On the Maximum Efficiency of Systems Containing Multiple Sources

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On the Maximum Efficiency of Systems Containing Multiple Sources Yoash Levron and Doron Shmilovitz

Abstract—This paper deals with the efficiency of systems consisting of multiple power sources. The operating point ensuring maximum efficiency is defined and solved analytically. Moreover, it is shown that, in the case of linear sources, the maximum efficiency is determined solely by the load power and the network’s maximum power, independently of the sources’ internal construction and their connectivity. The generic circuit theory treatment that is proposed applies to distributed power generation (smart grid, photovoltaic, fuel cells, etc.) and to distributed load systems (LED arrays, microprocessors). Index Terms—DC bus, distributed power system, power conservative circuit.

I. INTRODUCTION

I

N fueled power sources such as batteries, fuel cells and motor driven generators, there is a cost associated with energy loss, due to factors such as fuel cost in generators and reduced operation time for battery-powered systems. In addition to fuel savings, high efficiency systems have the advantage of being able to provide the same output power with less or smaller hardware (for instance, due to reduced cooling effort). So it is desirable to minimize energy consumption of fueled types of power sources. Since most practical loads demand a certain amount of power for adequate operation, an appropriate system design should establish an operating point at which maximum efficiency is attained while supplying the power demanded by the load. A reasonable criterion for designing a power system with multiple sources is that the system’s operating point supplies sufficient power to the load while maintaining the highest possible overall efficiency. This unique operating point is defined as the maximum efficiency point. Similar to the operation of solar cells at the maximum power point (MPP), the maximum efficiency point is regarded as the optimum operating point for fueled power sources. Thus, maximum efficiency, constrained by a given power output, is the desired goal for fueled systems, as opposed to maximum power in sustainable power systems. Systems that contain a single power source and the operational conditions needed for maximum efficiency or maximum power transfer have been widely explored. The matter of increasing power transfer is often referred to as ‘impedance

Manuscript received March 02, 2009; revised July 31, 2009; accepted November 11, 2009. Date of publication March 01, 2010; date of current version August 11, 2010.This paper was recommended by Asssociate Editor C. K. Tse. The authors are with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2009.2037846

matching’ and may also be extended to non-linear systems such as photovoltaic systems (where power converters are used to interface the non-linear photovoltaic source to the load while operating the photovoltaic power source at its maximum power point—MPP [1]–[4]). Assuming that the source and the load characteristics are fixed, a two port matching element is required in order to give the desired operating point, similar to impedance matching transformers in audio amplifiers. High efficiency power converters have been shown to be able to provide such matching [5]–[10]. This is due to the transformer-like average characteristics of high efficiency power converters which implies energy conservation, along with the fact that their output voltage is proportional to their input voltage and vice versa, i.e., their input current is proportional to their output current [8]–[10]. Indeed, the dc transformer is widely accepted for modeling dc-dc converters (for purposes such as conduction loss calculations and the modeling of dynamics [10], [11]). Some cases require large variations of the converter’s transfer ratio, such as in photovoltaic systems where matching is applied in order to extract maximum power under varying atmospheric conditions and load variations (known as maximum power tracking [1], [2]). Other examples of systems with highly variable transfer ratios include active power factor correction (APFC) and PWM inverters. In addition to the wide range of variation, these cases present challenges with modeling the large signal behavior and with difficult control schemes that ensure stability [12], [13]. It should be noted that other two-port power conserving controllable elements such as gyrators [10], [14]–[16], can also be used for impedance matching. In contrast to well known theories concerning the interaction between power sources and loads in single source systems, the literature regarding complex systems involving a plurality of power sources and loads is quite limited. The study of such systems becomes important todays due to the proliferation of concepts for distributed power systems and distributed generation driven by future applications of smart grid, distributed power generation (photovoltaic, fuel cells, etc.) and distributed load systems (LED arrays [17], microprocessors, etc.). An analysis that enables designers to find the maximum efficiency point of multiple source systems is provided and elaborated through an example. Moreover, we show that for linear or quasi-linear sources, the maximum attainable efficiency of any network is solely defined by its maximum power output, and is independent of the source array configuration or internal parameters. The theory presented in this paper is a direct extension of ‘power processing’ concepts, proposed by Singer and Erickson [18], [19].

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The LaGrange multiplier method is applied to solve for the power of every source. Using this method, a set of equations defines which solution is the minimum point

Since is constant and equation is simplified to

are independent variables, this

(2) Fig. 1. A general power-conditioning network, in which each power source is interfaced through a power converter.

II. MAXIMUM EFFICIENCY OF MULTIPLE POWER-SOURCE ENERGIZED NETWORKS We will show that a power-conditioning network in which each power source is interfaced by a power converter (Fig. 1) can operate at maximum efficiency for any given power demand. A. Finding the Operating Point for Maximum Overall Efficiency Consider an array of N sources. We use the following notation: —the power output of the th converter; —the efficiency of the th source and converter, as a function of power. An operating point is defined by the power generated by each source and cascading converter, . Knowing the powers and , allows the exact voltages corresponding efficiencies, and currents of every source to be calculated. For sources that may deliver the same output power with two different efficienis chosen as the branch of higher efficiency. The cies, total power is defined as the sum of source powers

The overall efficiency is defined as

These constitute equations with variables: the solutions and . are the powers at the operating points of . maximum overall efficiency— There are three possible cases when attempting to solve (2). • There is only one solution for the equation. In this case, the solution is the operating point for maximum efficiency. • There is more than one solution. These solutions correspond to operating points for minimum and maximum efficiency. In this case, the solution with the highest efficiency should be selected. • There is no solution. When there is no solution, the maximum efficiency is achieved at the boundary of some of the variables; that is, for at or . The interpretation least one of the sources is that at least one source is either so inefficient that it is not : the other sources can supply the power worth using it demand without using this source at all. Alternatively, the source is always more efficient than all the other sources, so . it should constantly supply its maximum power In both cases, a solution can be obtained by omitting that source from the set of (2), and then solving the reduced set of equations. It should be noted that the formal conditions for the existence and uniqueness of solutions are more complex. Systems with many or no solution are generally ill-behaved, and will not normally occur in practical systems that are designed to achieve best performance at minimum cost. B. Maximum Efficiency of Two Fuel Cell Stacks—Example

(1) In terms of the system, minimization of the primitive energy consumption while supplying the load demand implies opera, under the constraint of tion at maximum overall efficiency being delivered to the load. total power is regarded as a constant for this optimization problem. So, maximum efficiency can be found by minimizing the denominator

We demonstrate how to choose the point for maximum efficiency in two different fuel cell stacks. Two power processing schemes are compared. The first scheme consists of two stacks connected in series. The stacks are matched to the load by a single converter. In the second scheme, two converters match each stack to its optimum operating point. The second scheme can achieve a higher efficiency than the first scheme, while delivering the same amount of power to the load. In addition to the inherent losses of the stacks, we consider the converter losses and the losses in the pumps used to supply hydrogen to the stacks. We use the model of a polymer electrolyte membrane (PEM) fuel cell described by Kulikovsky [20]. The voltage-to-current density ( - ) curve of the cell is given in Fig. 2(a). It is important

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Fig. 3. Fuel cell stacks.

consists of 50 fuel cells connected in series. The stacks are made up of cells with different cross-sectional areas. cm and the The cross-sectional areas of stack 1 is cm . The stacks are cross section area of stack 2 is illustrated in Fig. 3. The stack parameters are summarized in Table I. The characteristic equations of the fuel cell stacks are given by (4) [22].

Fig. 2. Fuel cell’s characteristics. (a) Voltage versus current density. (b) Efficiency versus current density.

(4) to note that we have no simple equation with which we can extrapolate the curve so all calculations are done numerically. According to [21], [22] and [23], the fuel cell’s efficiency can be approximated by

where: output voltages of the stacks [Volt]; currents of the stacks [Amp];

(3)

power outputs of the stacks [Watt]; efficiencies of the stacks;

where cell’s efficiency, defined as the ratio of electrical output power to potential power stored in the cell by ingoing hydrogen; output voltage; open-circuit voltage ;

. For this fuel cell

fuel utilization coefficient. We choose a common . For simplicity, we assume value of is constant, hence it is not affected by that changes in or ; sum of internal current losses. We choose mA/cm . The efficiency versus current density curve is shown in Fig. 2(b). We use this basic cell to build two stacks. Each stack

actually , the voltage-to-current density function. The voltage-current curves and efficiency-power curves are evaluated according to (4) and depicted in Fig. 4. The power converters’ efficiency depends on its power throughput. We approximate this dependence by a third order polynomial, given by

(5) where is the converter’s output power, is its efficiency, is its nominal power (which typically corresponds to and the point of maximum efficiency). This power-efficiency curve

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Fig. 5. Converters’ efficiency versus power (assuming the efficiency peaks at the nominal power).

TABLE II NOMINAL CONVERTER POWERS

a power of powers is given by Fig. 4. Fuel cell stacks characteristics. (a) Voltage versus current. (b) Efficiency versus power.

TABLE I STACK PARAMETERS

W. Thus, the sum of stack

(6) The series connection results in an equal current through each stack (7) Combining (6) and (7), and substituting the basic stack equations (4), results in a single variable, , with dependence (8)

is shown in Fig. 5. With no loss of generality, we assume similar characteristics for both converters, except for their nominal power. The nominal power of each converter is given in Table II. 1) Maximum Efficiency of Scheme 1—Series Connection With a Single Converter: In scheme 1 (Fig. 6(a)), both stacks are connected in series, feeding a single converter. are the output powers of the fuel cell stacks. is the power absorbed by the load and is also the converter’s output is the converter’s efficiency, given by (5), where power. W. We assume that the pumps require a constant W for their operation. The load requires power of

The solution of this equation, obtained numerically, defines and the operating points of both stacks, see Fig. 7. 2) Maximum Efficiency of Scheme 2—Optimal Network Employing Two Converters: In scheme 2, Fig. 6(b), each stack is matched by a converter. The converters’ outputs are connected and are the output powers in parallel to feed the load. and are the converters’ output powers. of the two stacks. is the power consumed by the load. and are the efficiencies of the fuel cell stacks. and are the converters’ efficiencies, given by (5). The nominal converters’ powers are listed in Table II. The pumps W. The load requires a require a constant power of W. The fuel cell stack and cascading conpower of verter are viewed as a joint enhanced source. Thus, the power

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Fig. 7. Stacks’ operating points for P = 420 W, for the series connected stacks and the optimally connected stacks. (a) Operating points on the voltagecurrent curve (b) Operating points on the power-efficiency curve.

Fig. 6. Two power schemes. (a) Scheme 1: Series connection of the stacks to a single converter. (b) Scheme 2: Optimal network where each stack is connected to its own converter.

We assume that the load requires a power of W and solve the general equations for maximum efficiency, given by (2). Here it is written in explicit form

output of each enhanced source is the power output of its converter: or respectively. The joint efficiency of each source and . By definition is denoted by

(9)

where and are determined by the ingoing hydrogen flow rate and its potential chemical energy. The converters’ and stacks’ efficiencies are defined by the ratio of powers

This set of equations contains three equations and three vari, and in this case has one unique solution, which ables defines the operating point for maximum overall efficiency. The set of equations is solved numerically. The operating point is displayed in Fig. 7. 3) Efficiency Comparison of the Two Schemes: For both schemes, the overall efficiency is given by

The efficiencies of the enhanced sources result from a simple substitution of the former ratios

It is calculated by substituting the operating point obtained for each scheme % % Both schemes deliver the same output power, W, to the load, but scheme 2 has a higher overall efficiency.

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may include a single array or any number of arrays, and each array may include any number of power-sources. We claim that the overall maximum efficiency of such a network is given by (10) where maximum system efficiency; sum of generated powers; maximum power the network can supply.

Fig. 8. General topology of a multiple linear source system where each source array is coupled to the load by an ideal converter.

Note that the specific values of the internal voltages and series resistances of the power sources are irrelevant for our determination of the overall maximum efficiency; the latter is fully deand . Thus, when designing such a power termined by source array, the maximum efficiency is given independently of the actual parameters of the power sources used or their internal topology. The efficiency versus total supplied power is plotted in Fig. 9; the function is unaffected by the specific structure of the sources or network topology. The efficiency must obey: % % These results are consistent with the familiar efficiency-power relation in single linear source systems. The theorem proposed herein provides a generalization that applies to systems energized from multiple linear power sources. Another outcome of this theorem is that, for a given set of linear sources, maximum efficiency can be achieved by choosing a topology for which the maximum power is greatest. is held constant as increases, Specifically, if also increases.

Fig. 9. Dependence of the maximum efficiency on the total output power of systems that feature multiple linear power sources.

The efficiency attained by scheme 2 is 7.6% higher than the one attained by scheme 1 (relative increase). III. MAXIMUM EFFICIENCY OF ARRAYS OF MULTIPLE LINEAR SOURCES We analyze the particular case of systems where all the power sources are linear, that is all sources have linear - characteristics. Each source may be represented by a voltage source with a series connected resistor or by a current source with a parallel resistor. Within a bounded region of linearization, the results may be extended to non-linear sources as well. Let us consider a multiple source-multiple load system in which arrays of linear sources are coupled to a lossless connection network, and to the loads through ideal converters, Fig. 8. Each power source array may consist of either non-ideal voltage sources connected in series or non-ideal current sources connected in parallel, but not a combination of both. The network

A. Proof of the Theorem—Linear Sources First, note that a Thevenin equivalent circuit cannot be applied to prove this theorem, since it would not predict the power dissipated internally in the source array. Let us consider a group of linear non-ideal voltage sources, i.e., all sources exhibit linear - characteristics and are connected in series, as depicted in Fig. 10. We define equivalent voltage; equivalent resistance; —power output; —maximum power; —internal power loss; —internal power; —efficiency. The same equations hold for a group of non-ideal current sources connected in parallel; thus, they apply to all group types that we are considering. Consider now a network containing

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We apply the Lagrange multipliers method to find the minimum. This results in the following set of equations:

(16) And can be simplified to Fig. 10. Series connection between linear voltage sources.

(17)

multiple groups. The groups are denoted with a running index. The power output and efficiency of each group are given by

and rearrange (17)

We define a new variable to obtain

(18) (11) Substituting (18) into (17) yields We want to choose an operating point that result in maximum efficiency, under the constraint of supplying total power to the loads. This is an optimization problem, which can be solved using the Lagrange multiplier method. The constraint of the problem is:

(19) By reordering, we obtain

(12) The target function for maximization is the overall network efficiency

(20) Note that , where is the maximum power of the array. Substituting this into (20) results in

(13) Within this optimization problem, is fixed, since it is the load’s power consumption (imposed by the load). Thus, maximizing the expression (13) corresponds to minimizing its denominator. So an equivalent problem is to obtain the minimum of the denominator under the same constraint, as in

(21) Solving this quadratic equation and choosing the negative solution to obtain a minimum result in (22) (22) (23)

— —

(14)

We reformulate the problem with the as the free variables, thus creating a solvable optimization problem containing only variables

Substituting efficiency

into the expression for network yields (24)

Finally, substituting (23) into (24) yields

(15)

(25)

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Fig. 13. Maximum efficiency versus total output power (the power is normalized to the maximum power of the series network). The lowest graph is the efficiency of the series network. The other two lines represent the maximum efficiencies attainable by the optimum network for two selections of ratios  and .

The maximum power of the two-array energized network (optimum network) is given by the sum of the maximum powers of each array Fig. 11. An example of two linear power-source energized systems.

The ratios of voltage sources and resistances are defined as

Fig. 12. Two linear power-source systems for comparison. (a) Serial network (b) optimal network.

Note that means that the two power sources are identical. In this case, the optimum network has no advanfrom (1, 1) is tage over the serial one. The deviation of a measure of the power source mismatch. It is shown that, the higher the asymmetry, the higher the relative advantage of the optimum network over the series network in terms of maximum efficiency. The ratios and determine the ratio of maximum powers for the two networks:

B. Multiple Linear Source Example Two linear power-source energized systems are shown in Fig. 11. Although the two networks are completely different and contain different sources and loads, the maximum efficiency is determined similarly by the output power and maximum network power, according to (10). Another illustrative numeric example is shown in Fig. 12. One network is energized by a single array consisting of two serially connected linear sources, while, in the second network, each of the sources forms an independent array. The maximum power of the single array energized network for matched loads is given by

These ratios determine the maximum efficiencies. For both networks, the maximum efficiency is given by (10), and plotted in Fig. 13. The maximum efficiency of the optimum network is plotted twice for different selections of and . IV. CONCLUSION Systems containing multiple power sources are analyzed with basic circuit theory. The set of (2) defines the point of maximum efficiency for any group of sources. Showing some similarity to the Maximum Power Point (MPP) in sustainable power systems,

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the maximum efficiency point is the desired operating point for fueled systems, as at this point the consumption of stored energy is minimum for any power demand. For the example of two fuel-cell stacks, we showed that the scheme using a converter for each stack consumes 8% less hydrogen for powering the same load. Furthermore, the power loss within the fuel cells is reduced by 20% (relatively) by application of this scheme. That may be quite significant since, in a practical system, the cooling effort would be reduced accordingly. Using a converter for each stack can be advantageous because it enables an optimum working point for each stack, thus enabling the maximum efficiency point to be attained. It was shown that in networks energized from multiple linear sources, the maximum system efficiency is determined solely by the maximum network power, according to (10), and is independent of the array configuration or internal parameters of the source. The power source array may be regarded as a ‘black box’ which output behavior is entirely determined by its maximum power. From the design point of view, this implies that given a set of sources one should aim for a topology that guarantees the highest maximum power; this configuration will also exhibit maximum efficiency for any power demand. This result may be applied to practical systems even if they consist of non-linear sources, allowing for a swift evaluation of its maximum possible efficiency. This may be performed by linearization of the power sources in the vicinity of the operating point. The improvement of efficiency saves fuel, and therefore it can save on cost and weight. On the other hand, an increased number of converters would increase the systems complexity. So, the tradeoff between fuel saving and hardware complexity should be considered. By a first order approximation, the cost (or weight) of converters is proportional to their nominal power, so if overall power consumption remains constant, no additional cost is imposed due to conversion capacity splitting. Therefore, cost should not change drastically due to the additional complexity of hardware. Yet, fuel is saved, making the source splitting a viable option. It should be noted that this is a analysis general and should be regarded as a first order approach. Detailed cost-tradeoffs are necessary for any specific power system in order to determine the cost effective level of splitting. As regards the power sources, it seems that modular power sources that are composed of multiple smaller power sources in the first place may be beneficially splinted into smaller sources (which would be bounded by the reduced efficiency due to lower voltages). Examples of such modular sources are solar power sources include photovoltaic generators, which can already be seen to attract the splitting concepts [24]. On the other hand the splitting may not be beneficial for large (utility scale) synchronous generators as their efficiency will deteriorate due to splitting and their cost per Watt will increase. Practical multiple-source systems are proliferating in real life engineering situations such as in distributed generation ac power systems, photovoltaic generators (by means of multiple modules where power accumulates on a common bus), and energy harvesting systems employing multiple transducers. Other future applications include smart grids, large battery banks (such as in utility and vehicular applications) and distributed loads such as high power LED arrays and microprocessor

loads. This study provides an insight and bears an impact on the design and optimization of such systems. Other multiple-source power-critical systems to which this article may contribute are battery powered wireless sensor networks [25]. REFERENCES [1] N. Femia, D. Granozio, G. Petrone, G. Spagnuolo, and M. Vitelli, “Predictive & adaptive MPPT perturb and observe method,” IEEE Trans. Aerosp. Electron. Syst., vol. 43, no. 3, pp. 934–950, Jul. 2007. [2] R. Leyva, C. Alonso, I. Queinnec, A. Cid-Pastor, D. Lagrange, and L. Martinez-Salamero, “MPPT of photovoltaic systems using extremum—Seeking control,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 1, pp. 249–258, Jan. 2006. [3] P. Maffezzoni and D. D’Amore, “Compact electrothermal macromodeling of photovoltaic modules,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 2, pp. 162–166, Feb. 2009. [4] F. Boico, B. Lehman, and K. Shujaee, “Solar battery chargers for NiMH batteries,” IEEE Trans. Power Electron., vol. 22, no. 5, pp. 1600–1609, Sep. 2007. [5] B. D. Anderson, D. A. Spaulding, and R. W. Newcomb, “Time variable transformers,” Proc. IEEE, vol. 53, pp. 634–635, 1965. [6] G. W. Wester and R. D. Middelbrook, “Low-frequency characterization of switched Dc-Dc converters,” IEEE Trans. Circuits Syst., vol. CAS-9, no. 5, pp. 376–385, May 1973. [7] R. W. Newcomb, “The semistate description of non linear time-variable circuits,” IEEE Trans. Circuits Syst., vol. CAS-28, no. 2, pp. 62–71, Feb. 1981. [8] V. Vorperian, “Simplified analysis of PWM converters using the model of the PWM switch: Parts I & II,” IEEE Trans. Aerosp. Electron. Syst., vol. 4, no. 2, pp. 205–214, Apr. 1989. [9] D. Czarkowski and M. K. Kazimierczuk, “Linear circuit models of PWM flyback and buck/boost converters,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 39, no. 8, pp. 688–693, Aug. 1992. [10] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2nd ed. New York: Kluwer, 2001. [11] B. Bryant and M. K. Kazimierczuk, “Voltage loop of boost PWM DC-DC converters with peak current-mode control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, no. 1, pp. 99–105, Jan. 2006. [12] R. Giral, L. Martinez-Salamero, R. Leyva, and J. Maixe, “Sliding-mode control of interleaved boost converters,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl/, vol. 47, no. 9, pp. 1330–1339, Sep. 2000. [13] D. Biel, F. Guinjoan, E. Fossas, and J. Chavarria, “Sliding-mode control design of a boost-buck switching converter for AC signal generation,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 8, pp. 1539–1551, Aug. 2004. [14] A. N. Willson and H. J. Orchad, “Realization of ideal gyrators,” IEEE Trans. Circuits Syst., vol. CAS-21, no. 11, pp. 729–732, Nov. 1974. [15] E. M. Husain and I. B. Mo, “Power converters as natural gyrators,” IEEE Trans. Circuits Syst., vol. 40, no. 12, pp. 946–949, Dec. 1993. [16] I. S. Uzunov, “Theoretical model of ungrounded inductance realized with two gyrators,” IEEE Trans. Circuits Syst. II, vol. 55, no. 10, pp. 981–985, Oct. 2008. [17] J. Patterson and R. Zane, “Series input modular architecture for driving multiple LEDs,” in Proc. IEEE Power Electron. Specialists Conf., Jun. 15–19, 2008, pp. 2650–2656. [18] S. Singer, “Canonical approach to energy processing network synthesis,” IEEE Trans. Circuits Syst., vol. CAS-33, no. 8, pp. 767–774, Aug. 1986. [19] S. Singer and R. W. Erickson, “Canonical modeling of power processing circuits based on the POPI concept,” IEEE Trans. Power Electron., vol. 7, no. 1, pp. 37–43, Jan. 1992. [20] A. Kulikovsky, “The voltage current curve of a PEM fuel cell: Analytical and numerical modeling,” Nanotech 2003 vol. 3 [Online]. Available: www.nsti.org [21] M. Jeferson, A. Felix, N. Luciane, and G. Marcelo, “An electrochemical-based fuel-cell model suitable for electrical engineering automation approach,” IEEE Trans. Indust. Electron., vol. 51, no. 5, pp. 1103–1112, Oct. 2004. [22] A. V. da Rosa, Fundamentals of Renewable Energy Processes, 1st ed. New York: Elsevier, 2005. [23] F. Barbir, PEM Fuel Cells: Theory and Practice, 1st ed. New York: Elsevier, 2005.

LEVRON AND SHMILOVITZ: ON THE MAXIMUM EFFICIENCY OF SYSTEMS CONTAINING MULTIPLE SOURCES

[24] L. Linares, R. W. Erickson, S. MacAlpine, and M. Brandemuehl, “Improved energy capture in series string photovoltaics via smart distributed power electronics,” Proc. IEEE APEC 2009, pp. 904–910, Feb. 15–19, 2009. [25] R. Senguttuvan, S. Sen, and A. Chatterjee, “Multidimensional adaptive power management for low-power operation of wireless devices,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 55, no. 9, pp. 162–166, Sep. 2008. Yoash Levron received the B.Sc. degree from the Technion, Haifa, Israel, and the M.Sc. degree from Tel-Aviv University, Israel, in 2001 and 2006, respectively. He is currently pursuing the Ph.D. at Tel-Aviv University. His current research focuses on optimization of power processing systems.

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Doron Shmilovitz (M’98) was born in Romania in 1963. He received the B.Sc., M.Sc., and Ph.D. degrees from Tel-Aviv University, Tel-Aviv, Israel, in 1986, 1993, and 1997, respectively, all in electrical engineering. During 1986–1990, he worked in R&D for the IAF where he developed programmable electronic loads. During 1997–1999, he was a Post-Doctorate Fellow at New York Polytechnic University, Brooklyn. Since 2000, he has been with the Faculty of Engineering, Tel-Aviv University, where he established a state-ofthe-art power electronics and power quality research laboratory. His research interests include switched-mode converters, topology, dynamics and control, power quality, and power conversion for alternative energy sources, and general circuit theory.