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

Nonlinear Power Sharing Controller for Double-Input H-Bridge Based Converters Reza Ahmadi, Hassan Zargarzadeh, and Mehdi Ferdowsi Electrical and Computer Engineering Department Missouri University of Science and technology Rolla, MO, USA {rahd6, hzr47, ferdowsi}@mst.edu Abstract—This work designs a nonlinear controller with power sharing control capability for a double input buck-buckboost converter. First it reviews the principles of operation of the buck-buckboost converter and finds the nonlinear model of this converter. Next, it proposes a Lyapunov based nonlinear controller which is adaptive against input voltage and load disturbances. Finally, it provides sufficient experimental results to verify the operation of the designed controller.

I.

INTRODUCTION

Harnessing renewable sources of energy such as wind and solar has gained much attention lately, resulting to extensive research and development on new power electronic solutions. Particularly, multi-port converters are amongst the most anticipated power electronic devices to become the standard mean of combining energy from different sources with intermittent nature [1-7]. As a result, several classes of multiport converters have been proposed and studied in the literature lately [8, 9], but there has been less attention to controller design methods for this type of converters. Generally, the conventional design methods rely on a small-signal model of the dc-dc converter linearized about a specific operating point for the control design [10]. This linearized small-signal model is only valid when the converter is operating around that specific operating point and becomes less reliable in transients or when it is desired for the converter to operate around a new operating point. On the other hand, the power sharing capability of multi-port converters, which is the ability of the multi-port converters to vary the ratio of the power drawn from each input source, makes the operating point of these converters variable, thus, making it very difficult to control a multi-port converter with a conventional linear controller. The aim of this work is to design a nonlinear controller, independent of the operating point of the converter, for a double-input buck-buckboost converter. In addition to normal functions of conventional controllers such as line and load regulation and disturbance rejection, this controller has the capability of handling the power sharing mechanism, which means shifting the operating point of the converter in order to vary the amount of power drawn from each input source while keeping the converter stable. The designed controller is simple enough that can be implemented using a typical digital signal processor.

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Fig. 1. The buck-buckboost converter circuit diagram.

II. BUCK-BUCKBOOST CONVERTER The circuit diagram of the buck-buckboost converter is shown in Fig. 1. The principles of operation of this converter are studied in detail in [11] and are reviewed in here to give a deeper insight about the proposed control method. The buckbuckboost converter is a multi-port (double-input, singleoutput) converter with two input sources and , two controllable switches and , and one output port. The two controllable switches, and can be turned ON and OFF independently resulting to four modes of operation for this converter. The four modes of operation of this converter are shown in Fig. 2. In mode 1 (Fig. 2(a)), only is ON and energizes the inductor, in mode 2 (Fig. 2(b)), both and are ON and both sources energize the inductor, in mode 3 (Fig. 2(c)), only is ON and energizes the inductor, and finally in mode 4 (Fig. 2(d)), both switches are OFF and the inductor is in discharging cycle. The general switching pattern of and is shown in Fig. 3. In Fig. 3, the duty cycle of is labeled as , the duty , and the period in which both cycle of is labeled as and are ON is labeled as (with being the switching period). The duration of each mode ( , , , and ) in terms of duty cycles and switching period can then be expressed as,

200

0

0 0

0

(a) (b) (c) (d) Fig. 2. Four modes of operation of the buck-buckboost converter. The inductor voltage, capacitor current and input currents in each mode are shown based on the circuit parameters.

t t t

t t 1

d3(t)T d12(t)T

(1) t

t

t

S1

The duty cycles are generally considered time variant but for given constant duty cycles , , the equilibrium values of the output voltage and the inductor current of this converter are found in [11], 1 1 1

(1)

1

(2) (3) (4)

(1)

(2) (3) (4)

(1)

(2) (3) (4)

d2(t)T S2

(2)

1

d1(t)T

(3)

(1)

(2) (3) (4)

T1

T2

T3

T4

T

For the same constant duty cycles, the amount of power drawn from each input source is calculated in [11] as well, (4) (5) , are In [11] it is shown that when the input voltages is intended to remain fixed and the output voltage constant in order to deliver constant power to the load (in case of a constant load), there are infinite number of possible pairs of duty cycles ( and ) which can be set in (2) to generate the same output voltage. Thus, in contrary to classical single input converters where a specific output voltage can be generated only with a certain value of duty cycle, in doubleinput converters, a specific output voltage can be generated with different arrangements of the duty cycles , .

Fig. 3. The general switching pattern of and . Switches are ON when voltage is high and OFF when the voltage is low (zero).

On the other hand, (4) and (5) suggest that in case of fixed input voltages , and constant output voltage and thus constant inductor current, the ratio of the power drawn from the two sources (defined later on as ) depends on the ratio of the duty cycles of switches and . This last statement reveals the key ability of the buck-buckboost converter in power sharing management: in a buck-buckboost 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 by changing the ratio of duty cycles. Nonetheless, it is worth reminding that the amount of the output power is always equal to the sum of the powers drawn from the two sources. III. NONLINEAR CONVERTER MODELING The nonlinear model of the buck-buckboost converter can be developed by averaging inductor voltage and

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capacitor current over a switching period. The average of any converter signal over one switching period can be found by the evaluation of the following equation, 1

.

(6)

denotes average of over switching period Where T. Therefore, the average of the inductor voltage over one switching period can be found by the evaluation of (6) for the inductor voltage during 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 (6), the basic approximation of removing the high-frequency switching ripple from the inductor voltage should be carried-out first. The inductor voltage and capacitor current equations for each mode of operation of the converter are shown in Fig. 2. To carry-out the high-frequency switching ripple elimination, the instantaneous values of the capacitor voltage and inductor current ( and ) are replaced with their low-frequency components , . Consequently, the equations shown in Fig. 2 for the four modes of operation are transformed to the following equations respectively,

Using the inductor voltage equations in (7)-(10) which give the slow-varying component of the inductor voltage in each mode of operation and the duration of each mode from (1), it is possible to evaluate (6) to find the average inductor voltage: 1

(11)

Substituting (1) in (11) and simplifying the resulting equation gives the final result, 1

(12)

Similarly, the average capacitor current can be found by evaluation of (6) for the capacitor current signal, 1

(13)

Equations (12) and (13) form the nonlinear model of the buckbuckboost converter. The average current drawn from each source can be found likewise by the evaluation of (6) for the input current equations in (7)-(10),

(7) 0

(14) (15) (8)

Thus the average power drawn from each source is equal to, (16) (17)

(9)

IV. NONLINEAR CONTROLLER DESIGN

0

This section carries out the design of a nonlinear control scheme for the converter model described in (12) and (13) which is applicable to an experimental test bed. The aim is to design a nonlinear controller with the following specifications: (10) 0 0

• •

202

It is able to globally asymptotically regulate the output, meaning that it can navigate the output voltage from any initial condition to any desired output voltage. It is robust against the disturbances, particularly the variations of the input voltages and load.

• •

It is able to adjust the ratio of the power drawn from the sources to a desired value i.e. (Power sharing control capability). It is able to produce a restricted switching ratio such that, 0

,

1.

(18)

Therefore, no saturation block is needed to restict duty cycles in the actual controller. A. Open-Loop Controller The controller design procedure begins with finding the equilibrium point of the system. The equilibrium point of the system can be found by setting the left hand side of (12) and (13) equal to zero,

1

1

1 1

1

,

With this terminology by substituting (25) and (26) in (16) and (17) the average power drawn from each source can be formulated as, (27) (28) By denoting,

(22) (23)

and 1

(26)

(29) as the desired ratio of the power drawn from the sources, γ can be found from (27) and (28), (30) Since the desired output voltage should coincide with the equilibrium point of the capacitor voltage , in (19) we set, .

(24)

can be changed over time while the sum of the terms and therefore, the equlibrium values in the left hand side of (19) and (20) remain constant. The converter model in (12) and (13) is an over actuated , and one system because it has two inputs output . The aim is to regulate the output voltage while

(31)

Then by substituting (25) and (26) in the resulting equation, the general solution for is found,

(21) ,

,

and

(19)

represents the equilibrium point value of In (19) and (20), the capacitor voltage (which is equal to the output voltage) and represents the equilibrium point value of the inductor current. Equations (19) and (20) are similar to (2) and (3) respectively except that in (19) and (20) the duty cycles are considered to be time variant. As it was mentioned in section II, a particular value of the output voltage of the buckbuckboost converter can be generated using different pairs of the duty cycles; therefore, the output voltage can be kept constant while the dutycycles are varying along a path which keeps the output voltage constant. As a result, the duty cycles and therefore the summing terms in the right hand side of (19) and (20) which are

1

(25)

(20)

1

1

the ratio of the power drawn from the two sources is equal to a desired value of λ. If we choose

(32) Equation (32) along with (25) and (26) yield an open-loop solution of the output voltage control, because for a desired power ratio , can be found from (30), and therefore, is found from (32) and the required duty cycles are found from (25) and (26). Varying the power ratio can be carried out by varying in (32) and calculating the new duty cycles from (25) and (26). In this situation, the output voltage remains constant equal to the desired output voltage. B. Closed-Loop Controller As it was mentioned before, (32), (25), and (26) yield an open-loop controller for the converter. However, the problem arises when the circuit parameters such as input voltages or load vary due to the disturbance. In fact, it is more desired to have a closed-loop controller whose command is a function of the output error. In order to work out (32) to the desired closed-loop controller, in the first step, we need to show that

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the equilibrium point of the system is globally asymptotically stable. Lemma (the equilibrium point stability) - The equilibrium point of (19) and (20) is globally asymptotically stable. In other words, for any input , 0,1 , the converter output voltage globally asymptotically converges to a particular voltage of . Proof - Consider the following positive definite and radially unbounded Lyapunov function [12], 1 1 (33) 2 2 Taking the derivative along the system dynamics gives the following result,

(34) As it was mentioned above, and are constant, therefore, no differentiation is necessary on them. Substituting (12) and (13) in the generated derivative terms and simplifying the results using (19) and (20) yields, 1

2

.

Ψ t

Ψ 0

(37)

1

The two parameters , 0 are convergence speed parameters. Proof - Consider the following positive definite radially unbounded Lyapunov function, Ψ

2

Ψ

(38)

By taking the derivative along the system dynamics and using the result of the Lemma for from (35) we have, 1 Substituting (26) yields,

ΨΨ

(39)

from (19) into (39) and considering (25) and

1

ΨΨ

1

Since we defined Ψ t simplified to,

(35)

The result is a negative semi-definite function which implies that the system is stable in the sense of Lyapunov and therefore convergers to by evolution of time. On the other hand, with , (13) has only one solution, i.e., . Therefore, using the Barbalat’s lemma [12], 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 , while disturbances in , , and the load are unknown but slow varying ( 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 is estimated and updated the value of Ψ t by the closed-loop controller as the system evolves. The initial value of Ψ t is set to Ψ 0 where and are the nominal values of the two input voltages without disturbances. Theorem (adaptive nonlinear controller) - Assume that it is desired to regulate the output voltage to a desired voltage . Assume that Ψ t is unknown and is estimated by Ψ t . The following controller globally asymptotically stabilizes the estimation and output errors (Ψ Ψ t respectively). Ψ t and

2

, (40) can be

1

Ψ t

1 Using the fact that Ψ

(40)

Ψ

ΨΨ

(41)

Ψ, (41) can be rewritten as,

1

Ψ t

1

(42) Ψ t

1

Ψ

Ψ t

By choosing, (43)

Ψ

1

in (42), it can be rewritten as, 1

Ψ

1 2 1

Ψ

(44)

ΨΨ which can be simplified by choosing,

Ψ t

(36)

Where Ψ t is found in each step by evolution of the following integral,

Ψ

2

Substituting (45) in (44) yields,

204

1

(45)

1

Ψ

1

TABLE I POWER STAGE PARAMETERS OF THE DESIGNED CONVERTER

(46)

Parameter

Which is a negative definite function that shows the system is stable in the sense of Lyapunov. As it was shown in the above steps, in order to make a negative definite function as of (46), the two key equations (43) and (45) must hold, therefore these two equations form the nonlinear controller of the system. If (43) is manipulated such that is found based on the rest of the parameters, we will have, (47)

Value 24 36 50 30Ω 320 100

which is the same as (36) in the theorem. On the other hand solving (45) for Ψ t yields the following result,

the input power supplied by each source. The input power supplied by each source can be calculated from the average input currents using (16) and (17) by multiplying the value of the current drawn from the source by its voltage,

(48)

(49)

Ψ t

Ψ t

Ψ 0

2

1

which is the same as (37) in the theorem. In other words, choosing and Ψ t as (36) and (37) guarantees that the system is stable in the sense of Lyapunov, therefore (36) and (37) form the closed-loop controller. To apply (36) and (37) on the system, in each sampling is measured and Ψ t is calculated time, output voltage from (37) using the from the previous step. Then, the new is calculated by substituting the resulting Ψ t and into (36). Finally the values of the new the measured duty cycles are calculated form (25) and (26). V. EXPERIMENTAL RESULTS To validate the theoretical outcomes, the designed controller is employed to control a prototype double-input buck-buckboost converter. The power stage parameters of the buck-buckboost converter are listed in Table I. The controller is implemented using a digital signal processor (DSP). The DSP reads in the output voltage from the built in analog to digital converter (ADC). Knowing the current value of the output voltage and the reference voltage to be tracked, (36) and (37) are evaluated by the DSP code and the current and Ψ are found. Finally and are found form (25) and (26) based on the specified . The resulting duty cycles are commanded to the switches through the pulse width modulator (PWM) unit of the DSP. Several experiments are carried out on the closed-loop converter in order to inspect the performance of the proposed controller for line regulation, load regulation, power sharing management, and output voltage tracking capabilities. The results of the experiments are shown in Fig. 4. In this figure each scope shot illustrates the dynamic response of the closed-loop converter in case of a step change in one of the parameters. In each scope shot, the top blue trace is the average input current drawn from the first source, the top red trace is the average input current drawn from the second source, and the bottom trace is the output voltage. The input currents are plotted in order to calculate

(50) Figure 4(a) features the power sharing management capability of the proposed controller. To generate this scope shot, , the ratio of the power drawn from the two sources is stepped-down from 2 to 0.5 so that the amount of the power drawn from the first source is decreased by half and the amount of power drawn from the second source is increased by two times. The current wave forms clearly show that the current supplied by the first source (the blue trace) has decreased by half and the current drawn from the second source (the red trace) is increased by two times. The output voltage waveform (the bottom trace) suggests that the transient in the output voltage is very fast with minimal undershoot/overshoots. Figure 4(b) presents the load regulation of the closed-loop converter. To generate this scope shot, the load is stepped down from 30Ω to 25Ω. The current wave forms show that the input currents have increased and the output voltage wave form suggests negligible voltage transient in this situation. Figure 4(c) shows the line regulation of the closed-loop converter. To generate this scope shot, is stepped down from 25V to zero volts. The output voltage wave form shows acceptable transients, especially with such a huge step change in the input voltage. The current drawn from the second source shows large oscillations which is mainly because of the large step change in . Figure 4(d) shows the output voltage command tracking of the proposed controller. To generate this scope shot, the output voltage is commanded to step-up from 50V to 60V. The output voltage wave form shows fast response and small overshoot.

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(a)

(b)

(c) (d) Fig. 4. Experimental Results; in each scope shot the top blue trace is the input current drawn from the first source which is proportional to the power drawn from the first source ( ), the red trace is the input current drawn from the second source which is proportional to the power drawn from the second source ( ), and the bottom dark blue trace is the output voltage. (a) Power sharing action; is stepped down from 2 to 0.5, (b) Load regulation; the load is stepped down from 30Ω to 25Ω, (c) Line regulation; is stepped down from 25V to zero, and (d) The output voltage is stepped up from 50V to 60V.

[6]

VI. CONCLUSION A nonlinear power sharing controller for a double input buck-buckboost converter was designed in this work. First, the principles of operation of the buck-buckboost converter were reviewed and its nonlinear model was derived based on the converter operating modes. Then, the nonlinear controller was designed for the nonlinear model of the converter based on the Lyapunov theory. The designed controller was implemented on a prototype buck-buckboost converter and sufficient experimental results were provided to verify the design. REFERENCES [1] [2] [3] [4] [5]

K. P. Yalamanchili and M. Ferdowsi, "Review of multiple input dc-dc converters for electric and hybrid vehicles," in Proc. VPPC'05, Sept. 2005, pp. 160-163. K. P. Yalamanchili, M. Ferdowsi, and K. Corzine, "New double input dc-dc converters for automotive applications," in Proc. VPPC'06, Sept. 2006, pp. 1-6. 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. PESC'08, June 2008, pp. 2819-2823. K. Gummi and M. Ferdowsi, "Derivation of new double-input dc-dc converters using h-bridge cells as building blocks," in Proc. IECON'08, Nov. 2008, pp. 2806-2811. B. G. Dobbs and P. L. Chapman, "A multiple-input dc-dc converter topology," IEEE Trans. Power Electron. Lett., vol. 1, pp. 6-9, March 2003.

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, pp. 625-631, June 2004. [7] H. Li, Z. Du, K. Wang, L. M. Tolbert, and D. Liu, "A hybrid energy system using cascaded h-bridge converter," in Proc. Industry Applications Conference'07, Oct. 2006, vol. 1, pp. 198-203. [8] Y. Li, X. Ruan, D. Yang, F. Liu, and T. C. K., "Synthesis of multipleinput dc/dc converters," IEEE Trans. Power Electron., vol. 25, pp. 2372-2385, Sept. 2010. [9] Y. Li, D. Yang, and X. Ruan, "A systematic method for generating multiple-input dc/dc converters," in Proc. VPPC'08, Sept. 2008, pp. 16. [10] B. Bryant and M. K. Kazimierczuk, "Modeling the closed-current loop of pwm boost dc-dc converters operating in ccm with peak currentmode control," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, pp. 2404-2412, Nov. 2005. [11] R. Ahmadi and M. Ferdowsi, "Double-input converters based on hbridge cells: Derivation, small-signal modeling, and power sharing analysis," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. PP, pp. 1-14, 2011. [12] H. K. Khalil, Nonlinear systems, 3rd ed. New Jersey: Prentice Hall, 2001.

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