3 illustrates the hybrid trajectory (x,q) of the 2IBC in the currents phase plane ðiL1 ; iL2 Þ for different duty cycle values and a fixed desired phase shift ðϕ ¼ πÞ of the input signals. One can remark that the periodic behavior of the input signals leads to a cyclic hybrid trajectory, which is one of the important nonlinear phenomena that may be exhibited by DC–DC power converters [31]. Note that each phase shift leads to a specific limit cycle where the optimal one, with respect to input current ripple, is shown in Fig. 3. Therefore, it is more natural to consider the stabilization problem of the 2IBC as follows. Fast optimal limit cycle stabilization: Given the set of discrete modes Q ¼ fq1; q2; q3; q4g, how should they be selected to guaran- tee the fast stabilization of the predefined optimal limit cycle depicted in Fig. 3 with a fast transient property? This problem is not directly discussed using the continuous dynamical systems theory. In the next section, we will solve the above problem efficiently using hybrid systems theory. 3. Hybrid control design: main results The formulated stabilization problem in the previous section can be restated based on the closed loop hybrid model (3) as fol- lows: (1) For a given initial condition x0, how one should select the convenient discrete modes among the set Q to reach fast the neighborhood of the optimal limit cycle shown in Fig. 3, and fast balance of the input current among the two phases. (2) When the neighborhood of the optimal limit cycle is reached, how to select the discrete modes to ensure its local stabilization. In the following, we design an adaptive hybrid controller, illu- strated in Fig. 4, with two parts: A transient state automaton for fast transition and a steady-state automaton for local asymptotic Table 1 The discrete modes of the two interleaved boost converter and their equilibrium points. Discrete modes Input signals Currents evolution Equilibrium point u1 u2 iL1 iL2 x qi eq q1 0 0 ↘ ↘ vi 2Rþr vi 2Rþr 2Rvi 2Rþr !T q2 1 0 ↗ ↘ vi r vi Rþr Rvi Rþr !T q3 0 1 ↘ ↗ vi Rþr vi r Rvi Rþr !T q4 1 1 ↗ ↗ vi r vi r 0 h iT Iref1 Iref2 1q 1q 2q 3q Iref1 Iref2 2q 3q Iref1 Iref2 3q 2q 4q 4q Fig. 3. Projection of the desired limit cycle in the currents plane. Fig. 4. The proposed adaptive hybrid control scheme for the interleaved boost converter. M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 427
  4. 4. stability of the desired limit cycle. Besides, an adaptive rule for the load resistance will be proposed using an algebraic approach. 3.1. Transient automaton controller design In order to achieve a fast convergence of the inductance cur- rents to a neighbor of their references Iref1 and Iref2 , one can dis- tinguish two cases: x0 AΩ2 with Ω2 ¼ fxAX∣iL2 4iL1 g: The selected discrete mode in this region should ensure a decreasing value of the current iL2 and an increasing one for the current iL1 to direct the converter trajectory toward the balancing surface S1ðxÞ ¼ 0, with S1ðxÞ ¼ iL2 ÀiL1 . From the currents evolution in Table 1, it can be noticed that mode q2 is the only appropriate choice in this region. x0 AΩ3 with Ω3 ¼ fxAX∣iL2 oiL1 g: One should select the discrete mode that guarantees an increasing value of the current iL2 and a decreasing one for the current iL1 to reach the balancing sur- face S1ðxÞ ¼ 0. From Table 1, the mode q3 is the only suitable one for this purpose. Once the converter trajectory reaches the surface S1ðxÞ ¼ 0, a transition should be made to another discrete mode to bring the converter currents (iL1 ; iL2 ) toward the reference surface S2ðxÞ ¼ 0, with S2ðxÞ ¼ iL1 ÀIref1 . Two cases can be distinguished: xðtsÞAΩ4 with ts is the reaching time of the converter trajectory to the balancing surface and Ω4 ¼ fxAX∣iL1 ðtsÞoIref1 g: In order to force the converter currents to reach the surface S2ðxÞ ¼ 0 and preserving the invariance of the balancing surface, the discrete mode q4 is the one suitable. xðtsÞAΩ1 with Ω1 ¼ fxAX∣iL1 ðtsÞ4Iref1 g: In this case, the dis- crete mode q1 is the appropriate one to reach S2ðxÞ ¼ 0 while preserving the invariance of the balancing surface as listed in Table 1. Fig. 5 shows a graphical representation, in the currents phase plane, of the transient automaton controller. The converter tra- jectory evolution can be highlighted as follows: the first initial condition x01 belongs to the region Ω3 where mode q3 should be activated. The converter trajectory x(t) evolves through the dynamics _x ¼ f q3 ðxÞ until hitting the balancing surface S1ðxÞ ¼ 0 at the instant ts. In this case, the state xðtsÞ belongs to the region Ω4 and hence mode q4 is selected to charge the inductors currents until reaching the reference surface S2ðxÞ ¼ 0. After that, a steady- state controller must be activated to ensure the local stability of the desired cyclic behavior. Fig. 6 illustrates the hybrid automaton of the developed hybrid controller with both transient and steady control parts, where each node represents a discrete mode and the arrows indicate the possible discrete transitions. The transient automaton controller corresponds to the above analysis where the elements ðG; T; InitÞ are defined in the following proposition: Proposition 1. Under the transient automaton controller shown in Fig. 6 and defined by the guard conditions (6)–(8), the input current ðIinÞ of the 2IBC reaches its desired value ðIref Þ in finite time and within two commutations: q3q4; q3q1; q2q4, or q2q1. The initial dis- crete mode depends on the initial continuous state as follows: Init ¼ ðInitðq1Þ Â q1Þ [ ðInitðq2Þ Â q2Þ [ ðInitðq3Þ Â q3Þ [ðInitðq4Þ Â q4Þ ð6Þ with Initðq1Þ ¼ fxAΩ1 ðS1ðxÞ ¼ 0Þg; Initðq2Þ ¼ Ω2 Initðq4Þ ¼ fxAΩ4 ðS1ðxÞ ¼ 0Þg; Initðq3Þ ¼ Ω3 The guard conditions corresponding to the fast balance of the input current are given by: Gt ðT21Þ ¼ fxAX∣ðiL2 riL1 Þ4ðiL1 4Iref1 Þg Gt ðT24Þ ¼ fxAX∣ðiL2 riL1 Þ4ðiL1 oIref1 Þg Gt ðT31Þ ¼ fxAX∣ðiL2 ZiL1 Þ4ðiL1 4Iref1 Þg Gt ðT34Þ ¼ fxAX∣ðiL2 ZiL1 Þ4ðiL1 oIref1 Þg ð7Þ The guard conditions corresponding to the convergence of the inductor currents to their references and the transition to the steady- state controller are defined as follows: Gts 1 ðT12Þ ¼ fxAX∣ðiL1 rIref1 Þ4ðVref o2viÞg Gts 2 ðT12Þ ¼ fxAX∣ðiL1 rIref1 Þ4ðVref ¼ 2viÞg Gts 3 ðT12Þ ¼ fxAX∣ðiL1 rIref1 Þ4ðVref 42viÞg Gts 1 ðT42Þ ¼ fxAX∣ðiL1 ZIref1 Þ4ðVref o2viÞg Gts 2 ðT42Þ ¼ fxAX∣ðiL1 ZIref1 Þ4ðVref ¼ 2viÞg Gts 3 ðT42Þ ¼ fxAX∣ðiL1 ZIref1 Þ4ðVref 42viÞg ð8Þ Proof. See the Appendix. 3.2. Steady state automaton controller design The steady-state controller must operate in a cyclic fashion to guarantee the periodic hybrid motion depicted in Fig. 3, which corresponds to a three distinct converter operations depending on the desired output voltage Vref as follows: Vref o2vi: The limit cycle is characterized by the discrete sequence q2q1q3q1 where the spent time in mode qi before the transition to mode qj is denoted tà ij and given by: tà 21 ¼ tà 31 ¼ dT; tà 12 ¼ tà 13 ¼ ð0:5ÀdÞT ð9Þ with T ¼ tà 21 þtà 12 þtà 31 þtà 13 is the period of the limit cycle, which can be generally fixed depending on the switching frequency f¼1/T of the controlled semiconductor devices (sw1 and sw2). Fig. 7a illustrates the hybrid trajectory waveforms of the 2IBC in this case where the current ripples are listed in Table 2. Vref ¼ 2vi: This case is a transition one between case 1 and case 3 where the discrete sequence is q2q3. The input current ripple is equal to zero and the other ripples are provided in Table 2. balancing Fig. 5. State space partition of the 2IBC under the proposed transient state controller. M. Bougrine et al. / ISA Transactions 66 (2017) 425–436428
  5. 5. Fig. 6. The proposed hybrid automaton controller of the 2IBC. Fig. 7. Projection of the desired limit cycle in the currents plane. M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 429
  6. 6. The spent time in each mode is given by: tà 23 ¼ tà 32 ¼ 0:5T; ð10Þ The converter trajectory in steady-state is presented in Fig. 7b. Vref 42vi: The converter trajectory waveforms are illustrated in Fig. 7c where the limit cycle is defined by the discrete sequence q2q4q3q4 with the following spent times: tà 24 ¼ tà 34 ¼ ð1ÀdÞT; tà 42 ¼ tà 43 ¼ ðdÀ0:5ÞT ð11Þ The corresponding current ripples are reported in Table 2. From the above analysis, we note that the desired limit cycle is completely defined by the discrete sequence and the spent times, which depend on the desired switching frequency f, the input voltage and the desired output voltage. From the fact that the desired limit cycle has three geometric shapes, one can conclude that the steady-state controller must have three parts depending on the desired voltage Vref for a given switching frequency and input voltage as proposed in Fig. 6. In each part, we have the desired discrete sequence defined by a closed unidirectional automaton where the guard conditions, denoted Gs ðTijÞ, should be designed to guarantee the local asymptotic stability of the desired limit cycle. Transitions between the three parts, denoted ci with iAf1; 2; 3g, occur if the reference voltage is changed and does not meet the actual part. The condi- tions c1; c2 and c3 are valid if the reference output voltage verifies the conditions Vref o2vi; Vref ¼ 2vi and Vref 42vi respectively. Proposition 2 outlines the steady-state controller of the 2IBC. Proposition 2. Under the proposed steady-state controller in Fig. 6 with the guard conditions (12)–(14), the desired limit cycle of the 2IBC depicted in Fig. 3 is locally asymptotically stable. Part 1 ðVref oviÞ: Gs ðT21Þ ¼ fxAX∣iL1 ZIref1 þΔi1g Gs ðT13Þ ¼ fxAX∣Iin rIref ÀΔig Gs ðT31Þ ¼ fxAX∣iL2 ZIref2 þΔi2g Gs ðT12Þ ¼ fxAX∣Iin rIref ÀΔig ð12Þ Part 2 ðVref ¼ viÞ: Gs ðT23Þ ¼ fxAX∣iL2 rIref2 ÀΔi2g Gs ðT32Þ ¼ fxAX∣iL1 rIref1 ÀΔi1g ð13Þ Part 3 ðVref 4viÞ: Gs ðT24Þ ¼ fxAX∣iL2 rIref2 ÀΔi2g Gs ðT43Þ ¼ fxAX∣Iin ZIref þΔig Gs ðT34Þ ¼ fxAX∣iL1 rIref1 ÀΔi1g Gs ðT42Þ ¼ fxAX∣Iin ZIref þΔig ð14Þ Proof. See Appendix. The guard conditions (12)–(14) are obtained from the limit cycle trajectory computation based on the dynamics of each dis- crete mode. For example, let us consider the guard conditions design in part 1 where the discrete sequence is q2q1q3q1. For the discrete mode q2, the inductor current iL1 increases (charging case) where an upper constraint ðIref1 þΔi1Þ is chosen. Besides, the inductor current iL2 decreases (discharging case) but it is not used due to the necessity of other intermediate ripple cal- culation, which can be remarked in Fig. 7a. Furthermore, the input current (Iin) is avoided because its dynamic depends on the con- verter state. Using the same reasoning and by symmetry, the guard condition Gs ðT31Þ is obtained. In mode q1, the input current decreases, which allows us to add the lower constraint ðIref ÀΔiÞ for input current ðIinÞ in Gs ðT12Þ and Gs ðT13Þ. The same thought is applied to part 2 and part 3 for the guards design. The suggested hybrid controller shown in Fig. 6 and defined by the guard conditions (6)–(8) for the transient part and (12)–(14) for the steady part requires only one condition to be verified at each instant to determine the next discrete mode, which is attractive for practical implementations. 3.3. Adaptive rule based on an algebraic approach From the guard conditions, the hybrid controller can be clas- sified as a current control loop with the aim to maintain the output voltage at a desired level. In fact, the used reference current depends on the load resistance, which may affect the output vol- tage regulation. For that purpose, an online estimation of the load resistance should be developed for online adjustment of the hybrid control scheme. We opt for algebraic techniques for the load resistance estimation, which can be reliably achieved in real time that depends only on the arithmetic precision of the used processor [32–34]. Let us consider the output capacitor voltage dynamic: voðtÞ 1 R ¼ ð1Àu1ðtÞÞiL1 ðtÞþð1Àu2ðtÞÞiL2 ðtÞÀCo dvo dt ðtÞ Using the Laplace transformation for each discrete mode and taking into account that the control inputs are constant (u1ðtÞ ¼ u1 and u2ðtÞ ¼ u2), one can obtain: 1 R voðsÞ ¼ ð1Àu1ÞiL1 ðsÞþð1Àu2ÞiL2 ðsÞÀCoðsvoðsÞÀvoð0ÞÞ with s is the Laplace variable. To get rid of the initial condition voð0Þ, we differentiate both sides with respect to s as follows: 1 R dvo ds ðsÞ ¼ ð1Àu1Þ diL1 ds ðsÞþð1Àu2Þ diL2 ds ðsÞÀCoðvoðsÞþs dvo ds ðsÞ Time differentiations in this expression, i.e. multiplication by s in the operational form, can be avoided by multiplying both sides by a give power of sÀ1 . Moreover, as the integral plays the role of a low-pass filter, one seeks relation in which every time-function appears inside an integral in the time domain. By multiplication by sÀ2 , one obtains: 1 s2 dvo ds ðsÞ |fflfflfflfflffl{zfflfflfflfflffl} nðsÞ 1 R ¼ ð1Àu1Þ 1 s2 diL1 ds ðsÞþð1Àu2Þ 1 s2 diL2 ds ðsÞÀCo 1 s2 voðsÞþ 1 s dvo ds ðsÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dðsÞ Returning to time domain using the inverse Laplace transforma- tion, one gets an algebraic estimator of the load resistance: Table 2 Different state ripples of 2IBC. Ripples Δi1 Δi1 Δi Case 1 vi ÀrIref1 2L dT vi ÀrIref2 2L dT Vref Àvi ÀrIref1 L ð0:5ÀdÞT Case 2 vi ÀrIref1 4L T vi ÀrIref2 4L T 0 Case 3 Vref þrIref1 Àvi 2L ð1ÀdÞT Vref þrIref2 Àvi 2L ð1ÀdÞT vi ÀrIref1 L ðdÀ0:5ÞT M. Bougrine et al. / ISA Transactions 66 (2017) 425–436430
  7. 7. identification: ~RðtÞ ¼ nðtÞ dðtÞ ð15Þ with: nðtÞ ¼ Z T 0 ðT ÀσÞðÀσÞvoðtÀσÞdσ ð16aÞ dðtÞ ¼ Z T 0 ðT ÀσÞ ðÀσÞ ð1Àu1ÞiL1 ðσÞþð1Àu2ÞiL1 ðσÞ À Á ÀCovoðσÞ Â Ã þCoσvoðσފdσ ð16bÞ This estimator is not asymptotic, it can be evaluated for any t4T and give a real-time estimation of R. In practice, the integrals appearing in (16) are calculated by discrete methods (e.g. the trapezium rule) using a finite number of samples of iL1 , iL1 and vo in the time-interval ½tÀT; tŠ. 4. Simulation results For practical purposes, designing the converter parameters, especially passive components, is of utmost importance. To this end, it is necessary to reveal the existing links between the switching frequency, the output voltage, the current and voltage ripples, and the passive components. This has already been dis- cussed in the literature. For brevity reasons, only the most important equations will be mentioned. In CCM operation, the minimal required values for the passive components are obtained by considering the worst case in terms of ripples, which corresponds to the duty cycles d¼0.25 and d¼0.75. These are given by Lmin ¼ vo 8ΔIin;CCM;max f ð17aÞ Cmin ¼ vo 4RminΔvo;CCM;max f ð17bÞ with the output voltage vo, the desired switching frequency f ¼ 1 T, and the maximum allowed current ripple ΔIin;CCM;max and voltage ripple Δvo;CCM;max. A typical converter with the parameters in Table 3 will be studied in closed loop under the hybrid controller presented in Fig. 6 and defined by the two propositions alongside the adaptive rule equations (16)–(17). The initial value of the load estimator R0 is fixed at 15 Ω where the evaluation starts after ϵ¼0.5 ms and re-initialized each period Te ¼ 2 ms. Performance and robustness of the adaptive hybrid controller will be checked through different scenarios and tested with a fuel cell source in the next subsections. 4.1. Variable reference output voltage Fig. 8 illustrates the continuous state evolution of the 2IBC in closed loop for a variable reference output voltage. At startup, the reference input current Iref that corresponds to Vref ¼ 35 V is calculated using (4), which is perturbed by the wrong value of the estimated resistor load ðRe ¼ R0Þ. The hybrid controller forces the input current to converge rapidly to Iref ¼ 4:08 A, as it can be remarked from Fig. 8b, which justifies the overshoot in the output voltage. After ϵ ¼ 0:5 ms, the algebraic estimator pro- vides the real value of the load resistance, as illustrated in Fig. 9, to the hybrid controller. The latter overcomes the situation and stabilizes quickly the converter state around its reference xref ¼ ½1:53 A 1:53 A 35 VŠT . The reached limit cycle is depicted in Fig. 10a, which has the discrete sequence q2q1q3q1. The current ripples are obtained from Table 2 for a fixed frequency f¼10 Hz. At t¼0.01 s, the reference voltage Vref is decreased by 5 V. The hybrid controller allows us to bring the input current in finite time to the new desired value Iref ¼ 2:25 A and forces the converter trajectory to converge to the optimal limit cycle using part 1 of the steady-state controller as it can be observed from the used discrete modes in Fig. 10b. In the interval [0.02 s, 0.03 s], the reference voltage is set to 40 V. The hybrid controller uses part 2 of the steady-state controller to stabilize the optimal limit cycle in this case as shown in Fig. 10c, which contains only the discrete modes q2 and q3. It should be noted that the input current ripple is equal to zero as it can be remarked from the currents evolution in Table 3 Converter parameters. Parameter vi L; r Co f R0 Value 20 V 3.3 mH, 1 mΩ 23 μF 10 kHz 15 Ω Fig. 8. State evolution of the 2IBC for a variable load resistance. Fig. 9. Load estimation based on the algebraic approach. M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 431
  8. 8. Fig. 8b. In the last time interval, the hybrid controller drives the converter state to its new reference xref ¼ ½2:53 A 2:53 A 45 VŠT quickly where the reached limit cycle is depicted in Fig. 10d. The used discrete sequence is q2q4q3q4 because the reference output voltage Vref verifies the part 3 condition ðVref 42viÞ of the steady- state controller. In summary, the adaptive hybrid controller exhibits interesting properties such as: minimum setting time, zero steady-state error and a fixed operating frequency in steady-state (f¼10 kHz). Next, we will proceed with the load resistance variations for adaptive property test of the hybrid controller. 4.2. Variable load resistance Fig. 11 shows the obtained results for a desired output voltage Vref ¼ 35 V and a variable load resistance. The latter is increased by Fig. 10. Projection of the reached limit cycle in the currents plane ðiL1 ; iL2 Þ and the corresponding discrete mode evolution. Fig. 11. State evolution of the 2IBC for a variable load resistance. Fig. 12. Evolution of the real and the estimated load resistance value. Fig. 13. PEM fuel cell characteristics V–I and P–I. M. Bougrine et al. / ISA Transactions 66 (2017) 425–436432
  9. 9. 5 Ω from 20 Ω to 30 Ω each 10 ms and decreased by 10 Ω at the instant 30 ms. Fig. 9 shows the algebraic estimator response under these disturbances. The algebraic estimator tries to detect and reject any load variation by the re-initialization of the algebraic formulae each 2 ms, which is clear from the periodic small over- shoots in Fig. 12. Once the load resistance is estimated, the hybrid controller will be updated by a new corrected reference input current to reject the load perturbation. The obtained results illustrate the effectiveness and the robustness of the hybrid con- troller in spite of these fast variations. 4.3. Association of the 2IBC with a fuel cell source Fuel cell systems are increasingly used in the last decade in automotive applications, which are classified as clean energy with zero CO2 emission. In this part, we consider an example of 1.26 kW proton exchange membrane (PEM) fuel cell with the P–I and V–I characteristics shown in Fig. 13, which is provided in MATLAB s SIMPOWER toolbox. Fig. 14 illustrates the 2IBC state evolution for the output voltage reference and a variable load resistance. One can remark that the fuel cell Vref ¼ 80 V has a minimal ripple in different phases of load variations. The adaptive hybrid controller is able to drive the converter state x to its reference quickly and with a good tracking. The robustness of the controller despite load variations is clearly observed (Fig. 16). The closed loop maintains the control require- ments in spite of fuel cell stack voltage variations presented in Fig. 15. Table 4 shows a qualitative comparison between the proposed control scheme based on hybrid automata theory and some other existing control strategies for the conventional boost converter and the interleaved one. One may read the proposed controller as an extension for the interleaved case of the controller presented in [35] for the conventional boost converter. The problem of the phase shifting between the control signals that does not show up in the conventional boost has been solved implicitly for the two- interleaved converter by considering a limit cycle stabilization problem in CCM. However, the case of DCM is beyond the scope of this paper and will be dealt with in future work. 5. Conclusions and future works In this paper, a new adaptive hybrid control scheme is devel- oped for the asymptotic stabilization of the desired limit cycle of two phase interleaved boost converter. From practical point of view, this corresponds to fast transient and low input current achievements, which is advantageous for many real world appli- cations such as: fast MPPT with low fluctuations around the maximum power point, extended lifetime of fuel cell system and solar panels, high efficiency and enhanced thermal properties of the converter, etc. The controller is developed via a hybrid automaton formalism with two parts: (1) a transient state automaton for fast transient and (2) a steady-state automaton for local limit cycle stabilization. An online adaptive rule based on the algebraic approach is also investigated for robustness purpose. Mathematical proofs and simulation results confirm the effectiveness and the robustness of Fig. 14. State evolution of the 2IBC associated with fuel cell source under a variable load resistance. Fig. 15. Fuel cell stack voltage. Table 4 Qualitative comparison. Previous works Control Model nature Phase shift Studied modes Complexity Applicability Giral et al. [36] Sliding mode-based Instantaneous model Enforced in OL with latch CCM Medium Two-interleaved Hubber et al. [37] Open loop control Instantaneous model Enforced in OL with delay CCM/DCM Simple Two-interleaved Thammasiriroj et al. [38] Flatness-based Average model Enforced in OL with PWM CCM Very high N-interleaved Sreekumar and Agarwal [35] Limit cycle stabilization Hybrid model – CCM/DCM Simple Conventional DC–DC boost Proposed controller Limit cycle stabilization Hybrid model Controlled in CL by the limit cycle stabilization CCM Medium Two-interleaved M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 433
  10. 10. the suggested controller under varying load and input voltage. A PEM fuel cell source associated with the 2IBC is also highlighted where satisfactory results are obtained. As a future work, a generalization of the proposed hybrid controller to n-phase interleaved boost converter and other multilevel topologies will be considered. A complete study of systems including different renewable and clean energy hybrid sources will be investigated. An adaptive hybrid controller for continuous conduction mode (CCM) and discontinuous conduc- tion mode (DCM) operation of the converter for light loads will also be discussed. Appendix A Proposition 1 Proof. From the transient state automaton configuration and the developed guard conditions, the input current Iin reaches its reference Iref in finite time if the following conditions are verified: Finite time reachability of the balancing surface ðS1ðxÞ ¼ 0Þ for x0 AΩ2 [ Ω3. Invariance of the balancing surface. Finite time reachability of the reference surface ðS2ðxÞ ¼ 0Þ for xðtsÞAðS1ðxÞ ¼ 0Þ. Roughly speaking, for any continuous initial condition in X, the input current has to reach the balancing surface in finite time (condition 1) and remain there (the invariance condition). Besides, in the balancing surface, we must choose a discrete mode that exhibits finite time reachability of the inductor currents to their references. By taking into consideration that all the state matrices of the discrete modes are Hurwitz, the first and the last conditions are equivalent to the following statements proof: x q2 eq AΩ3; x q3 eq AΩ2; x q1 eq AΩ4 ðS1ðxÞ ¼ 0Þ; x q4 eq AΩ1 ðS1ðxÞ ¼ 0Þ: Indeed, if for example an initial condition x0 belongs to Ω2 then the discrete mode q2 will be activated, and from the fact that the stable equilibrium point is in the complementary region of Ω2, the converter trajectory must pass through the balancing surface, which is equivalent to finite time reachability of the balancing surface ðS1ðxÞ ¼ 0Þ from one side. The same thought is valid in the other side for x0 AΩ3. Once the converter trajectory hits the bal- ancing surface a transition to mode q1 or mode q4 will be done. Fortunately, from the dynamics of these modes, it is clear that the balancing surface is invariant. It remains to prove that the equili- brium point of mode q1 is on the balancing surface and belongs to the opposite region (Ω4) with the same thought for mode q4 to conclude with the finite time attractivity of the reference surface. Let us start by the first statement: x q2 eq AΩ3⟹ieq L2 oieq L1 From Table 1, we have: ieq L2 ¼ vi Rþr oieq L1 ¼ vi r which proves the first statement. The second statement x q3 eq AΩ2 can be verified with same manner. The third statement can be proved as follows: x q1 eq AΩ4 ðS1ðxÞ ¼ 0Þ⟹ieq L2 ¼ ieq L1 oIref1 From Table 1, one can remark that ieq L2 ¼ ieq L1 ¼ vi 2R þr and it remains to prove only the following condition: vi 2Rþr oIref1 ¼ 1 2 V2 ref Rvi Let us begin by the boost property Vref 4vi and multiplying both sides by Vref 2Rvi , one can get: 1 2 V2 ref Rvi 4 Vref 2R 4 Vref 2Rþr 4 vi 2Rþr Hence the statement x q1 eq AΩ4 ðS1ðxÞ ¼ 0Þ is proved. The last statement can be checked by taking into consideration that the parasitic resistance r is negligible where a reference input current cannot be bigger than vi r . □ 0 0.01 0.02 0.03 0.04 40 45 50 Time (s) LoadResistance() R R e Ω Fig. 16. Evolution of the real and the estimated load resistance value. Fig. 17. Eigenvalues of the Jacobian of the Poincaré map in case 1. M. Bougrine et al. / ISA Transactions 66 (2017) 425–436434
  11. 11. Proposition 2 Proof. The local stability of the desired limit cycle can be proved using the following theorem [39], and the reader may refer to [40,41,35] for more detail and for various applications to multi- level and classical power converters. Theorem 1. Consider a general piecewise linear system with the form given by (1). Assume that there exists a limit cycle γ with period T, and defined by the discrete sequence q1q2…qN. Guard condition between mode qi to mode qj, denoted GðTijÞ, is given by a hyperplane of dimension nÀ1 as follows: GðTijÞ ¼ fxAX∣Cijxþdij ¼ 0g ð18Þ with Cij AR1Ân and dij AR. The stability analysis of the limit cycle γ can be checked by looking at the eigenvalues of the Jacobian of the Poincaré map P as follows: dP ¼ dPk1…dP34dP23dP12 with dPij ¼ IÀ f qi ðxà ijÞCij Cijf qi ðxà ijÞ ! eAità ij ð19Þ where xà ij is the switching state between mode qi and mode qj, which belongs to the guard condition GðTijÞ. The local stability of the limit cycle γ can be checked as follows: If the map dP has all its eigenvalues inside the unit circle, then the limit cycle γ is locally asymptotic stable. If at least one of the eigenvalues is outside the unit circle, then the limit cycle γ is unstable. We will use this theorem to prove the local stability of the desired limit cycle under the proposed hybrid steady-state con- troller in the following three cases. Case 01 ðVref o2viÞ: In this case, the Jacobian of the Poincaré map that corresponds to the discrete sequence q2q1q3q1 is dP ¼ dP12dP31dP13dP21. Transitions between the discrete modes of the discrete sequence occurs in 2-dimensional hyperplanes respecting (18), which can be defined using (12) as follows: C12 ¼ ½1 1 0Š; d12 ¼ ÀðIref ÀΔiÞ C31 ¼ ½0 1 0Š; d31 ¼ ÀðIref2 þΔi2Þ C13 ¼ ½1 1 0Š; d13 ¼ ÀðIref ÀΔiÞ C21 ¼ ½1 0 0Š; d21 ¼ ÀðIref1 þΔi1Þ 8 : The switching points between the discrete modes of the desired limit cycle are given as follows: xà 12 ¼ ½Iref1 ÀΔi1; Iref2 þΔi1 ÀΔi; Vref þΔvŠT xà 31 ¼ ½Iref1 ÀΔi2 þΔi; Iref2 þΔi2; Vref ÀΔvŠT xà 13 ¼ ½Iref1 þΔi2 þΔi; Iref2 ÀΔi2; Vref þΔvŠT xà 21 ¼ ½Iref1 þΔi1; Iref2 ÀΔi1 þΔi; Vref ÀΔvŠT 8 : Replacing these results and the corresponding discrete mode dynamic in (19), one can get the Jacobian of the Poincaré map. Using the converter parameters in Table 3, the eigenvalues of dP are evaluated numerically for a variable reference voltage ðVref o 2viÞ and load resistance, which are depicted in Fig. 17. All the eigenvalues are inside the unit circle, which proves the local sta- bility of the desired limit cycle under the proposed steady-state controller in this case. Case 02 ðVref ¼ 2viÞ: In this case, the discrete sequence q2q3 has the Jacobian of the Poincaré map dP ¼ dP32dP23. The switching surfaces in the form (18) and the switching points are given as Fig. 18. Eigenvalues of the Jacobian of the Poincaré map in case 2. Fig. 19. Eigenvalues of the Jacobian of the Poincaré map in case 3. M. Bougrine et al. / ISA Transactions 66 (2017) 425–436 435
  12. 12. follows: C32 ¼ ½1 0 0Š; d12 ¼ ÀðIref1 ÀΔi1Þ C23 ¼ ½0 1 0Š; d31 ¼ ÀðIref2 ÀΔi2Þ ( xà 32 ¼ ½Iref1 ÀΔi1; Iref2 þΔi2; Vref ŠT xà 23 ¼ ½Iref1 þΔi1; Iref2 ÀΔi2; Vref ŠT ( Fig. 18 illustrates that all the eigenvalues of the dP are inside the unit circle for varying load resistance, which proves the local stability of the desired limit cycle in this case. Case 03 ðVref 42viÞ: The desired limit cycle has the discrete sequence q2q4q3q4 where its Jacobian of the Poincaré map is dP ¼ dP42dP34dP43dP24. From the guard conditions given by (14), one can get the following switching surfaces parameters: C42 ¼ ½1 1 0Š; d42 ¼ ÀðIref þΔiÞ C34 ¼ ½1 0 0Š; d34 ¼ ÀðIref1 ÀΔi1Þ C43 ¼ ½1 1 0Š; d43 ¼ ÀðIref þΔiÞ C24 ¼ ½0 1 0Š; d24 ¼ ÀðIref2 ÀΔi2Þ 8 : The switching points between the discrete modes of the desired limit cycle are given as follows: xà 42 ¼ ½Iref1 þΔi1; Iref2 ÀΔi1 þΔi; Vref ÀΔvŠT xà 34 ¼ ½Iref1 ÀΔi1; Iref2 þΔi1 ÀΔi; Vref þΔvŠT xà 43 ¼ ½Iref1 ÀΔi2 þΔi; Iref2 þΔi2; Vref ÀΔvŠT xà 24 ¼ ½Iref1 þΔi2 ÀΔi; Iref2 ÀΔi2; Vref þΔvŠT 8 : Fig. 19 shows that all the eigenvalues of the dP are inside the unit circle for different values of Vref 42vi and varying load resistance, which proves the local stability of the desired limit cycle in this case. References [1] Dlugokencky E, Tans P. Trends in atmospheric carbon dioxide. National Oceanic Atmospheric Administration (NOAA); 2015. URL 〈http://www.esrl. noaa.gov/gmd/ccgg/trends/global.html〉. [2] Khosroshahi A, Abapour M, Sabahi M. 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