2912 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 6, JUNE 2013 …ecee.colorado.edu/~rwe/references/GPO2013.pdf · 2016-07-01 · 2912 IEEE TRANSACTIONS ON POWER ELECTRONICS,

2912 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 6, JUNE 2013

Probabilistic Analysis of a Generalized Perturband Observe Algorithm Featuring Robust

Operation in the Presence of Power Curve TrapsDavid C. Jones, Member, IEEE, and Robert W. Erickson, Fellow, IEEE

Abstract—Local maxima in the measured power curve of a pho-tovoltaic (PV) panel, called traps, are caused by switching tran-sients and other deterministic phenomena in PV power convert-ers. Traps exist even in the absence of shading, and degrade theperformance of perturb and observe (PO) maximum power pointtrackers. A generalized PO (GPO) algorithm is proposed whicheliminates this degradation. PO/GPO algorithms in the presenceof random noise are shown to be modeled by multidimensionalMarkov chains. These models quantify the tendency for PO at lowrandom noise levels to become stuck in traps following insolationchanges, thereby decreasing tracking performance and reducingharvested power. By contrast, GPO tracks insolation changes un-hindered by traps. All theoretical results are experimentally veri-fied using a hardware PV converter.

Index Terms—Markov chains, maximum power point track-ing (MPPT), perturb and observe (PO), photovoltaic (PV) powerelectronics.

I. INTRODUCTION

THE steady-state power delivered by a photovoltaic (PV)panel is a function of the panel characteristics, the tem-

perature and insolation applied to each constituent PV cell, andthe external circuit to which the panel is connected. With theexternal circuit unspecified, the panel output current and powercan be expressed as functions of the panel terminal voltage V .The output power P (V ) achieves a global peak value PMP atthe voltage VMP , both of which are calculable from given quan-tities [1], [2]. The external circuit is commonly controlled bya maximum power point tracking (MPPT) circuit or algorithm,whose goal is to set and maintain the PV panel output voltageat V = VMP .

MPPT controllers have been a very active area of researchfor the last 15 years, with a large number of distinct circuitsand algorithms proposed in more than 100 conference andjournal articles during that time. Overviews of this body ofwork are given in [3]–[5]. A nonexhaustive list of the most

Manuscript received May 15, 2012; revised July 16, 2012, August 15,2012, and September 6, 2012; accepted October 2, 2012. Date of current ver-sion December 7, 2012. Recommended for publication by Associate EditorM. Veerachary.

The authors are with Phobos Energy, Inc., Menlo Park, CA 94025 USA, andalso with the Department of Electrical, Computer, and Energy Engineering, Uni-versity of Colorado, Boulder, CO 80309 USA (e-mail: [email protected];[email protected]).

Digital Object Identifier 10.1109/TPEL.2012.2224378

common MPPT techniques, along with a few of the support-ing references, include perturb and observe (PO) [5]–[13], in-cremental conductance/resistance [14]–[17], ripple correlationcontrol [18]–[20], fraction of open circuit [21], [22], steepestdescent [23], parameter-based modeling [24], extremum seek-ing, sliding mode, or voltage-oriented control [25]–[29], aDIRECT algorithm [30], particle swarm optimization [31],and fuzzy logic, neural network, or chaos theory tech-niques [32]–[34]. A thorough appreciation of the overallset of MPPT techniques to date is attainable through theoverviews of [3]–[5] and the complete set of papers referencedtherein.

This paper focuses on one of the simplest and most popularMPPT controllers, the perturb and observe algorithm. A PO-based controller causes the PV voltage to take the sequence ofvalues Vk where

Vk+1 = Vk + Δk ; Δk ={

Δk−1 , Pk ≥ Pk−1

−Δk−1 , Pk < Pk−1; Δ0 = Δ

(1)

and where |Δk | = Δ is the PO step size, and Vk and Pk de-note the panel output voltage and measured power sequences,respectively. PO works by stepping the panel voltage along theP (V ) curve, using measurements of the power taken at eachstep in an attempt to always move toward VMP . The algorithmtime step is sufficiently long to allow converter dynamics tosettle, and to allow filtering/averaging of the measured powersignal, following each perturbation.

In a switching power converter, what is commonly (andloosely) called “noise” is actually electromagnetic interfer-ence (EMI) generated by the switching process. Power transis-tor switching stimulates converter parasitic LC step responseswhich are visible as ringing and spikes in measured waveforms.In consequence, the sensed feedback signals which drive thePO algorithm are “noisy.” This EMI actually consists of bothdeterministic and stochastic components. The EMI can be par-tially reduced by filtering of the feedback signal; however, theEMI includes deterministic low-frequency components that arecorrelated with the pulsewidth modulation control and hence arenot removed by low-pass filtering. As shown in Section II, thesecorrelated components can create small-scale local maxima inthe measured power curve.

This paper introduces the critical role that these small localmaxima in the measured value of P (V ), called “traps,” canplay in the performance of the PO algorithm. The effect oftraps is closely related to that of power measurement stochastic

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JONES AND ERICKSON: PROBABILISTIC ANALYSIS OF A GPO ALGORITHM FEATURING ROBUST OPERATION 2913

noise, but the two are distinct phenomena arising from sepa-rate physical processes. An understanding of the existence oftraps, and the nature of their relationship to power measure-ment stochastic noise, explains commonly observed PO be-havior in a way that random noise alone cannot. In the ab-sence of traps, it is well known that PO tracking speed andsteady-state power harvesting percentage are both increased ifthe power measurement random noise is decreased, e.g., byapplying cleaner analog circuit design or power measurementfiltering.

In the presence of traps, however, lower random noise can beobserved to actually worsen both performance characteristics,by increasing the probability that the system becomes “stuck,”for many seconds or even minutes at a time, in one or more ofthese off-peak traps. Designers observing such behavior mightreasonably respond by attempting to lower measurement ran-dom noise further, but this only makes the problem worse. Thispaper discusses the origin of traps in power measurement cir-cuits, and proposes a simple generalized perturb and observe(GPO) algorithm designed to overcome their effects. A newprobabilistic model for both the PO and GPO algorithms isalso derived, showing that both algorithms are examples ofmultidimensional Markov chains. Inclusion of the measuredpower curve of a prototype dc/dc PV converter in this modelyields results which accurately predict the laboratory converterbehavior.

In the existing literature, the PO algorithm has long beenknown to suffer from two imperfections: 1) an inherent trade-off between steady-state power harvesting efficiency on theone hand, and the speed with which insolation changes canbe tracked on the other; and 2) the potential to become stuckat any local maxima produced in P (V ) by partial shading ofthe panel’s cells. If not addressed, these imperfections can de-grade the reliability of the overall PV power processing sys-tem [35]. As a result, much of the recent research on the POalgorithm proposes enhancements to address these shortcom-ings. For example, [5]–[7] confront the tradeoff between effi-ciency and tracking speed through adaptation of the PO stepsize, while [8], [9] approach the same problem through opti-mization of PO parameter values. The efficiency/speed tradeoffis addressed in [10] using a PO modification which separatesthe effect of insolation change from that caused by the pertur-bation. In [11], the problem of local maxima due to shadingis addressed through extensive augmentation of the PO algo-rithm. (MPPT techniques more complex than PO also proposesolutions to these imperfections [15], [23]–[34].) These PO en-hancements reduce or eliminate the algorithm’s partial shadingand efficiency/speed tradeoff problems, but they do not addressthe system impairment due to traps introduced here. While thetechniques of [5]–[11] can be applied in combination with theGPO algorithm proposed here, traps are a distinct impairmentwhich merit separate consideration.

Fig. 1 shows a plot of P (V ) for two example thin-filmPV panels wired in series and operating at uniform insola-tion of 1000 W/m2 , with VMP = 61.74 V and PMP = 282 W.These panels consist of copper-indium-gallium-selenide (CIGS)cells supplied by a major U.S. manufacturer, with the shown

Fig. 1. Power versus voltage for two series thin-film PV panels at 1000 W/m2 ,with and without a measurement trap.

Fig. 2. Closeup of Fig. 1 at trap location.

volt/ampere ratio a consequence of the large cell size used bythis manufacturer. Superimposed on this curve near V = 67 Vis a small multiplicative cubic polynomial, which produces animaginary local maximum at 67.4 V. This artificial trap is high-lighted in the closeup of Fig. 2. The curve with trap in Figs. 1 and2 represents the measured P (V ) in the absence of random noisebut including a deterministic error caused by the power mea-surement circuit. The trap of Figs. 1 and 2 is not meant to reflectthe characteristics of any particular measurement circuit, but isan artifice presented here for discussion purposes only. Underthe assumption that the power measurement circuit delivers theP (V ) shown, in the absence of random measurement noise thePO algorithm could easily become “stuck in the trap” at thelocal maximum and thereby significantly reduce the harvestedpower during that time.

This paper asserts that traps are a common, if not widelyacknowledged, feature of power measurement circuits. As anexample, consider the measured P (V ) curve shown in Fig. 3.This curve shows the measured power from a prototype dc/dcconverter used to connect PV panels to a high-voltage dc bus.In Fig. 4, a closeup of the peak in Fig. 3 reveals numerous traps,including a particularly large one at 49.4 V. The traps in thiscircuit are only visible through the use of heavy 32-bit aver-aging of the measured power value at each voltage, a processdetailed in Section II. Note also that the starting points for thevertical axes are well above zero, which is also necessary torender the traps visible. This paper asserts that these traps arenot indicative of a poor power measurement circuit designinthis particular converter, but rather that traps are endemic topower measurements made on many switching power electronic


Fig. 3. Averaged power versus voltage from hardware PV dc/dc convertermeasurement circuit.

Fig. 4. Closeup of Fig. 3 curve peak.

circuits. Section II of this paper describes the means by whichtraps in such circuits commonly arise. Common sources of trapsare shown to include changes in shape and sample phase of ring-ing waveforms, analog-to-digital converter (ADC) quantizationerror, and transition between converter modes.

With the common existence of traps identified, it becomespossible to design MPPT controllers which work well in theirpresence. Certainly, those schemes in the existing literature de-signed to find and track the global maximum in the presence oflocal maxima from partial shading are viable approaches. But asthese techniques are typically of much higher implementationcomplexity than PO, their use in cost-sensitive applications isquestionable. As a result, Section III of this paper develops ageneralization of the PO algorithm, the GPO, specifically de-signed to avoid “entrapment” while incurring only a very smallincrease in algorithm complexity. The GPO works by comparingthe measured power at the current iteration to a power measure-ment stored from a past iteration, but not necessarily the mostrecent past. The current iteration power is also compared to thestored past power multiplied by a value α where α < 1, α ∼= 1.The updating of the stored past power value, and the directionin which to step the voltage for the next iteration, depend onthe outcome of these two comparisons. The GPO is designedto progress unhindered through any trap whose local valley-to-peak ratio is greater than α. Section III shows that at α = 1the GPO algorithm becomes the PO, so that GPO is indeed ageneralization of PO.

The presence of power measurement random noise in thePO/GPO algorithm makes the resulting voltage and power

trajectories stochastic processes. A probabilistic model for thePO/GPO algorithms in the presence of random noise is derivedin the Appendix, which shows that both algorithms are describedby multidimensional Markov chains. These Markov chains areapplied in Section IV in the analysis of algorithm tracking time.Tracking time refers to the duration required to return the con-verter to the peak power operating point following a changein atmospheric conditions. Prior work describes the impact ofconverter dynamics on tracking time, and methods wherebyPO parameters can be tuned for best performance in a givenconverter [9], [16]. This paper assumes preapplication of suchtechniques, and decouples all analysis from specific converterdynamics by expressing tracking time in units of algorithm it-erations. Section IV shows that GPO tracks changes faster thanPO under all conditions. It is also shown that in the absence oftraps, tracking time decreases for both algorithms as the randommeasurement noise is reduced, an intuitively pleasing result. Inthe presence of traps, however, a decrease in random noise be-yond a critical point dramatically increases tracking time for thePO algorithm. The reason is that lower random noise decreasesthe probability that a large noise sample will eject the algorithmfrom a trap. By contrast, GPO tracking time is, by design, almostcompletely unaffected by the presence of traps.

In the existing MPPT literature, only four previous workstouch closely on the new ideas described in this paper. First,in [36], a PO derivative with improved robustness is presentedwhich shares some similarity with the GPO. However, by asso-ciating tracking problems with high measurement EMI alone,rather than with the relationship between traps and random noiselevels both high and low, the algorithm of [36] makes updatingdecisions based on different thresholds than those used in GPO.As a result, it can be shown that GPO has superior trackingand/or power harvesting performance compared to the algo-rithm of [36]. Second, in [37], the potential impact of measure-ment bias on PO behavior is considered. However, by treatingmeasurement bias as simply a constant, rather than as a func-tion of PV voltage, [37] does not anticipate the existence oftraps and their deleterious effects. Finally, in [37]–[39], statis-tical analyses of the PO algorithm in the presence of randomnoise are conducted. In [37], [38], error probabilities for thesign of the PO increment are derived, but the fundamental un-derlying Markov chain structure, derived here in the Appendix,is not recognized. This Markov analysis is necessary in orderto properly compute statistics for PO tracking time. In [39], theMarkov structure of PO is recognized, but an inaccurate 1-DMarkov model is proposed. In the Appendix, it is shown thatthe correct Markov model for PO is 2-D, and for GPO is 3-D.In the context of this paper, the distinction is important. It canbe shown that while the model of [39] yields approximately ac-curate results for a unimodal power curve P (V ), it dramaticallyunderestimates tracking time for P (V ) with traps or other localmaxima.

The remainder of this paper is organized as follows. Section IIdescribes the origin of traps in switching power circuits whileSection III describes the GPO algorithm designed to deal withthem. Sections IV and V present theoretical and experimental re-sults on PO/GPO tracking time in the presence of random noise.


Fig. 5. Buck–Boost-DCX dc–dc converter connecting PV panel to a high-voltage dc bus, with a PO/GPO-based digital controller.

Section VI offers a summarizing conclusion. The probabilisticmodel for the PO/GPO algorithms is derived in the Appendix.

II. ORIGIN OF TRAPS IN POWER MEASUREMENT CIRCUITS

Traps in a measured power curve can arise from any circuitnonideality which imparts a measurement error whose valuevaries with the PV voltage. The key to understanding traps is todistinguish the effects of truly random noise from deterministic“noise-like” signals on the measured power value. In this sec-tion, ringing waveforms from switching transients are evaluatedas an example of the latter. When deterministic sources combineto produce P (V ) measurement errors which are not monotonicin V , a trap can result.

An illustrative example system is used to guide the discus-sion in this section, and to provide a platform for experimentalresults. In this system, illustrated in Fig. 5, a PV panel deliverspower to a regulated dc bus of voltage Vbus through a dc–dcconverter. The converter consists of a noninverting buck–boostfollowed by a 1-to-8 step-up dc transformer (DCX) [40]. Thepanel output voltage V is regulated by a digital controller, withthe reference Vref for the voltage control loop set by the PO/GPOalgorithm. Since Vbus is externally regulated in this example,a convenient technique for power measurement here is to mea-sure the converter output current Ibus and to simply assume thatVbus is equal to its regulated value [41]. Measurement of Ibusoccurs via a current sense transformer, rectifier, resistor, andfilter, applied in series with the DCX ac current waveform [40].In order to control the effects of random noise, Navg consecu-tive samples of Ibus , taken at the converter switching frequency1/Ts , are averaged. The PO/GPO power input sample Pk is thenthe product of this average and the known value of Vbus . Whilethis architecture has been chosen to match that applied in theexperimental results of Section V, the discussion here is widelyapplicable to the measurement of power in other architecturesas well.

Random and deterministic influences on measured power aredistinguished through averaging. For P (V ) the actual powercurve, define PM (V ) as the measured power curve in the ab-sence of random (e.g., thermal) noise. Define PM,j (V ) as asingle measurement of P (V ) in the presence of such noise, i.e.,

PM,j (V ) = PM (V ) + ξj (V ) (2)

Fig. 6. Current sensor output, times Vbus , for DBoost = 0.20, 0.21, 0.22,along with switch control waveforms. The bottom (next to bottom) waveformis high at those sampling phases for which a trap minimum (peak) occurs.

where ξj (V ) is a sample of a zero-mean ergodic stochasticprocess whose statistics may depend on V . For example, mea-surements made on the hardware used in the experiments ofSection V suggest ξj (V ) has standard deviation

σξ =√

a + bP 2(V ) (3)

where a = 0.21136 W2 and b = 2.5909 × 10−5 for measuredpower in watts. The power measurement input Pk to thePO/GPO algorithm averages several power measurements toeffectively reduce this random noise, as in

Pk =1

Navg

Nav g∑j=1

PM,j (V ) = PM (V ) +1

Navg

Nav g∑j=1

ξj (V ).

(4)PM (V ) can then be determined from

PM (V ) = limNavg → ∞

1Navg

Nav g∑j=1

PM,j (V ). (5)

All deterministic phenomena are captured in PM (V ), which isnot a random variable. The power measurement error

ε(V ) = PM (V ) − P (V ) (6)

reflects the presence of deterministic phenomena which causethe measured and actual power curves to differ, even in the ab-sence of random noise. Prominent examples of such phenomenaare described in Sections II-A and II-B.

A. Ringing From Switching Transients

Switching transients are frequently thought of as “noise,” butin fact they are deterministic waveforms which are periodicunder steady-state conditions. Fig. 6 shows the ac portion ofthree measured current sensor waveforms, prior to ADC sam-pling, corresponding to three steady-state boost duty cycles


DBoost = 0.20, 0.21, 0.22. These switching transient wave-forms arise from the excitation of system parasitic LC stepresponses by periodic transistor ON/OFF switching. Each wave-form has been averaged by the oscilloscope to remove randomnoise and multiplied by Vbus to be in units of watts. At thetop of Fig. 6 are the three corresponding boost gate driver sig-nals. (The output current waveforms are also affected by DCXswitching transitions at t = nTs/2, not shown in the figure.)The average values of the output current waveforms (multi-plied by Vbus) over a full switching period are 201.2, 200.8,and 200.4 W for DBoost = 0.20, 0.21, 0.22, respectively. Aver-age power increases as DBoost decreases because the operatingpoint voltage for these DBoost values is below VMP . However,Fig. 6 shows how the contribution of switching transients tomeasurement error can produce traps. The waveform secondfrom the bottom in Fig. 6 is high for ADC sample phases atwhich the sampled power for DBoost = 0.21 is greater thanthat for both DBoost = 0.21 ± 0.01. If one of these samplephases is used in measuring power, the peak of a trap willoccur at the DBoost = 0.21 operating point. Similarly, the bot-tom waveform in Fig. 6 is high for sample phases which willproduce a local minimum at the DBoost = 0.21 operating point.In general, a trap peak will occur at an operating point V ifPM (V ) > PM (V − Δ) and PM (V ) > PM (V + Δ) are bothtrue. From (6), this becomes

(Trap Peak at V ) ↔ (ε(V ) − ε(V − Δ) > P (V −Δ)−P (V ))

and (ε(V ) − ε(V + Δ) > P (V + Δ) − P (V )). (7)

In words, a trap peak occurs at V if the negative of the changein measurement error due to a perturbation in V is greaterthan the resulting change in actual power, on both sides ofV . Fig. 6 shows that for the system of Fig. 5, small duty cycleperturbations about DBoost = 0.21 are accompanied by signif-icant changes to the ringing waveforms present in the mea-surement of Ibus . Whether or not these changes create a trappeak depends on the ADC sampling phase, and on the valuesof P (V + Δ) − P (V ) and P (V − Δ) − P (V ). The changes inswitching waveforms due to perturbations about other voltage/duty cycle values will result in traps over a potentially differentrange of sampling phases. Trap peaks will appear in PM (V ) atthose voltages whose trapped sampling phase region includesthe fixed sampling phase used by the hardware.

The creation of traps in this way is by no means limited tothe prototype hardware represented by Fig. 5. Fig. 7 shows theoutput current waveform of a simulated boost converter operat-ing at DBoost = 0.21, whose only departure from the dc valueof 0.5 A is small residual ripple and damped single-frequencyringing from switch transitions. Perturbations to DBoost willshift the phase of the ringing waveform for t ∈ [DBoostTs, Ts ],creating a change in measurement error that may be significantfor sampling phases in that range. Fig. 8 shows a closeup ofthe ringing waveforms for DBoost = 0.21 and 0.22 indicatinga potential value for ε(V ) − ε(V − Δ) as large as 1.0 W. Asimilar change in error exists for DBoost = 0.20. From this and(7), we conclude that the formation of traps via switching tran-sients is likely for even this simple circuit example. In passing,

Fig. 7. Example residual ripple and switching transient ringing in outputcurrent waveform for DBoost = 0.21.

Fig. 8. Closeup of Fig. 7 showing both DBoost = 0.21, 0.22.

it is noted that traps due to switching transients can in theorybe avoided by placing the sampling instant at a point wheredifferences in ringing at different duty cycles are guaranteed tobe negligible. However, as switching transients are generated onboth the rising and falling edges of the duty cycle signal, such asampling phase will likely not exist for converters designed tooperate over a wide range of duty cycles. Other implementationdetails can restrict the choice of sampling phase as well, e.g.,the need to perform a certain number of computations betweeneach sampling instant and the beginning of the next switchingperiod.

B. Other Sources

Other sources of traps that have been identified include con-verter mode changes, insufficient accommodation for perturba-tion settling time, and quantization error with low random noise.An example of the first of these is the large trap at 49.4 V inFig. 4, which is caused by the brief cessation of buck/boostswitching (and the associated ringing) as the converter entersa pass-through mode [43]. A switch between buck and boostmodes can also produce a significant change in ringing wave-forms. For converters which can operate in both continuousconduction mode (CCM) and discontinuous conduction mode(DCM), a switch between these modes will also be accompaniedby large changes in ringing as diode turn-off transitions are ei-ther added or removed. All of these mode changes can producetraps through abrupt, voltage-sensitive changes in deterministicpower measurement error.


Fig. 9. Block diagram of the GPO and PO algorithms.

III. GPO ALGORITHM

Traps in the measured power curve PM (V ) pose the risk ofa reduction in harvested power due to intervals of PO algorithmentrapment. This section presents a generalized version of PO,the GPO, specifically designed to avoid this problem. A blockdiagram of the GPO algorithm is shown in Fig. 9. In Fig. 9,Pk and Δk are the measured power and voltage step, respec-tively, for the algorithm at iteration number k, as in (1). Rk

is equal to a past power measurement Pk−L , where L = 1 forthe PO algorithm while L ≥ 1 for GPO. The nongeneralizedPO algorithm is implemented by the Fig. 9 blocks numbered1–5, with the value of α set to 1, and with the switch set tothe “α = 1 (PO)” position. The GPO algorithm generalizesα to any value α < 1, α ∼= 1, and also uses blocks numbered6–7 with the switch thrown to “α ≤ 1 (PO/GPO).” In fact, theswitch can be left at the PO/GPO position and the PO switchpath removed without altering the flowchart functionality, forin the α = 1 case the path through block #6 only occurs whenPk = Rk , making blocks 4 and 7 equivalent for that case. Theswitch merely highlights the exact and slight additional com-plexity involved in adding the generalized functionality of theGPO.

For the case α < 1, the system behavior under the control ofGPO is illustrated in Fig. 10. Fig. 10 shows the peak of an exam-ple trap-free power curve PM (V ), and also the correspondingrandom noise-free system behavior in terms of trajectories forvoltage V (t) and the past peak power value Rk (t) stored in thealgorithm. As seen in Figs. 9 and 10, GPO functions by rampingthe voltage alternately up or down, with changes in ramp slopetriggered by detection of the condition Pk < αRk . Rk storesthe value of maximum power measured since the last time thePk < αRk condition was detected. Detection of Pk < αRk up-dates Rk to the current measured power value and initiates aninversion of the voltage ramp slope that persists until the nexttime Pk < αRk is detected.

The largest advantage to this behavior compared to that ofPO occurs in the presence of measured power curve traps. Inthe absence of random noise, on its way to the global peak, theGPO passes unhindered through the example trap of Fig. 11 ifthe condition

α <Pt,min

Pt,max↔ Unhindered noise - free GPO trap passage

(8)

Fig. 10. Example trap-free PM (V ) and associated trajectories for voltageV (t) and stored past peak power Rk (t) under low-noise GPO control.

Fig. 11. Example trap in PM (V ). In the absence of noise, GPO passes throughthis trap if α < Pt,m in /Pt,m ax , while PO becomes stuck at Pt,m ax .

is satisfied. By contrast, the PO algorithm becomes stuck atPt,max until freed by a sufficiently large random noise burst orthe intervention of some other algorithm. This behavioral dif-ference yields tremendous tracking performance superiority forthe GPO in the presence of any traps which satisfy (8), espe-cially in low random noise conditions. As a result, followingdetermination of PM (V ) via the averaging procedure of (5), αshould be set at the largest value satisfying (8) for each trap vis-ible in PM (V ). The probabilistic analysis of Section IV showsthat GPO maintains a tracking advantage, albeit reduced, evenin the absence of traps.

Note that by virtue of the decision/branching blocks of Fig. 9,the processing performed by GPO is a highly nonlinear functionof its input and state. It is for this reason that the effective track-ing bandwidth is increased, relative to that for PO, even with theintroduction of delay into the past power measurement sampleRk . Another issue that arises is the presence and extent of anytradeoffs which exist between tracking speed and steady-stateperformance for the two algorithms. Section IV addresses thisas well. In the absence of both traps and changes in insolation,the superior tracking ability of GPO is of little consequence andPO will have higher steady-state efficiency E, defined as theaverage steady-state harvested power under the control of theMPPT algorithm divided by that which would be harvested ifthe system were operated at the fixed voltage V ≡ VMP . Underthe much more realistic case where both traps and insolationchanges are common, one approach is to adapt α, establishing a


value α < 1 when the need for robust tracking is detected, thenletting α → 1 as steady-state conditions return. One such adap-tive α GPO algorithm has been developed with results shown inSection V. In general, however, for α either fixed or adaptive, asdiscussed in Section IV the efficiency topic is a subtle one andrequires a statistical approach.

IV. PROBABILISTIC ANALYSIS OF PO AND GPO ALGORITHMS

In the presence of stochastic power measurement noise, thebehavior of PO/GPO algorithms becomes randomized and aprobabilistic analysis is required to quantify performance. Inthe Appendix, it is shown that the probabilistic behavior of bothalgorithms is described by multidimensional Markov chains. InSection IV-A, these models are applied in the determination ofstatistics for algorithm tracking behavior as a function of randommeasured noise level. In Section IV-B, issues related to MPPTefficiency for both algorithms are discussed.

A. Tracking Time Probability

Tracking time refers to the number of PO/GPO iterations re-quired for the algorithm to locate the new VMP following anabrupt change in insolation. In this section, tracking time statis-tics are evaluated by applying the Markov probability models tothe power curves of Figs. 1–4 for a starting voltage 10 V aboveVMP . An algorithm step size of Δ = 50 mV is applied, implyinga minimum tracking time of 10/0.05 = 200 iterations. For thecase of Figs. 1 and 2, results are generated with and without theartificial trap present. The effect of random measurement noiseis considered by applying the model of (2)–(4) and varying thevalue of Navg . Tracking time statistics are presented through thecumulative distribution function (CDF), which for each valueof k plots the probability that the algorithm has reached VMP ink or fewer iterations.

The results are presented in Figs. 12–22. Eight of these figurespresent sample CDF plots, each taken for a specific choice ofpower curve and power measurement averaging value Navg .Each CDF plot shows results for three different values of α: 1.0(PO), 0.994 (GPO), and 0.988 (GPO). The α = 0.994 value isthe largest α for which the GPO will pass completely unhinderedthrough all traps in Figs. 1–4 in the absence of random noise.The α = 0.988 value is shown for comparative purposes. Theremaining three figures plot tracking time 99th percentile foreach α value and on each power curve as a function of Navg .(Note that as used here Navg < 1 simply represents an increase,rather than a decrease, in the variance of ξj (V ) described by(3).)

The results can be summarized by the following three facts:1) GPO tracking time performance is superior to that for PO withor without power curve traps and at all averaged random noiselevels; 2) GPO tracking time improves monotonically as thenoise averaging level Navg is increased, both with and withouttraps, tending toward the minimum value of 200. The sameis also true for PO, but only in the absence of traps; 3) Inthe presence of traps, PO tracking time improves as Navg isincreased, but only for Navg below a critical value. Any increasein Navg above this critical value causes a very large and abrupt

Fig. 12. Tracking time CDF for Fig. 1 power curve without traps, Navg = 1.

increase in PO tracking time. This is due to the PO algorithmbecoming stuck at the trap peak for long periods, and indicatesa catastrophic failure mode of the PO algorithm at low-averagedrandom noise levels.

Figs. 12–22 support this summary of the modeling resultsas follows. Figs. 12–15 show results without traps, in whichGPO reaches VMP about 2–15 times faster than PO, dependingon values for α and Navg . For both algorithms, tracking timedecreases monotonically as Navg is increased. The introductionof the artificial trap in Figs. 16–19 further improves the trackingadvantage for GPO, to an extent that depends on random noiselevel. Comparing Figs. 13 and 16, we see that for Navg = 8 thetrap barely increases GPO tracking time, but doubles that forPO. For Navg = 32 in Figs. 14 and 17 the trap has no impacton GPO tracking time, which is now at the minimum possiblevalue of 200, but the trap increases PO tracking time from 3000iterations to 120 000 iterations, a factor of 40 increase. ForNavg further increased to 128 in Fig. 18, the PO tracking failureis complete. While GPO tracking time is still at the minimumpossible value of 200 iterations, the PO is stuck permanently inthe trap. This catastrophic PO failure at low-averaged randomnoise levels is also shown in Fig. 19 by the vertical asymptote forPO tracking time above Navg = 32. (Because of computationallimitations, the tracking time calculation is halted at 150 000iterations.)

The overall conclusion is the same in Figs. 20–22, where theactual hardware power curve of Fig. 3 is applied. The significanttracking advantage for GPO at higher averaged random noiselevels becomes a tremendous advantage as the noise is reducedto the point where PO becomes stuck in one or more traps for along period. The asymptote in Fig. 22 for this case also occursnear Navg = 32, with Fig. 21 suggesting a PO tracking timeon the order of 2 million iterations while that for GPO is theminimum value of 200.

B. MPPT Efficiency

By definition, MPPT efficiency E involves measurement ofpower in steady state. This implies not only that E is to bemeasured under constant environmental conditions but also thatany system start-up transients be allowed to decay prior tomeasurement initiation. In the realistic case where traps arepresent, GPO can have a measured MPPT efficiency which ishigher or lower than that for PO, depending on random noiselevel and which set of off-peak traps the PO system wanders




Fig. 15. Tracking time 99% versus Navg for Fig. 1 power curve withouttraps.

Fig. 16. Tracking time CDF for Fig. 1 power curve with traps, Navg = 8.



Fig. 19. Tracking time 99% versus Navg for Fig. 1 power curve with traps.

Fig. 20. Tracking time CDF for Fig. 3 power curve, Navg = 8.

Fig. 21. Tracking time CDF for Fig. 3 power curve, Navg = 32.

Fig. 22. Tracking time 99% versus Navg for Fig. 3 power curve.


Fig. 23. Steady-state voltage PDF for Fig. 3 power curve with Navg = 32.

between during the course of the measurement. In a modelingenvironment, the presence of traps induces related difficulties, asthe steady-state solution tends toward multiple stationary pointsas the average random noise is reduced. This is illustrated inFigs. 23 and 24, which show steady-state voltage probabilitydensity functions (PDFs) for the hardware power curve of Fig. 3at Navg = 32 and 256. Fig. 23 shows the likelihood for PO-based systems to wander from one trap to the next even withenvironmental conditions held fixed. Fig. 24 shows the emer-gence of a second stationary point at the off-peak trap of 49.4 V,indicating the potential for stable long-term operation there.

In effect, use of PO in the presence of traps produces a systemwhich becomes nonstationary and nonergodic as the averagedrandom measurement noise is reduced. For such a system, ac-curate modeling and measurement of steady-state features suchas MPPT efficiency become problematic. Steady-state perfor-mance of a system which moves randomly between differentstationary points can only be meaningfully assessed through theincorporation of models of weather-induced insolation changes.Such an approach is beyond the scope of this paper.

V. EXPERIMENTAL RESULTS

A hardware prototype of the buck–boost-DCX converter in-troduced in Fig. 5 has been designed and implemented for the PVhigh-voltage dc bus application. Measurements on this converterare used to test the theoretical ideas developed in this paper. Thebuck–boost and DCX use inductance values of 27 and 3.4 μH,respectively, and 16 μF capacitors. Both are run at a switch-ing frequency of 100 kHz. The control functionality shown inFig. 5 has been implemented on a Freescale MC56F8006 32-MHz digital signal controller. Including power averaging andperturbation settling time, the achieved PO/GPO iteration timestep is approximately 15 ms. The PO/GPO step size is Δ = 50mV. Statistically rigorous histogram verification of the modeledCDF curves in Figs. 12–22 is impractical, as each CDF wouldrequire hundreds of experimental runs with each taking up to38 min to reach the 150 000 iteration limit. Fortunately, suchexhaustive testing is not required to verify the theoretical ideasproposed in this paper. The existence of traps and their effecton the behavior of a PO/GPO-controlled PV converter are bothreadily verified via a smaller number of experiments.

The experimental setup is shown in Fig. 5, except with thePV panel replaced by a 1 kW voltage source in series with an

Fig. 24. Steady-state voltage PDF for Fig. 3 power curve with Navg = 256.

approximately 9-Ω power resistor. This change allows for mea-surements to take place under stable and repeatable conditions.The voltage source is set to 90 V, which results in VMP = 45 Vand PMP = 219 W. The 400-V regulated dc output bus is aninverter operating in a constant voltage mode. The measuredpower curve PM (V ) is produced by stepping Vref in Fig. 5, andat each voltage averaging 1024 10-bit samples of the quantity400Ibus using 32 bit accuracy. The result is shown in Figs. 3and 4, which as already discussed clearly showed the presenceof multiple traps.

The effect of these traps on hardware converter tracking per-formance is studied by first regulating the converter input volt-age to the value 55 V, 10 V above and 13 W below the peak,then switching to normal PO/GPO control at the beginning ofthe test. Experimental results in terms of measured instanta-neous output power P (t) are collected for both α = 1.0 (PO)and α = 0.994 → 1.0 (adaptive α GPO) over a range of Navgvalues. More is said about the adaptive α GPO case below. Forboth algorithms and each value of Navg , multiple P (t) traces arecollected and overlaid on a single figure. Only ten P (t) traceson each figure are required to clearly identify the underlyingprobabilistic behavior of each algorithm. Each figure shows anadditional trace which starts the test at V = VMP , as this isuseful in some of the PO cases for identifying the peak powervalue.

The results are shown for the PO algorithm in Figs. 25–29and for GPO in Figs. 30–34. For each figure, dynamic MPPTefficiency equals the value plotted in each curve divided bythe value 219.2 W. The adaptive α GPO traces of Figs. 30–34all exhibit static MPPT efficiencies between 99.8–99.9%, whilethe PO traces of Figs. 25–29 exhibit static MPPT efficienciesover the much wider range 96.3–99.9%. This demonstrates theimprovement in robustness achieved by the GPO algorithm com-pared to PO. For PO, the presence of traps is not obvious forNavg = 16 and Navg = 32 but at Navg = 64 their presence ap-pears in the form of traces which become stuck at 213, 217, and218 W. One of the traces is still stuck at 217 W 90 s after the be-ginning of the test. At Navg = 128, escaping the traps is takinglonger, with five of the ten traces still stuck at least 1 W belowthe peak at the end of the test. By Navg = 512, the PO algorithmhas essentially failed the tracking test. Ninety seconds after thebeginning of the test, all ten traces are at least 2 W below peakpower, and several are 9 W below.


Fig. 25. Ten P (t) traces for α = 1.000 (PO) and Navg = 16.





Fig. 30. Ten P (t) traces for α = 0.994 → 1.000 (GPO) and Navg = 16.






In comparison with results from the probabilistic model ofSection IV, the onset of prolonged entrapment in the PO exper-imental results occurs somewhat less abruptly and at a highervalue of Navg . This is likely caused by negative correlation be-tween consecutive averaged random noise samples ηk−1 and ηk

which contaminate Pk−1 and Pk , respectively. In the Appendix,such correlation is not considered, but can be shown to be equiv-alent to a lower value of Navg . With this minor exception, the PObehavior in Figs. 25–29 is clearly consistent with that predictedby theory. Below the critical value of Navg = 32, increases inNavg produce modest improvement in PO tracking and steady-state performance. Above Navg = 32, further increase in Navgcauses entrapment for prolonged periods, posing the potentialfor a dramatic decrease in harvested power.

In Figs. 30–34 the P (t) traces for adaptive α GPO are shown.During 0 ≤ t ≤ 10 s, the adaptive algorithm holds α fixed atthe starting value α = 0.994 until the return of stable conditionsis detected. During 10 ≤ t ≤ 20, a slow ramp to the value α =1.0 occurs. The 20 s α < 1.0 interval represents a period ofincreased robustness following a significant change in powerand could be made as long or short as desired. Comparison ofFigs. 30–34 to Figs. 25–29 demonstrates the two performanceadvantages of GPO over PO: 1) GPO tracking time is faster evenwith traps masked by higher averaged random noise. For allcases, the GPO system reaches VMP within 3 s of the start of thetest, while for Navg = 16 or 32, PO takes 10–20 s; 2) unlike PO,GPO tracking time is not increased by traps for Navg greater thana critical value. Tracking time remains at 3 s for all GPO traces,but exceeds 90 s for many of the PO traces for Navg ≥ 64. Witha minor exception likely due to unmodeled noise correlation,the experimental results are in complete agreement with theproposed theory.

VI. CONCLUSION

PV converters containing PO-based controllers are frequentlyobserved to remain off peak for random intervals followingchanges in insolation. This paper shows that this puzzling be-havior is due to local maxima, called traps, in the power curvemeasured by the converter. Traps are caused by errors betweenthe measured and actual power curve which vary with PV volt-age in a nonmonotonic fashion. A PO controller in a dynamicenvironment is prone to getting stuck in traps, especially in de-signs which provide low random noise power measurementsas input to the algorithm. Attempts to address this behavior bylowering the random power measurement noise only make theproblem worse.

Measurements and simulations both show that traps can becreated by sampled switching waveforms which vary with con-verter duty cycle. Changes in converter mode, e.g., betweenbuck and boost or between CCM and DCM, have a similar ef-fect. Trap formation is also assisted by ADC quantization in alow noise environment. As these hardware characteristics canbe difficult to avoid, this paper has proposed a GPO algorithmspecifically designed to deal with traps. At each iteration step,GPO makes two comparisons between the power measured at

the current voltage and a power measured while at a previousvoltage, not necessarily the most recent past. The results of thetwo comparisons are used to update PV voltage and past storedpower for the next iteration. GPO is shown to be a generalizationof PO, requiring only a slight increase in implementation com-plexity. From a plot of the measured power curve with randomnoise averaged out, the designer can select the GPO parameter αto insure that the system will pass unhindered through all traps.

In the presence of random noise, both PO and GPO algorithmsare shown to form multidimensional Markov chains. Probabilis-tic modeling verifies traps as the source of PO’s puzzling behav-ior. Above a certain random noise threshold, tracking behaviorfor PO improves with reductions in measurement noise level.However, below that threshold PO tracking fails catastrophi-cally as noise samples large enough to dislodge the algorithmfrom a trap become increasingly rare. By contrast, GPO passesunhindered through traps and has performance which improvesmonotonically as random noise is decreased. Even in the ab-sence of traps, GPO exhibits significant tracking superioritycompared to PO.

Both PO/GPO algorithms have been implemented in a PVconverter using a 32-MHz digital signal controller. Plots ofmeasured instantaneous output power verify both the role oftraps in PO behavior and the superior tracking performance ofthe proposed GPO algorithm.

APPENDIX

MULTIDIMENSIONAL MARKOV CHAIN MODELS

FOR PO AND GPO

A Markov process is one in which the future is independent ofthe past if the present is given [44]. In the case of a discrete-timestochastic process X0 ,X1, , . . . , Xk ,Xk+1 , . . . , this means thatthe process is Markov if and only if

fXk + 1 |Xk ,...,X 0 (xk+1 |xk , . . . , x0) = fXk + 1 |Xk(xk+1 |xk )

(9)

where fA |B (a |b) denotes the PDF of the random variable Aconditioned on the random variable B having the value b. Theapplicability of Markov models to the PO and GPO algorithmsis handled separately in the next two sections.

A. PO Algorithm

The sequence of voltages Vk under the control of the POalgorithm is not a (1-D) Markov process, as the step directionfrom Vk to Vk+1 depends not just on the power measured atV = Vk but also on that at V = Vk−1 . However, if we defineXk as the row vector

Xk = [VkVk−1 ] (10)

then the process Xk satisfies (9) and we conclude that under thecontrol of PO the voltage sequence forms a 2-D Markov process.For the case where the step size Δ is fixed and the voltagelimited to the range v1 ≤ V ≤ vN ,Xk forms the Markov chainillustrated in Fig. 35. In this figure, the set of possible values


Fig. 35. PO algorithm 2-D Markov chain.

{x1 , x2 , . . . , x2N } for Xk are shown as two rows of states, withthe top row containing states for which Vk = Vk−1 + Δ and thebottom row those for which Vk = Vk−1 − Δ. All other possiblevalues for Xk have zero probability and need not be considered.The two states in each column share the same first componentvi , and represent arriving at that voltage from either a lower (toprow) or higher (bottom row) voltage. The transition probabilityπi,j from state numbers i to j is independent of time and denotesthe conditional probability

πi,j = Pr[Xk+1 = xj |Xk = xi ] (11)

where Pr[Ω] denotes the probability of the event Ω. If we definethe time-dependent row vector of state probabilities

Γ(k) = [Pr(Xk = x1) Pr(Xk = x2) . . . Pr(Xk = x2N )](12)

then from [44]

Γ(k) = Γ(k − 1)Π (13)

where Π is the matrix with elements πi,j . The transition proba-bilities along the top row of Fig. 35 are

π2i−1,2i+1 = Pr[PM (vi) + ηi ≥ PM (vi−1) + ηi−1 ]

= Q[(PM (vi−1) − PM (vi)) /

√σ2

i + σ2i−1

]∀i, 2 ≤ i ≤ N − 1 (14)

where (see (2)–(5)) PM (vi) is the random noise-free measuredpower at V = vi and

ηi =1

Navg

Nav g∑j=1

ξj (vi) (15)

is the averaged value of the random noise sequence{ξ1(vi), . . . , ξNav g (vi)} which disturbs that measurement. ηi

is zero-mean and as shown in the second line of (14) we as-sume that it is Gaussian with variance σ2

i , a valid assumptionfor Navg sufficiently large even if the {ξ1(vi), . . . , ξNav g (vi)}are not independent [45]–[48]. For the noise model of (2)–(4),we have

σ2i = [a + bP 2

M (vi)]/Navg . (16)

We also assume that Navg and/or the algorithm settling time aresufficiently large for averaged noises from adjacent PO iterations

Fig. 36. Alternative representation of the GPO algorithm, showing how Δk /Δk−1 and Rk+1 are determined by comparing Pk to both Rk and αRk .

ηi and ηi±1 to be independent. Finally, in (14), the function Qis by definition

Q(x) =12

[1 − erf

(x√2

)]=

1√2π

∫ ∞

x

e−t2 /2dt. (17)

Similarly, the bottom row transitions of Fig. 35 have probability

π2i,2i−2 = Pr[PM (vi) + ηi ≥ PM (vi+1) + ηi+1]

= Q

[(PM (vi+1) − PM (vi)) /

√σ2

i + σ2i+1

]

∀i, 2 ≤ i ≤ N − 1 (18)

with the diagonal transitions given by

π2i−1,2i−2 = 1 − π2i−1,2i+1 , 2 ≤ i ≤ N − 1

π2i,2i+1 = 1 − π2i,2i−2 , 2 ≤ i ≤ N − 1. (19)

Tracking time statistics are computed by iterating (13) with allcomponents of Γ(0) set to zero except for the starting state,which is set to 1. The CDF can be conveniently computed as thesum of the components in Γ(k) corresponding to VMP , providedthat (14) and (18) are artificially modified so that transitions outof the VMP states are inhibited. Calculation of MPPT efficiencyis based on Γ(∞), the process steady-state PDF, and the givenpower curve PM (V ). Γ(∞) can be computed as the eigenvectorof the unity eigenvalue equation (see (13)) Γ(∞) = Γ(∞)Π.

B. GPO Algorithm

The GPO algorithm of Fig. 9 can also be described by Fig. 36.In Fig. 36, the measured power during iteration k is comparedto both the stored past power measurement Rk and αRk . Thevalues of Δk/Δk−1 ∈ {±1} and Rk+1 are both set as shown inthe figure according to the outcome of these two comparisons.In Fig. 36, the random variable Ak denotes the voltage at whichthe stored power measurement Rk was made, i.e., Ak = Vk−L


Fig. 37. GPO 3-D Markov chain states. States for which Vk – Vk−1 = Δ arediamonds and states for which Vk – Vk−1 = –Δ are circles.

Fig. 38. State transitions from interior states y2N (n−1)+2m −1 andy2N (n−1)+2m ;1 <= m <= N, 2 <= n <= N – 1.

for some L ≥ 1. The value of Vk+1 thus depends on the pow-ers measured while at voltage Vk and at voltage Ak = Vk−L .Vk+1 also depends on Vk−1 , for knowledge of the previous stepdirection Δk−1 = Vk − Vk−1 , in combination with knowledgeof Δk/Δk−1 , is required to determine Vk+1 . Hence, Vk+1 isdependent on {Vk , Vk−1 , Vk−L} and since L > 1 is possible,Xk = [VkVk−1 ] is no longer Markov according to (9). InsteadYk = [VkVk−1Ak ] satisfies (9) and thus the GPO algorithm isdescribed by a 3-D Markov chain.

The GPO 3-D Markov chain can be drawn in two dimen-sions by noting that, given Vk , Vk−1 must equal one of the twovalues Vk−1 = Vk ± Δ. Hence, in Fig. 37, the GPO Markovchain is represented as a pair of discrete-valued finite planes,with possible values for Vk along the horizontal axis and pos-sible values for Ak = Vk−L along the vertical axis. The twoplanes are drawn overlaid on a single page with states fromthe Vk − Vk−1 = Δ plane drawn as diamonds and with statesfrom the Vk − Vk−1 = −Δ plane drawn as circles. State transi-tions with nonzero probability from states with first componentVk ∈ {v2 , v3 , . . . , vN −1} are shown in Fig. 38, while those fromthe remaining end states are shown in Fig. 39. Then, for the stateprobability vector

Φ(k) = [Pr(Yk = y1) Pr(Yk = y2) ... Pr(Yk = y2N 2 )](20)

we have

Φ(k) = Φ(k − 1)Λ (21)

Fig. 39. State transitions from end states y2m −1 , y2m , y2N (N −1)+2m −1;and y2N (N −1)+2m , 1 < = m < = N .

where Λ is the GPO state transition probability matrix withelements λi,j . From Figs. 37–39, the complete set of nonzeroelements of Λ is

λ2m−1,2N +1 = λ2m,2N +1 = 1

λ2N (N −1)+2m−1,2N (N −1) = λ2N (N −1)+2m,2N (N −1) = 1 (22)

λ2N (n−1)+2m−1,j =

⎧⎪⎨⎪⎩

ρ1(m,n), j = 2n(N + 1) − 1ρ2(m,n), j = 2nN + 2m − 1ρ3(m,n), j = 2n(N + 1) − 4N

(23)

λ2N (n−1)+2m,j =

⎧⎪⎨⎪⎩

ρ1(m,n), j = 2n(N + 1) − 4N

ρ2(m,n), j = 2nN + 2m − 4N

ρ3(m,n), j = 2n(N + 1) − 1(24)

ρ1(m,n) = Pr [PM (vn ) + ηn ≥ PM (vm ) + ηm ]

= Q

[PM (vm ) − PM (vn )√

σ2n + σ2

m

](25)

ρ3(m,n) = Pr [PM (vn ) + ηn < α (PM (vm ) + ηm )]

= Q

[PM (vn ) − αPM (vm )√

σ2n + α2σ2

m

](26)

ρ2(m,n) = 1 − ρ1(m,n) − ρ3(m,n) (27)

where (22)–(27) are valid∀m = 1, 2, . . . , N and∀n = 2, 3, . . . ,N − 1. The computation of GPO tracking time and MPPT ef-ficiency is the same as described for PO except with Γ and Πreplaced by Φ and Λ, respectively.

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David C. Jones (M’85) received the B.S. degree fromBradley University, Peoria, IL, and the M.S. degreefrom the University of Illinois at Urbana-Champaign,Urbana-Champaign, in 1985 and 1986, respectively,both in electrical engineering. Since 2007, he hasbeen working toward the Ph.D. degree in electri-cal engineering from the University of Colorado,Boulder.

In 1984, he was a Technical Staff Member at TRWElectronics Systems Group where he was involved inthe analysis of satellite antennas. From 1985 to 1994,

he was a Technical Staff Member at Bell Communications Research (Bellcore),where he performed research in the field of digital subscriber lines. From 1994 to1997, he was a Technical Staff Distinguished Member at US WEST AdvancedTechnologies, where he researched broadband communications networks. From1997 to 2001, he was a Technical Director at Broadcom, Inc., where he devel-oped technology for very high speed digital subscriber lines. Since 2011, hehas been a Lead Engineer with Phobos Energy, Inc., Menlo Park, CA, where hedevelops photovoltaic power converters. His research interests include powerelectronics, renewable energy systems, signal processing, and algorithms forcommunications and control.

Robert W. Erickson (M’84–SM’99–F’00) receivedthe B.S., M.S., and Ph.D. degrees in electrical engi-neering from the California Institute of Technology,Pasadena, in 1978, 1980, and 1982, respectively.

Since 1982, he has been a member of the facultyof electrical and computer engineering at the Univer-sity of Colorado, Boulder, where he was a Depart-ment Chair from 2002 to 2006. Additionally, he is aCTO of Phobos Energy, Inc., Menlo Park, CA. Heis the author of the textbook Fundamentals of PowerElectronics (New York: Springer, 2001). His current

research interests include modeling and control of power conversion systems,modular/multilevel converter systems, and power electronics for renewable en-ergy sources (wind and solar).

Dr. Erickson received the IEEE Power Electronics Society Transactions PrizePaper Award, for the paper “Nonlinear Carrier Control for High-Power-FactorBoost Rectifier” in 1996.

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