Nonlinear Power Sharing Controller for a Double-Input H-Bridge ...

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Nonlinear Power Sharing Controller for a Double-Input H-Bridge-Based Buckboost–Buckboost Converter Reza Ahmadi, Hassan Zargarzadeh, Student Member, IEEE, and Mehdi Ferdowsi, Member, IEEE

Abstract—This paper describes the design of a nonlinear controller with power sharing control capabilities of a double input buckboost–buckboost converter. First, this converter is introduced, and its principles of operation and the equations describing the converter circuit are reviewed. Next, the nonlinear model of the buckboost–buckboost converter is found by averaging the state equations. Then, a Lyapunov-based nonlinear controller, which is adaptive against input voltage and load disturbances (to provide line regulation and load regulation) is proposed. Finally, several simulation and experimental results from a prototype buckboost– buckboost converter operating under the proposed controller are reported to verify the operation of the designed controller. Index Terms—Buckboost–buckboost converter, Lyapunovbased nonlinear controller, double-input converter, power sharing.

I. INTRODUCTION ARNESSING renewable energy sources, such as wind and solar energy, recently has gained much attention, resulting in the extensive research and development of new power electronic solutions. Particularly, multiport converters are among the most anticipated power electronic devices with the potential to become the standard means by which to combine energy from different sources with intermittent natures [1], [2]. Employing multiport converters is a substitute for the conventional approach of interfacing each energy source to the dc bus using a separate single-input, single-output converter. Fig. 1 demonstrates a basic block diagram of a system with two energy sources linked to a double-input converter that combines the energy from the two sources and transfers it to the load. In addition to the evident advantages of utilizing a doubleinput converter similar to the one shown in Fig. 1, such as reduced component count, potential weight reduction, and source

H

Manuscript received April 12, 2012; revised June 24, 2012; accepted July 22, 2012. Date of current version November 22, 2012. This work was partially supported by the National Science Foundation under Grant 0640636. Recommended for publication by Associate Editor P. Mattavelli. R. Ahmadi and M. Ferdowsi are with the Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409-0810 USA (e-mail: [email protected]; [email protected]). H. Zargarzadeh was with the Electrical and Computer Engineering Department Missouri University of Science and Technology, Rolla, MO 65409-0810 USA. He is now with the Department of Industrial and Engineering, Southeast State Missouri University, Cape Girardeau, MO 63701 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2012.2211620

integration flexibility, one crucial advantage termed the “power sharing management capability” stands out. Power sharing management is the ability of the double-input converter to vary the ratio of the power drawn from the two input sources while keeping the total output power constant [3]–[8]. As shown in Fig. 1, in a double-input converter, Pout = P1 + P2

(1)

where P1 and P2 represent the power drawn from the two input sources, respectively, and Pout is the total power transferred to the load. The power sharing management principle states that in the case of constant load and thus constant output power Pout , when needed, the double-input converter can alter the amount of power drawn from the two energy sources (P1 and P2 ) such that (1) always holds. An example of such a situation is when a double-input converter is used in a hybrid electric vehicle to combine the power from the fuel cell system and the battery and feed it to the traction motor. When the battery’s state of charge drops below a certain limit, the converter should start to reduce the amount of power drawn from the battery and increase the amount of power drawn from the fuel cell system in order to avoid complete depletion of the battery. Meanwhile, assuming the traction motor requires constant power (no acceleration or deceleration happening, no road bumps, no wind speed change, etc.) the total power fed to the traction motor should remain constant in order to maintain the vehicle’s performance. Given all of the aforementioned advantages, several classes of multiport converters recently have been proposed and studied in the literature [1], [9]–[31]. In [20] and [21], a generalized systematic method to derive multi-input converters based on the building blocks of conventional single-input converters is presented. In [16], synthesis of a family of double-input converters based on a single-pole triple-throw switch is proposed. In [17], another family of double-input converters based on Hbridge cells is introduced. In [27], a new group of bidirectional multiport converters designed by combination of a dc link and a magnetic coupling is proposed. In [9], a type of multi-input converter with the advantage of less part count is designed. In [13], multiple-input converters based on a multiwinding transformer are presented. Moreover, [10] and [26] propose specific types of multi-input converters for vehicular applications. Similarly, [12] and [18] present multiple-input converters suitable for solar applications. Despite the considerable number of papers about multiport converter topologies in the literature, less attention has been paid to controller design methods for these types of converters. In general, conventional controller design methods

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Fig. 1. Power sharing capability in a double-input converter. The two sources cooperate to supply the required output power. The double-input converter can alter the ratio of the power supplied by the two sources.

rely on a small-signal model of the dc–dc converter linearized around a specific operating point [4], [32]–[36]. This linearized small-signal model is valid only when the converter is operating around that specific operating point; it becomes less reliable in transients or when the converter needs to operate around a new operating point, making controllers designed based on this model less reliable. On the other hand, the power sharing capability of multiport converters causes the operating point of these converters to vary, thus making it very difficult to control a multiport converter with a conventional linear controller. The aim of this paper is to design a controller for a doubleinput buckboost–buckboost converter that is independent of the converter’s operating point. In addition to the normal functions of conventional controllers, such as line and load regulation and disturbance rejection, this controller has the ability to manage power sharing, which entails shifting the converter’s operating point in order to vary the amount of power drawn from each input source while maintaining the converter’s stability. The designed controller is simple enough to be implemented using a normal digital controller. The operation and specifications of the buckboost–buckboost converter under investigation are reviewed in Section II. Section III is devoted to acquiring the nonlinear model of the buckboost– buckboost converter, which is essential to the controller design. Section IV explains the controller design procedure and derives the nonlinear adaptive controller with power sharing management capability. Simulation and experimental results are provided in Section V to verify the performance of the designed controller. Section VI offers conclusions drawn from this paper.

II. BUCKBOOST–BUCKBOOST DOUBLE-INPUT CONVERTER The buckboost–buckboost converter is shown in Fig. 2. The principles of operation for this converter are studied in detail in [4] and are reviewed here to provide deeper insight into the proposed control method. The buckboost–buckboost converter is a multiport (double-input, single-output) converter with two input sources, V1 and V2 , three controllable switches, S1 , S2 ,

Fig. 2.

Buckboost–buckboost converter circuit diagram.

and S3 and one output port. Two of the controllable switches, S1 and S2 , can be turned ON and OFF independently, while the conduction status of S3 depends on that of S1 and S2 . Switch S3 should be turned ON only when S1 is ON and S2 is OFF (S3 = S1 × S¯2 ); thus, modes of operation of this converter depend only on the status of conduction of S1 and S2 , resulting in totally four modes of operation, which are shown in Fig. 3. In mode 1 [see Fig. 3(a)], only S1 is ON, and V1 energizes the inductor (S3 is ON). In mode 2 [see Fig. 3(b)], both S1 and S2 are ON, and both sources energize the inductor (S3 is OFF). In mode 3 [see Fig. 3(c)], only S2 is ON, and V2 energizes the inductor (S3 is OFF). Finally, in mode 4 [see Fig. 3(d)], both switches are OFF, and the inductor is in the discharging cycle (S3 is OFF). The general switching pattern of S1 and S2 are shown in Fig. 4, in which the duty cycle of S1 is labeled d1 (t), the duty cycle of S2 is labeled d2 (t), and the period in which both S1 and S2 are ON is labeled d12 (t)T (with T being the switching period). The duration of each mode (T1 , T2 , T3 , and

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Fig. 3. Four modes of operation of the buckboost–buckboost converter. The inductor voltage, capacitor current, and input currents in each mode are determined based on the circuit parameters (state equations).

T4 ) in terms of duty cycles and switching period then can be expressed as ⎧ T1 ⎪ ⎪ ⎪ ⎨T 2 ⎪ T3 ⎪ ⎪ ⎩ T4

= (d1 (t) − d12 (t))T = (d12 (t))T = (d2 (t) − d12 (t))T = (d3 (t))T = (1 − (d1 (t) + d2 (t) − d12 (t)))T.

(2)

The duty cycles generally are considered time variant, but for given constant duty cycles (D1 , D2 , D12 ), the equilibrium values of the output voltage and the inductor current of this converter are found in [4] (the output voltage has a negative value because of how the polarity of the output voltage is defined in Fig. 2) VO = −

D1 D2 V1 − V2 1 − (D1 + D2 − D12 ) 1 − (D1 + D2 − D12 ) (3)

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by evaluating the following equation: x(t)T =

1 T



t+T

x(τ )dτ

(7)

t

where x(t)T denotes the average x(t) over switching period T. Therefore, the average inductor voltage over one switching period can be found by evaluating (7) for the inductor voltage throughout a switching period. The inductor voltage in this converter is composed of a fundamental slow varying component related to the dynamics of the converter and a high-frequency component related to the switching ripple. In order to be able to calculate (7), we should first employ the basic approximation of removing the high-frequency switching ripple from the inductor Fig. 4. General switching pattern of the buckboost–buckboost converter. voltage. The inductor voltage and capacitor current equations for each mode of operation of the converter are shown in Fig. 3.  To eliminate the high-frequency switching ripple, the instantaD1 1 IL = V1 2 neous values of the capacitor voltage and inductor current, [iL (t) R (1 − (D1 + D2 − D12 )) and vc (t)], are replaced with their low-frequency components,  D2 v c (t)T and iL (t)T . Consequently, the equations shown in + V2 . (4) (1 − (D1 + D2 − D12 ))2 Fig. 3 for the four modes of operation are transformed into the For the same constant duty cycles, the amount of power drawn following equations, respectively from each input source is calculated in [4] as well ⎧ diL (t) ⎪ ⎪ ⎨ vL (t) = L dt = V1 (5) P1 = V1 × I1 = D1 V1 IL P2 = V2 × I2 = D2 V2 IL .

(6)

In [4], it is shown that in a buckboost–buckboost converter an infinite number of triples of (D1 , D2 , D12 ) can be applied in (3) to generate the same intended output voltage. As a result when the input voltages (V1 , V2 ) are fixed and the output voltage (VO ) is intended to remain constant in order to deliver constant power to the load, an infinite number of triples of (D1 , D2 , D12 ) can be set into the converter circuit. Thus, in contrast to classical single-input converters, in a buckboost–buckboost converter, a specific output voltage can be generated by setting different arrangements of the duty cycles (D1 , D2 ) for the two controllable switches and adjusting the proper D12 corresponding to each arrangement. On the other hand, (5) and (6) suggest that in the case of fixed input voltages (V1 , V2 ) and a constant output voltage, and thus a constant inductor current, the ratio of the power drawn from the two sources (later defined as l) depends on the ratio of the duty cycles of switches S1 and S2 . This relationship reveals the key ability of the buckboost–buckboost converter in power sharing management, namely that in this type of converter, it is possible to change the ratio of the amount of power drawn from the two input sources while keeping the output voltage and thus the total output power constant just by changing the ratio of duty cycles. Nonetheless, it is worth noting that the amount of output power always equals the sum of the power drawn from the two sources. III. NONLINEAR CONVERTER MODELING The nonlinear model of the buckboost–buckboost converter can be developed by averaging the inductor voltage vL (t) and capacitor current ic (t) over a switching period. The average of any converter signal x(t) over one switching period can be found

vc (t)T dvc (t) ⎪ ⎪ =− ⎩ ic (t) = C dt R i1 (t) = iL (t)T i2 (t) = 0 ⎧ diL (t) ⎪ ⎪ ⎨ vL (t) = L dt = V1 + V2 dvc (t) ⎪ v (t) ⎪ =− cR T ⎩ ic (t) = C dt i1 (t) = iL (t)T i2 (t) = iL (t)T

(8)

(9)

⎧ diL (t) ⎪ ⎪ ⎨ vL (t) = L dt = V2 dvc (t) ⎪ v (t) ⎪ =− cR T ⎩ ic (t) = C dt i1 (t) = 0 i2 (t) = iL (t)T

(10)

⎧ diL (t) ⎪ ⎪ ⎨ vL (t) = L dt = vc (t)T dvc (t) ⎪ v (t) ⎪ = −iL (t)T − c R T ⎩ ic (t) = C dt

i1 (t) = 0 i2 (t) = 0.

(11)

Using the inductor voltage equations in (8)–(11), which yield the slow-varying component of the inductor voltage in each mode of operation, and the duration of each mode from (2), it is

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possible to evaluate (7) to yield the average inductor voltage  1 t+T diL (t)T vL (t)T = L = vL (τ )dτ dt T t =

1 ((V1 )(T1 ) + (V1 + V2 )(T2 ) T + (V2 )(T3 ) + (vc (t)T )(T4 )).

(12)

Substituting (2) in (12) and simplifying the resulting equation yields the final result L

diL (t)T = vc (t)T (1 − d1 (t) − d2 (t) dt + d12 (t)) + d1 (t)V1 + d2 (t)V2 .

(13)

Similarly, the average capacitor current can be found by evaluating (7) for the capacitor current signal C

dvc (t)T = −iL (t)T (1 − d1 (t) − d2 (t) dt vc (t)T . + d12 (t)) − R

(14)

Equations (13) and (14) form the nonlinear model of the buckboost–buckboost converter. The average current drawn from each source can be found likewise by evaluating (7) for the input current equations in (8)–(11) i1 (t)T = d1 (t) × iL (t)T

(15)

i2 (t)T = d2 (t) × iL (t)T .

(16)

Thus, the average power drawn from each source is

Fig. 5. Switching pattern of the buckboost–buckboost converter when the rising edges of the two switch command signals are synchronized in order to reduce the complexity of the proposed controller.

morphs between two models based on the values of d1 (t) and d2 (t). If d1 (t) is smaller than d2 (t) ⎧ di (t) L T ⎪ = vc (t) (1 − d2 (t)) + d1 (t)V1 + d2 (t)V2 ⎨L dt ⎪ ⎩ C dvc (t)T = −i (t) (1 − d (t)) − vc (t)T . L T 2 dt R (21) If d2 (t) is smaller than d1 (t) ⎧ di (t) L T ⎪ = vc (t) (1 − d1(t)) + d1(t)V1 + d2 (t)V2 ⎨L dt ⎪ ⎩ C dvc (t)T = −i (t) (1 − d (t)) − vc (t)T . L T 1 dt R (22)

P1 (t)T = V1 × i1 (t)T = d1 (t) × V1 × iL (t)T (17) P2 (t)T = V2 × i2 (t)T = d2 (t) × V2 × iL (t)T . (18) In practice, to reduce the complexity of the proposed controller, the rising edges of the two switch command signals are synchronized so that both S1 and S2 are always turned ON simultaneously in each switching period. Fig. 5 shows the switching pattern of the two switches in this case. As a result, the time period during which both switches are ON (d12 (t)) always equals the minimum value of d1 (t) and d2 (t) d12 (t) = min {d1 (t), d2 (t)} .

(19)

This transforms the nonlinear model of the converter into  ⎧ 1 − d1 (t) − d2 (t) diL (t)T ⎪ ⎪ ⎪ = vc (t) ⎪L ⎪ +min {d1 (t), d2 (t)} dt ⎪ ⎨ +d1 (t)V1 + d2 (t)V2  ⎪ ⎪ ⎪ 1 − d1 (t) − d2 (t) ⎪ dvc (t)T vc (t)T ⎪ ⎪ = −iL (t)T . − ⎩C +min {d1 (t), d2 (t)} dt R (20) Based on (19), if d1 (t) is smaller than d2 (t) then d12 (t) equals d1 (t), and if d2 (t) is smaller than d1 (t), then d12 (t) equals d2 (t); therefore, during operation, the nonlinear model of the converter

IV. NONLINEAR CONTROLLER DESIGN This section describes the process of designing a nonlinear control scheme for the converter model, shown in (21) and (22), which is applicable to an experimental test bed. The aim is to design a nonlinear controller with the following specifications: able to globally asymptotically regulate the output, meaning that it can navigate the output voltage from any initial condition to any desired output voltage; robust against disturbances, particularly variations in the input voltages and load; able to adjust the ratio of the power drawn from the sources to a desired value, i.e., l (power sharing control capability); able to produce a restricted switching ratio such that 0 < d1 (t), d2 (t) < 1.

(23)

Therefore, no saturation block is needed to restrict duty cycles in the actual controller; easy to implement using a digital signal processor (DSP). A. Open-Loop Controller The controller design procedure begins with finding the equilibrium point of the system, which is accomplished by setting the left-hand side of (21) and (22) equal to zero. In case d1 (t) is

AHMADI et al.: NONLINEAR POWER SHARING CONTROLLER FOR A DOUBLE-INPUT H-BRIDGE-BASED BUCKBOOST–BUCKBOOST

smaller than d2 (t)

By denoting

Vc∗ = VO = − IL∗

1 = R



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d1 (t) d2 (t) V1 − V2 1 − d2 (t) 1 − d2 (t)

d1 (t) d2 (t) V1 + V2 (1 − d2 (t))2 (1 − d2 (t))2

λ=

(24)  .

(25)

d1 (t) d2 (t) V1 − V2 = VO = − 1 − d1 (t) 1 − d1 (t)   d1 (t) 1 d2 (t) IL∗ = V + V 1 2 . R (1 − d1 (t))2 (1 − d1 (t))2

γ=λ (26) (27)

In (24)–(27), Vc∗ represents the equilibrium point value of the capacitor voltage, which equals the output voltage, and IL∗ represents the equilibrium point value of the inductor current. Equations (24)–(27) are similar to (3) and (4) except that their associated duty cycles are considered time variant. As noted in Section II, a particular value of the output voltage of the buckboost–buckboost converter can be generated using different pairs of duty cycles; therefore, the output voltage can be kept constant while the duty cycles vary along a path that keeps the output voltage constant. As a result, the duty cycles, and therefore the summing terms on the right-hand side of (24)– (27), which are d1 (t) V1 , 1 − d2 (t)

d1 (t) V1 (1 − d2 (t))2

(28)

d2 (t) V2 , 1 − d2 (t)

d2 (t) V2 (1 − d2 (t))2

(29)

d1 (t) V1 , 1 − d1 (t)

d1 (t) V1 (1 − d1 (t))2

(30)

d2 (t) V2 , 1 − d1 (t)

d2 (t) V2 (1 − d1 (t))2

(31)

can be changed over time, while the sum of the terms, and therefore, the equilibrium values on the left-hand side of (24)– (27), remain constant. The converter model in (21) and (22) is an overactuated system because it has two inputs, (d1 (t), d2 (t)), and one output, (vc (t)). The aim is to regulate the output voltage while the ratio of the power drawn from the two sources equals a desired value of λ. Therefore, we choose d1 (t) = γd(t)

(32)

d2 (t) = d(t).

(33)

and

With this terminology, by substituting (32) and (33) in (17) and (18), the average power drawn from each source can be formulated as P1 (t)T = γd(t) × V1 × iL (t)T

(34)

P2 (t)T = d(t) × V2 × iL (t)T .

(35)

(36)

as the desired ratio of the power drawn from the sources, γ can be determined using (34) and (35)

In case d2 (t) is smaller than d1 (t) Vc∗

P1 (t)T P2 (t)T

V2 . V1

(37)

Since the desired output voltage vcd should coincide with the equilibrium point of the capacitor voltage Vc∗ , so in (24) and (26) we set Vc∗ = vcd .

(38)

Then, substituting (32) and (33) in the resulting equations yields the general solutions for d(t). In case d1 (t) is smaller than d2 (t) d(t) =

−vcd . γV1 + V2 − vcd

(39)

In case d2 (t) is smaller than d1 (t) d(t) =

−vcd . γV1 + V2 − γvcd

(40)

Equations (39) and (40) along with (32) and (33) yield an openloop output voltage control solution because for a desired power ratio (λ) the value of γ is found using (37); therefore, d(t) can be found using (39) or (40). Finally, the required duty cycles are found using (32) and (33). The power ratio can be varied by varying γ in (39) or (40) and calculating the new duty cycles from (32) and (33). In this situation, the output voltage remains constant and equals the desired output voltage. B. Closed-Loop Controller As mentioned earlier, (39), (40), (32), and (33) yield an openloop controller for the converter. However, problems arise when the circuit parameters, such as input voltages or load, vary due to disturbances. In fact, having a closed-loop controller whose command is a function of the output error is more desirable. In order to calculate (39) and (40) to achieve the desired closedloop controller, in the first step, we need to show that the system’s equilibrium point is globally asymptotically stable. Lemma (Equilibrium Point Stability)–: The equilibrium point of (24) and (25) or (26) and (27) is globally asymptotically stable. In other words, for any input d1 , d2 ∈ [0, 1], the converter output voltage globally asymptotically converges to a particular voltage of Vc∗ . Proof: Consider the following positive definite and radially unbounded Lyapunov function [37]: L1 =

1 1 L(iL (t)T − IL∗ )2 + C(vc (t)T − Vc∗ )2 . 2 2

(41)

Taking the derivative along the system dynamics yields the following result:

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diL (t)T (iL (t)T − IL∗ ) L˙ 1 = L dt dvc (t)T (vc (t)T − Vc∗ ) . +C (42) dt As mentioned earlier, Vc∗ and IL∗ are constant; therefore, they require no differentiation. Substituting (21) or (22) in the generated derivative terms and simplifying the results using (24)and (25) or (26) and (27), respectively, yields 1 L˙ 1 = − (vc (t)T − Vc∗ )2 . (43) R The result is a negative semidefinite function, which implies that the system is stable in the sense of Lyapunov; therefore, vc (t) converges to Vc∗ over time. On the other hand, with vc (t) = Vc∗ , the inductor current equation in (21) and (22) has only one solution, i.e., iL (t) = IL∗ . Therefore, using Barbalat’s lemma [37], we conclude that the equilibrium point of the system is globally asymptotically stable. Recognizing the equilibrium point of the system as a globally asymptotically stable point, the below theorem proposes a controller that globally asymptotically regulates the output to a desired value vcd , while disturbances of V1 and V2 are unknown but slow varying (V˙ 1 ∼ = V˙ 2 ∼ = 0). In the below controller in order to compensate for the errors caused by the disturbances of the input voltages (line regulation), an extra parameter Ψ(t) is defined such that the value of Ψ(t) = γV1 (t) + V2 (t) is estimated and updated by the closed-loop controller as the system evolves. The initial value of Ψ(t) is set to Ψ(0) = γV1 + V2 , where V1 and V2 are the nominal values of the two input voltages without disturbances. Theorem (Adaptive Nonlinear Controller): Assume that one wishes to regulate the output voltage vc (t) to a desired voltage ˆ (t). vcd . Also assume that Ψ(t) is unknown and is estimated by Ψ The following controller globally asymptotically stabilizes the ˜ ˆ estimation and output errors, (Ψ(t) = Ψ(t) − Ψ(t) and vc (t) − d vc , respectively) when d1 (t) is smaller than d2 (t)

 α vc (t) − vcd − vc (t) d(t) = (44) ˆ Ψ(t) + α (vc (t) − v d ) − vc (t) c

ˆ where Ψ(t) is found in each step through the evolution of the following integral: t ˆ ˆ (0) + Ψ(t) =Ψ



−2β vc (τ ) − vcd

0

d(τ ) dτ. 1 − d(τ )

(45)

In case d2 (t) is smaller than d1 (t), the controller transforms into 

α vc (t) − vcd − vc (t) d(t) = (46) ˆ Ψ(t) + γα (vc (t) − v d ) − γvc (t) c

ˆ where Ψ(t) is found in each step through the evolution of t ˆ ˆ (0) + Ψ(t) =Ψ 0



−2β vc (τ ) − vcd

d(τ ) dτ. 1 − γd(τ )

(47)

The two parameters α and β > 0are convergence speed parameters. a) Proof for the Case in Which d1 (t) is Smaller Thand2 (t): Consider the following positive definite radially unbounded Lyapunov function: 2 α  ˆ . L2 = L1 + (48) Ψ−Ψ 2βR Taking the derivative along the system dynamics and using the result of the Lemma for L˙ 1 from (43) yields 1 α ˆ˙ ˜ ΨΨ. L˙ 2 = − (vc (t)T − Vc∗ )2 − R βR

(49)

Substituting Vc∗ from (24) in (49) and considering (32) and (33) yields 2  1 d(t) α ˆ˙ ˜ ˙ L2 = − (γV1 + V2 ) − ΨΨ. vc (t)T + R 1 − d(t) βR (50) We defined Ψ(t) = γV1 (t) + V2 (t), so (50) can be simplified to 2  1 d(t) α ˆ˙ ˜ L˙ 2 = − Ψ(t) − ΨΨ. (51) vc (t)T + R 1 − d(t) βR ˆ + Ψ, ˆ (51) can be rewritten as Because Ψ = Ψ 2  d(t) ˜ 1 d(t) ˆ L˙ 2 = − Ψ(t) + Ψ(t) vc (t)T + R 1 − d(t) 1 − d(t) α ˆ˙ ˜ Ψ(t)Ψ(t). (52) − βR By choosing vc (t)T +



d(t) ˆ Ψ = α vc (t)T − vcd 1 − d(t)

in (52), it can be rewritten as 2 α2 1 L˙ 2 = − vc (t)T − vcd − R R −



d(t) 1 − d(t)

(53)

2 ˜ 2 (t) Ψ

 d(t) 2 ˜− α Ψ ˆ˙ Ψ ˜ α vc (t)T − vcd Ψ R 1 − d(t) Rβ

(54)

which can be simplified by choosing ˆ˙ = −2β(vc (t)T − v d ) d(t) . Ψ c 1 − d(t) Substituting (55) in (54) yields α2 1 L˙ 2 = − (vc (t)T − vcd )2 − R R



d(t) 1 − d(t)

(55)

2 ˜ 2 (t) (56) Ψ

which is a negative definite function indicating that the system is stable in the sense of Lyapunov. As shown in the earlier steps, in order to make L˙ 2 a negative definite function as of (56), the two key equations, (53) and (55), must hold. Therefore, these two equations form the nonlinear controller of the system. The manipulation of (53) such that d(t) is found based on the rest of the parameters yields 

α vc (t) − vcd − vc (t) d(t) = (57) ˆ Ψ(t) + α (vc (t) − v d ) − vc (t) c

AHMADI et al.: NONLINEAR POWER SHARING CONTROLLER FOR A DOUBLE-INPUT H-BRIDGE-BASED BUCKBOOST–BUCKBOOST

Fig. 6.

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Implementation flow chart of the proposed nonlinear controller.

which is the same as (44) in the theorem. On the other hand, ˆ solving (55) for Ψ(t)yields t ˆ ˆ (0) + Ψ(t) =Ψ



−2β vc (τ ) − vcd

0

d(τ ) dτ 1 − d(τ )

TABLE I POWER STAGE PARAMETERS OF THE DESIGNED CONVERTER

(58)

which is the same as (45) in the theorem. In other words, choosˆ ing d(t)and Ψ(t)as in (44) and (45) guarantees the stability of the system in the sense of Lyapunov; therefore, (44) and (45) form the closed-loop controller. Proof for the case in which d2 (t) is smaller than d1 (t): The proof for this case is similar to that for the previous case, with the exception that Vc∗ is substituted into (49) from (26) rather than (24), which yields 1 L˙ 2 = − R



2

α ˆ˙ ˜ ΨΨ. βR (59) Similar to the previous case, (59) can be simplified to the following equation considering Ψ(t) = γV1 (t) + V2 (t): 1 L˙ 2 = − R

d(t) (γV1 + V2 ) vc (t)T + 1 − γd(t)



d(t) Ψ(t) vc (t)T + 1 − γd(t)

2 −

−

α ˆ˙ ˜ ΨΨ. (60) βR

ˆ + Ψ, ˜ (60) can be rewritten as Therefore, using Ψ = Ψ 2  d(t) ˜ 1 d(t) ˆ ˙ L2 = − Ψ (t) + Ψ(t) vc (t)T + R 1 − γd(t) 1 − γd(t) α ˆ˙ ˜ Ψ(t)Ψ(t). (61) − βR

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By choosing vc (t)T +

 d(t) ˆ Ψ = α vc (t)T − vcd 1 − γd(t)

(62)

in (61), it can be rewritten as 2 α2 1 L˙ 2 = − vc (t)T − vcd − R R −



d(t) 1 − γd(t)

2 ˜ 2 (t) Ψ

 d(t) 2 ˜− α Ψ ˆ˙ Ψ ˜ α vc (t)T − vcd Ψ R 1 − γd(t) Rβ

(63)

which can be simplified by choosing

 ˆ˙ = −2β vc (t)T − vcd Ψ

d(t) . 1 − γd(t)

(64)

Substituting (64) in (63) yields 2 α2 1 L˙ 2 = − vc (t)T − vcd − R R



d(t) 1 − γd(t)

2 ˜ 2 (t) Ψ

(65) which is a negative definite function that shows that the system in the sense of Lyapunov. Similar to the previous case, to make L˙ 2 a negative definite function as of (65), the two key equations, (62) and (64) must hold. Therefore, these two equations, which can be manipulated into (46) and (47), respectively, form the nonlinear controller of the system. To implement (44)–(47) in the system, in each sampling time ˆ calculated first the output voltage vc (t) is measured and Ψ(t)is using d(t) from the previous step. If d1 (t) from the previous step ˆ calculated from (45), is smaller than d2 (t) the value ofΨ(t)is otherwise it is calculated from (47). Next, the new d(t) is calcuˆ lated by substituting the resulting Ψ(t) and the measured vc (t) into (44) or (46). Again, if d1 (t) from the previous step is smaller than d2 (t) the value of d(t) is calculated from (44), otherwise it is calculated from (46). Finally, the values of the new duty cycles are calculated using (32) and (33). This implementation method is shown in the flow chart of Fig. 6. V. SIMULATION AND EXPERIMENTAL RESULTS To validate the theoretical outcomes, the designed controller is employed to control a prototype double-input buckboost– buckboost converter. The power stage parameters of this converter are listed in Table I. The converter and nonlinear controller are first simulated using MATLAB Simulink, and several simulation results are obtained by performing different experiments on the simulation model. Then, the converter is built in the lab, and the same experiments are performed on the real converter in order to compare the simulation and experimental results. The converter controller is implemented using a DSP. The DSP reads the output voltage of the converter using the built-in analog-to-digital converter (ADC). Obtaining the current value of the output voltage and the reference voltage to be tracked, (44)–(47) are evaluated by the DSP code, and the current d(t) and Y(t) are found. In the next step, d1 (t) and d2 (t) are determined using (32) and (33) based on the specified λ. Finally, the resulting duty cycles are commanded to the switches through the pulsewidth modulator unit of the DSP.

Fig. 7. Normal operation of the buckboost–buckboost converter: (a) experimental results; (b) simulation results. In (a) and (b), the top blue trace is the input current waveform of the first input, the red trace is the input current waveform of the second input, the magenta trace is the inductor current, and the green trace is the output voltage.

Fig. 7 illustrates the waveforms of the input currents and the inductor current, as well as the absolute value of the output voltage of the open-loop converter during normal operation. Fig. 7(b) is generated using the simulation model, and Fig. 7(a) shows the same waveforms in a scope shot obtained from the designed converter. These waveforms are provided in the first step in order to verify the operation of the power stage of the converter without the controller. In this case, the simulation and experimental results match closely except for the existence of some undamped ringing in the waveforms obtained from the experimental converter. This is normal because no attempt was made to dampen the ringing in the designed prototype converter in this stage of the design. Verifying the operation of the open-loop converter, the proposed controller is implemented, and several experiments are conducted on the closed-loop converter in order to inspect the performance of the controller for line regulation, load regulation, power sharing management, and output voltage tracking capability. The results of the experiments for both the simulation model and the designed converter are shown in Figs. 8–11,

AHMADI et al.: NONLINEAR POWER SHARING CONTROLLER FOR A DOUBLE-INPUT H-BRIDGE-BASED BUCKBOOST–BUCKBOOST

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Fig. 8. Line regulation: (a, b) experimental and (c, d) simulation results showing the dynamic response of the buckboost–buckboost converter when V 1 is stepped down from 24 to 18 V. (a) Average value of the input currents of the designed converter. (b) Output voltage waveform of the designed converter. (c) Average value of the input currents from the simulation. (d) Output voltage waveform from the simulation.

Fig. 9. Load regulation: (a, b) experimental and (c, d) simulation results showing the dynamic response of the buckboost–buckboost converter when the load resistance is stepped down from 20 to 10 Ω. (a) Average value of the input currents of the designed converter. (b) Output voltage waveform of the designed converter. (c) Average value of the input currents from the simulation. (d) Output voltage waveform from the simulation.

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Fig. 10. Output voltage command tracking: (a, b) experimental and (c, d) simulation results showing the dynamic response of the buckboost–buckboost converter when the output voltage is commanded to step up from 45 to 50 V. (a) Average value of the input currents of the designed converter. (b) Output voltage waveform of the designed converter. (c) Average value of the input currents from the simulation. (d) Output voltage waveform from the simulation.

Fig. 11. Power sharing management capability: (a, b) experimental and (c, d) simulation results showing the dynamic response of the buckboost–buckboost converter when the ratio of the power drawn from the two sources is stepped down from 2 to 0.5. (a) Average value of the input powers of the designed converter. (b) Output voltage waveform of the designed converter. (c) Average value of the input powers from the simulation. (d) Output voltage waveform from the simulation.

AHMADI et al.: NONLINEAR POWER SHARING CONTROLLER FOR A DOUBLE-INPUT H-BRIDGE-BASED BUCKBOOST–BUCKBOOST

each of which illustrates the dynamic response of the closedloop converter when a step change occurs in a parameter. In Figs. 8–11, in order to plot the average value of the input currents, the raw data for the input currents are transferred from the oscilloscope to the MATLAB software, and the moving average of the data is calculated and plotted. Similarly, in order to plot the average power supplied by each source based on (17) and (18), the moving average of each input current is multiplied by the voltage of the supplying source. Fig. 8 illustrates the line regulation of the closed-loop converter. Fig. 8(a) and (b) shows the experimental results, and Fig. 8(c) and (d) shows the simulation results when V1 is stepped down from 24 to 18 V. The ratio of the power drawn from the two sources (l) and the total output power does not change during the process, so the reduction of power being supplied by the first source due to decreased voltage is compensated for by increasing the current drawn from this source; therefore, as shown in Fig. 8(a) and (c), the average current drawn from the first source (blue trace) is increased, while the average current drawn from the second source remains constant. In this case, the resulting transient in the output voltage shown in Fig. 8(b) and (d) is acceptable. The simulation and experimental results closely match except that the dynamics of the transient in the simulation and experimental result are a bit different because the simulation model is built from ideal components while the designed converter is made with nonideal components. Fig. 9 presents the load regulation of the closed-loop converter. Fig. 9(a) and (b) shows the experimental results, and Fig. 9(c) and (d) shows the simulation results when the load resistance is stepped down from 20 to 10 Ω. The output power is increased by a factor of two, so the input power is doubled as well. Given that the ratio of the power drawn from the two sources (l) is constant, the current drawn from the two sources is doubled accordingly. In this case, as in the previous case, the resulting transient in the output voltage shown in Fig. 9(b) and (d) is rather negligible. Fig. 10 illustrates the output voltage command tracking of the proposed controller. Fig. 10(a) and (b) shows the experimental results, and Fig. 10(c) and (d) shows the simulation results when the output voltage is commanded to step up from 45 to 50 V. Similar to the previous case, because the output power is increased, the currents drawn from the two sources are increased accordingly. In this case, the output voltage transient in Fig. 10(b) and (d) displays a very low rise time and negligible overshoot. Fig. 11 features the power sharing management capability of the proposed controller. Fig. 11(a) and (b) shows the experimental results, and Fig. 11(c) and (d) shows the simulation results when the ratio of the power drawn from the two sources (λ) is stepped down from 2 to 0.5 so that the amount of power drawn from the first source is decreased by half and the amount of power drawn from the second source is increased by a factor of two. Based on the data in Table I, the output voltage of the converter equals to 45 V, and the load resistance equals 20 Ω; therefore, the output power equals to 101.25 W. As a result, the two input sources cooperate to supply 101.25 W of power to the load according to the specified power ratio (l). As shown in Fig. 11(c), in the simulation when the power ratio equals 2, the

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power drawn from the first source (P1 ) equals to 67.5 W, and the power drawn from the second source equals to 33.75 W. On the other hand, when the power ratio is stepped down to 0.5, the power drawn from the first source is decreased to 33.75 W, while the power drawn from the second source is increased to 67.5 W. Fig. 11(a) shows the experimental results equivalent to simulation results shown in Fig. 11(c). The experimental and simulation results match closely except that in the former, the amount of power drawn from the two sources is slightly more than that of the latter. In the designed converter, in contrast to the simulation, the input power should compensate for the losses (switching losses, copper loss, magnetic loss, etc.), in addition to supplying the required output power; therefore, the input power required by the converter to supply 101.25 W of power to the load is more than 101.25 W. Although the output voltage undershoot shown in Fig. 11(b) and (d) seems large, it is justifiable because the step change in the power ratio is rather large. VI. CONCLUSION A nonlinear power sharing controller for a double-input buckboost–buckboost converter was designed in this paper. First, the buckboost–buckboost converter’s principles of operation were reviewed, and the nonlinear model of the converter was derived based on the converter state equations. Then, the controller was designed for the nonlinear model of the converter based on the Lyapunov theory. The designed controller was implemented on a prototype buckboost–buckboost converter, and sufficient simulation and experimental results were provided to verify the design. REFERENCES [1] H. Matsuo, W. Lin, F. Kurokawa, T. Shigemizu, and N. Watanabe, “Characteristics of the multiple-input dc-dc converter,” IEEE Trans. Ind. Electron., vol. 51, no. 3, pp. 625–631, Jun. 2004. [2] K. P. Yalamanchili and M. Ferdowsi, “Review of multiple input dc-dc converters for electric and hybrid vehicles,” in Proc IEEE Vehicle Power Propulsion Conf., Sep. 2005, pp. 160–163. [3] N. D. Benavides and P. L. Chapman, “Power budgeting of a multipleinput buck-boost converter,” IEEE Trans. Power Electron., vol. 20, no. 6, pp. 1303–1309, Nov. 2005. [4] R. Ahmadi and M. Ferdowsi, “Double-input converters based on h-bridge cells: Derivation, small-signal modeling, and power sharing analysis,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 59, no. 4, pp. 875–888, Apr. 2012. [5] D. Somayajula and M. Ferdowsi, “Power sharing in a double-input buckboost converter using offset time control,” in Proc. IEEE Appl. Power Electron. Conf., Feb. 2009, pp. 1091–1096. [6] R. Ahmadi, N. Yousefpoor, and M. Ferdowsi, “Power sharing analysis of double-input converters based on h-bridge cells,” in Proc. Electric Ship Technol. Symp., Apr. 2011, pp. 111–114. [7] V. Prabhala, D. Somayajula, and M. Ferdowsi, “Power sharing in a doubleinput buck converter using dead-time control,” in Proc. Energy Convers. Congr. Expo., Sep. 2009, pp. 2621–2626. [8] R. Ahmadi, H. Zargarzadeh, and M. Ferdowsi, “Nonlinear power sharing controller for double-input h-bridge based converters,” in Proc. IEEE Appl. Power Electron. Conf., Feb. 2012, pp. 200–206. [9] B. G. Dobbs and P. L. Chapman, “A multiple-input dc-dc converter topology,” IEEE Trans. Power Electron. Lett., vol. 1, no. 1, pp. 6–9, Mar. 2003. [10] H. Behjati and A. Davoudi, “A multi-port dc-dc converter with independent outputs for vehicular applications,” in Proc. IEEE Vehicle Power Propulsion Conf., Sep. 2011, pp. 1–5. [11] H. Behjati and A. Davoudi, “A mimo topology with series outputs: An interface between diversified energy sources and diode-clamped multilevel inverter,” in Proc. IEEE Appl. Power Electron. Conf., Feb. 2012, pp. 1–6.

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[12] Y. M. Chen, Y. C. Liu, S. C. Hung, and C. S. Cheng, “Multi-input inverter for grid-connected hybrid pv/wind power system,” IEEE Trans. Power Electron., vol. 22, no. 3, pp. 1070–1077, May 2007. [13] Y. M. Chen, Y. C. Liu, and F. Y. Wu, “Multi-input dc/dc converter based on the multiwinding transformer for renewable energy applications,” IEEE Trans. Ind. Appl., vol. 38, no. 4, pp. 1096–1104, Aug. 2002. [14] Y. M. Chen, Y. C. Liu, F. Y. Wu, and Y. E. Wu, “Multi-input converter with power factor correction and maximum power point tracking features,” in Proc. IEEE Appl. Power Electron. Conf., Aug. 2002, vol. 1, pp. 490–496. [15] A. Di Napoli, F. Crescimbini, S. Rodo, and L. Solero, “Multiple input dc-dc power converter for fuel-cell powered hybrid vehicles,” in Proc. IEEE Power Electron. Spec. Conf., Nov. 2002, vol. 4, pp. 1685–1690. [16] K. Gummi and M. Ferdowsi, “Synthesis of double-input dc-dc converters using a single-pole triple-throw switch as a building block,” in Proc. IEEE Power Electron. Spec. Conf., Jun. 2008, pp. 2819–2823. [17] K. Gummi and M. Ferdowsi, “Derivation of new double-input dc-dc converters using h-bridge cells as building blocks,” in Proc. IEEE Ind. Electron. Conf., Nov. 2008, pp. 2806–2811. [18] K. Kobayashi, H. Matsuo, and Y. Sekine, “Novel solar-cell power supply system using a multiple-input dc-dc converter,” IEEE Trans. Ind. Electron., vol. 53, pp. 281–286, Feb. 2006. [19] A. Kwasinski, “Identification of feasible topologies for multiple-input dcdc converters,” IEEE Trans. Power Electron., vol. 24, no. 3, pp. 856–861, Mar. 2009. [20] Y. Li, X. Ruan, D. Yang, F. Liu, and C. K. Tse, “Synthesis of multipleinput dc/dc converters,” IEEE Trans. Power Electron., vol. 25, no. 9, pp. 2372–2385, Sep. 2010. [21] Y. Li, D. Yang, and X. Ruan, “A systematic method for generating multiple-input dc/dc converters,” in Proc. IEEE Vehicle Power Propulsion Conf., Sep. 2008, pp. 1–6. [22] C. Liu, K. Ding, J. R. Young, and J. F. Beutler, “A systematic method for the stability analysis of multiple-output converters,” IEEE Trans. Power Electron., vol. 2, no. 4, pp. 343–353, Oct. 1987. [23] D. Liu and L. Hui, “A zvs bi-directional dc-dc converter for multiple energy storage elements,” IEEE Trans. Power Electron., vol. 21, no. 5, pp. 1513–1517, Sep. 2006. [24] Y. C. Liu and Y. M. Chen, “A systematic approach to synthesizing multiinput dc-dc converters,” IEEE Trans. Power Electron., vol. 24, no. 1, pp. 116–127, Jan. 2009. [25] M. Rodriguez, P. Fernandez-Miaja, A. Rodriguez, and J. Sebastian, “A multiple-input digitally controlled buck converter for envelope tracking applications in radiofrequency power amplifiers,” IEEE Trans. Power Electron., vol. 25, no. 2, pp. 369–381, Feb. 2010. [26] L. Solero, A. Lidozzi, and J. A. Pomilio, “Design of multiple-input power converter for hybrid vehicles,” IEEE Trans. Ind. Electron., vol. 20, no. 5, pp. 1007–1016, Sep. 2005. [27] H. Tao, A. Kotsopoulos, J. L. Duarte, and M. A. M. Hendrix, “Family of multiport bidirectional dc-dc converters,” Electric Power Appl., vol. 153, pp. 451–458, May 2006. [28] Q. Wang, J. Zhang, X. Ruan, and K. Jin, “Isolated single primary winding multiple-input converters,” IEEE Trans. Power Electron., vol. 26, no. 12, pp. 3435–3442, Dec. 2011. [29] K. P. Yalamanchili, M. Ferdowsi, and K. Corzine, “New double input dcdc converters for automotive applications,” in Proc. IEEE Vehicle Power Propulsion Conf., Sep. 2006, pp. 1–6. [30] C. Onwuchekwa and A. Kwasinski, “A modified-time-sharing switching technique for multiple-input dc-dc converters,” IEEE Trans. Power Electron., vol. 27, no. 11, pp. 4492–4502, Nov. 2012. [31] Z. Ouyang, Z. Zhang, M. A. E. Andersen, and O. C. Thomsen, “Four quadrants integrated transformers for dual-input isolated dc-dc converters,” IEEE Trans. Power Electron., vol. 27, no. 6, pp. 2697–2702, Jun. 2012. [32] K. Gummi and M. Ferdowsi, “Double-input dc-dc power electronic converters for electric-drive vehicles-topology exploration and synthesis using a single-pole triple-throw switch,” IEEE Trans. Ind. Electron., vol. 57, no. 2, pp. 617–623, Feb. 2010. [33] R. Ahmadi and M. Ferdowsi, “Canonical small-signal model of doubleinput converters based on h-bridge cells,” in Proc. IEEE Energy Convers. Congr. Expo., Sep. 2011, pp. 3946–3953. [34] D. Somayajula and M. Ferdowsi, “Small-signal modeling and analysis of the double-input buckboost converter,” in Proc. IEEE Appl. Power Electron. Conf., Feb. 2010, pp. 2111–2115.

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Reza Ahmadi received the B.S. degree in electrical engineering from the Iran University of Science and Technology, Tehran, Iran, in 2009. He is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla. He is also a Graduate Research Assistant at the Missouri University of Science and Technology. His research interests include modeling, design and control of power electronic converters, electric-drive vehicles, and multi-input power converters.

Hassan Zargarzadeh (S’09) received the B.S. degree from Tehran Polytechnic, Tehran, Iran, in 2001, the M.S. degree from the Iran University of Science and Technology, Tehran, in 2009, both in electrical engineering, and the Ph.D. degree from the Missouri University of Science and Technology, Rolla, in August 2012. He is currently an Instructor in the Department of Industrial and Engineering Technology, Southeast State Missouri University, Cape Girardeau. His research interests include nonlinear, adaptive, and optimal control systems.

Mehdi Ferdowsi (S’02–M’04) received the B.S. degree in electronics from the University of Tehran, Tehran, Iran in 1996, the M.S. degree in electronics from the Sharif University of Technology, Tehran, in 1999, and the Ph.D. degree in electrical engineering from the Illinois Institute of Technology, Chicago, in 2004. He is currently an Associate Professor in the Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla. His research interests include electric-drive vehicles, multi-input power converters, multilevel power converters, and battery charge equalization. Dr. Ferdowsi received the NSF CAREER Award in 2007. He is an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS.