1

A Distributed Approach to Maximum Power PointTracking for Photovoltaic Sub-Module Differential

Power ProcessingShibin Qin, Student Member, IEEE, Stanton T. Cady, Student Member, IEEE, Alejandro D. Domınguez-Garcıa,

Member, IEEE, Robert C.N. Pilawa-Podgurski, Member, IEEE

Abstract—This paper presents the theory and implementationof a distributed algorithm for controlling differential powerprocessing converters in photovoltaic (PV) applications. Thisdistributed algorithm achieves true maximum power point track-ing (MPPT) of series-connected PV sub-modules by relyingonly on local voltage measurements and neighbor-to-neighborcommunication between the differential power converters. Com-pared to previous solutions, the proposed algorithm achievesreduced number of perturbations at each step and potentiallyfaster tracking without adding extra hardware; all these featuresmake this algorithm well-suited for long sub-module strings. Theformulation of the algorithm, discussion of its properties, as wellas three case studies are presented. The performance of thedistributed tracking algorithm has been verified via experiments,which yielded quantifiable improvements over other techniquesthat have been implemented in practice. Both simulations andhardware experiments have confirmed the effectiveness of theproposed distributed algorithm.

I. INTRODUCTION

IN photovoltaic (PV) energy systems, PV modules are oftenconnected in series for increased string voltage. However,

there is usually mismatch between the I-V characteristicsof the series connected PV modules; this is typically theresult of partial shading, manufacturing variability and thermalgradients. Since all modules in a series string share thesame current, the string output power may be limited byunderperforming modules. A bypass diode is often connectedin parallel with each PV module to mitigate this mismatchand prevent PV hot spotting, but the efficiency loss is stillsignificant when only a central converter is used to performMPPT on the PV string.

To address the mismatch problem, distributed power elec-tronics architectures that perform module-level MPPT, or evensub-module-level MPPT, have been proposed; the two mostdominant architectures are DC optimizers [1]–[4], and micro-inverters [5]–[7], as shown in Fig. 1a and Fig. 1b, respectively.The major limitation of these two solutions, however, is thatthe distributed converters are connected in series with the PVmodules and must process the full power output of everymodule. Even with highly efficient distributed converters, theoverall power losses of the system can still be significant.

The information, data, or work presented herein was funded in part by theAdvanced Research Projects Agency-Energy (ARPA-E), U.S. Department ofEnergy, under Award DE-AR0000217.The authors are with the Department of Electrical and Computer Engineering,University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA. E-mail:sqin3, scady2, aledan, pilawa@ILLINOIS.EDU.

Differential power processing (DPP) for PV applicationshas gained significant attention recently due to its substantialimprovements over conventional solutions in terms of effi-ciency, reliability, and cost. The structure of a DPP-basedsystem in shown in Fig. 1c; an overview of the DPP conceptcan be found in [8]. In contrast to DC optimizers or micro-inverters, DPP converters, or differential power processors(DPPs), are configured in parallel with the PV string. Thebulk power that is common to all PV modules is directlyprocessed by the central inverter without any intermediateconversion. DPP converters only need to process the powerdifference between series connected PV modules, which isoften just a small fraction of the bulk power. This resultsin high efficiency of the system, small size and low powerratings of the power electronics circuit components. Due tothese advantages, DPP has been further extended to apply tosub-module-level applications [9]–[13].

In order to compare the efficiency of a DPP-based systemwith micro-inverters or DC optimizers, we consider a systemwith 10 PV modules (i.e., 30 PV sub-modules) as an example.The irradiances on the PV sub-modules are set randomly bydrawing from a Gaussian distribution with a standard deviationequal to 10% of the mean; the resulting maximum powerof each PV sub-module is displayed in Fig. 2 using stripedred bars. The differential power, i.e., the power differencebetween the maximum power of each PV module and thepower common to all PV modules, is also shown in Fig. 2using dotted blue bars. In a DC optimizer-based system, theDC optimizers have to process the full power of all PV sub-modules (represented by the striped red bars), which adds upto 2218 W. In a DPP-based system, each DPP only needsto process the differential power (represented by the dottedblue bars), which only adds up to 242.5 W. Both the DCoptimizer-based system and the DPP-based system then usea string-level inverter to convert DC string voltage into ACvoltage. In a micro-inverter system, the PV module voltageis directly converted to AC, but the efficiency of a micro-inverter is typically lower and its per-watt cost is typicallyhigher than that of a string-level inverter or a central inverter.Table I summarizes the power losses of these three differentsolutions for the irradiance condition in this example. DCoptimizers are assumed to have an average efficiency of 96%,while that of DPP converters is assumed to be 92% sincethey often operate in light-load conditions. For both the DCoptimizer and DPP-based system, a string inverter efficiency

2

DC

DC

DC

DC

DC

DC

DC

AC

(a) DC optimizer

DC

AC

DC

AC

DC

AC

(b) Micro-inverter

DC

DC

DC

DC

DC

AC

(c) DPP-based system

Fig. 1: Three types of distributed power electronics solutions for PV systems.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

25

50

75

100

Index of PV sub-modules and DPPs

Pow

er[W

]

differential power sub-module power

Fig. 2: Maximum power of PV sub-modules and differential power of a 30 PV sub-module system with randomly generatedirradiance condition.

of 98% was assumed. The micro-inverters are assumed to havean efficiency of 95%, and since, in practice, they only interfacewith PV modules, and therefore cannot compensate mismatchbetween sub-modules, the effective efficiency is typically evenlower. As is evident from Table I, the DPP-based system yieldssignificant advantages in terms of overall system efficiency.

To apply the concept of differential power processingto PV systems, especially at the sub-module-level, severalarchitectures and corresponding control schemes have beenproposed [9]–[11], [13]–[17]. For example, the work presentedin [14], [17] uses a switched inductor topology; the workpresented in [9] uses a resonant switched capacitor topol-ogy; the work presented in [10] uses a transformer coupledtopology. All of these proposed solutions adopt a controlmethod commonly known as voltage equalization or “virtualparallel” operation, in which all distributed converters striveto equalize the voltages of all PV sub-modules in a string.The distributed converters in the system run in open loop,simplifying the control requirements while obviating the needfor communication among converters. However, the voltageequalization approach can only achieve near maximum power

point (MPP) operation without truly seeking the MPP ofeach individual sub-module. The effectiveness of this approachrelies on the fact that the MPP voltages of sub-modules ina string are very close even if their MPP currents differsignificantly. This does require all the sub-modules to havevery similar electrical characteristics, which in practice istypically guaranteed by a costly binning process performedby PV module manufacturers [18]. However, even modulesthat are carefully matched at installation will suffer from non-uniform degradation after several years of field exposure. Asillustrated in [19], the standard deviation of PV module MPPvoltages may increase by nearly four times over a twenty-yearperiod. Furthermore, field operating conditions that result inthermal gradients or severe irradiance mismatch along the PVstring can also cause the MPP voltages of sub-modules todrift apart. All of these factors limit the tracking efficiency ofthe voltage equalization approach; thus, in order to increasethe total energy harvested over the entire lifetime of the PVsystem, it is necessary to develop a scheme capable of trueMPPT.

On the other hand, the generation control circuit presented

3

TABLE I: Comparison of system efficiency of the DC optimizer-based system, the micro-inverters and the DPP-based systemXXXXXXXXXXLosses

SolutionsDC optimizer Micro-inverter DPP

Power processed by DC stage [W] 2218 N/A 242.5

Intermediate power losses [W] 2218× 4% = 88.7 N/A 242.5× 8% = 19.4

Power processed by inverter(s) [W] 2218− 88.7 = 2129.3 2218 2218− 19.4 = 2198.6

Inverter power losses [W] 2129.3× 2% = 42.6 2218× 5.0% = 110.9 2198.6× 2% = 44.0

Overall efficiency 94.08% 95.00% 97.14%

TABLE II: Comparison of DPP control approaches````````````Features

ApproachesVirtual Parallel [9], [10], [14] Centralized P&O [11], [15], [16] Multilevel PPT [13] This work

Tracking Near MPPT True MPPT True MPPT True MPPT

Distributed algorithm Yes No Yes Yes

Requiring local current sensing No Yes Yes No

Requiring communication No Centralized Synchronization Neighbor-to-Neighbor

in [15], [16] achieves true MPPT without any local currentsensing at each sub-module, but employs a control scheme thatrequires communication between all converters and a centralcontrol unit. The central control unit has to (i) command allthe DPP converters to exhaust every possible combination ofconverter duty ratio perturbations, and (ii) measure the stringvoltage respectively in sequence before making the next DPPtracking step. For a system with n DPP converters, the centralcontrol unit has to try 2n duty ratio perturbations duringeach DPP tracking step, rendering the algorithm slow andinfeasible for a large system. Moreover, with this approach, thereliability of the system is fundamentally limited, as a singlefailure of the central control unit, or a communication linkfailure, would result in the malfunction or complete loss of theoverall system. The work presented in [11] adopts a buck-boostconverter topology that overcomes some of the limitations inthe architecture of the generation control circuit, and achievestrue MPPT with a faster control scheme. However, this controlscheme still requires communication between all DPP convert-ers and the central converter. Recently, a distributed MPPTapproach for a DPP-based system was presented in [13],which significantly reduces the communication requirements.However, this approach requires measurements of all PVsub-module currents; this results in additional power losses.Moreover, this approach requires synchronization betweenpower converters, which still relies on some communication.

Table II provides a comparison of some key features of DPPcontrol approaches presented in previous work and the oneproposed in this paper. The information in this table impliesthat for DPP converters, either communication or local currentmeasurements are needed to acquire adequate information toperform true MPPT operation. Implementing local currentmeasurement impairs the system efficiency and increases thehardware cost, whereas implementing communication mayresult in a much smaller impact because communication hard-ware is often required for other purposes, including individual

PV module diagnostics and on/off capability. Therefore, oursuggested approach is to eliminate local current sensing whilepreserving communication, but to reduce the communicationrequirement to overcome the limitations of the centralized ap-proaches. In this paper, we present a distributed algorithm thatrequires only neighbor-to-neighbor communication betweenadjacent DPP converters to perform sub-module-level trueMPPT. In our approach, there is no need for a central controlunit or any local current sensing. Each DPP only perturbs itsduty ratio once per DPP tracking step, so for a system with nDPP converters, only n duty ratio perturbations are requiredper DPP tracking step (instead of 2n perturbations in previoussolutions [15]); each DPP converter observes only local volt-age changes (instead of the entire string voltage). Moreover,the algorithm has the potential for parallelization of the dutyratio perturbations as the sensitivity between non-adjacentDPP converters is low, which can reduce the communicationoverhead and speed up the tracking even further. Comparedto the centralized counterparts, no extra hardware is neededsince the algorithm can be implemented in the DPP convertermicro-controllers that already exist for converter local control.Furthermore, since a central control unit is unnecessary andeach micro-controller can make independent control decisions,there is no longer a single point of failure, increasing theoverall system reliability.

The remainder of this paper is organized as follows. Theformulation of the proposed distributed MPPT algorithm ispresented in Section II. Some properties of the proposed algo-rithm are discussed in Section III. Hardware implementation ofthe DPP converters and the experimental setup used to verifythe effectiveness of the MPPT algorithm are introduced inSection IV. Section V provides case studies involving a 3-DPP system, and a 5-DPP system, including both simulationand experimental results, followed with simulation results fora larger 31-DPP system. Finally, Section VI concludes thepaper.

4

Io

Sub-module

1V1

+

−

Sub-module

2V2

+

−

Sub-module

3V3

+

−

D1

D2

1

Store:x1[k], z1[k]

Send:x1[k]

z1[k]

Measure:u1[k]

2

Store:x2[k], z2[k]

Send:x2[k]

z2[k]

Measure:u2[k]

Fig. 3: Block diagram of a 3-sub-module, 2-DPP system.

II. MPPT ALGORITHM FORMULATION

In this section, we formulate a distributed iterative algorithmwhich, through the exchange of information among localcontrollers, and for a given string current, maximizes thepower extracted from an array of n series-connected PV sub-modules outfitted with n−1 DPP converters. We first formulatethe algorithm for a 3-sub-module, 2-DPP system, and thengeneralize it to a system of any size.

A. Algorithm Formulation for a 3-sub-module, 2-DPP System

A system comprising two DPP converters and three sub-modules is illustrated in Fig. 3. This system is based on thearchitecture presented in [11]. Each DPP is implemented asa bidirectional buck-boost converter that enforces a voltageratio between two adjacent PV sub-modules. The PV string isattached to a central converter (omitted in Fig. 3). The controlobjective of the MPPT algorithm is to maximize the powerextracted from the PV string, Po, i.e.,

maximizeD1,D2

Po = Vo × Io, (1)

where Io is the string current, and Vo := V1 + V2 + V3 is thestring voltage. Given an irradiance condition, each PV sub-module has a unique maximum power point (Vi,max, Ii,max),i = 1, 2, 3. As derived in [8], when each PV sub-module isoperating at its respective maximum power point, the entiresystem is operating at its global maximum power point, andthe corresponding string current and the string voltage canbe uniquely determined through KCL and KVL analysis asVo,max and Io,max. Moreover, the duty ratios of all DPPconverters, D1,max and D2,max, can be uniquely determined,such that Io,max, D1,max and D2,max form a set of variablesthat fully determines the system. The control objective thusentails tracking the unique combination of (Io,max, D1,max,D2,max) that corresponds to Po,max.

To this end, we separate the setting of Io and Vo into twocontrol loops. In a relatively slow control loop, the centralconverter is configured as a controllable current sink, and it

maximize Vo (DPP)

update Io (central inverter)

timeI Io,1 o,2Po,1 Po,2 Io,3 Po,3 Io,4

Fig. 4: Timeline of the “slow” control loop (central inverter)and “fast” control loop (DPP).

performs a conventional perturb and observe (P&O) algorithmto seek the string current Io that gives the largest Po. In thesecond, relatively fast control loop, all DPP converters adjusttheir duty ratios in each DPP tracking iteration and take manyiterations to maximize Vo. Since the perturbation interval ofthe central converter is much longer than the time constant ofthe DPP converter control loop (because the DPP convertersare switching at a much higher frequency, as will be discussedin Section IV), at any given time, the string current Io can beconsidered temporarily fixed for the DPPs. Given a fixed Io,it is easy to see that, regardless of the irradiance incident oneach of the sub-modules, maximizing the string voltage Vo, isequivalent to maximizing Po.

The timeline of Fig. 4 illustrates the operation of the twocontrol loops: in the slow control loop, the central inverter em-ploys a P&O algorithm, such that it first updates and enforces(“perturb”) a certain string current value, Io. Following this,the DPP converters take many iterations in the fast controlloop to maximize the string voltage Vo for this Io. The stringpower is then measured (“observed”) by the central inverter touse in its P&O algorithm. The slow and fast control loops canrepeat this pattern and find the optimal string current Io,max.Here, we assume that the slow loop implemented in the centralinverter uses a conventional P&O algorithm to seek Io,max,and we focus on the distributed control algorithm for DPPs inthe fast control loop.

In our system architecture, we observe that each DPPhas access to measurements of only two of the three sub-module voltages instead of the entire string. Thus, in orderto determine the duty ratios for which Vo is maximized, weassume that the local controllers of DPP1 and DPP2 canshare information through neighbor-to-neighbor communica-tion; this can be described by a directed graph, as shown onthe right of the block diagram in Fig. 3. Then, we tailor thedistributed optimization algorithm in [20] to our setting. [Thereader is referred to the Appendix for a brief overview on themathematics of this algorithm.]

The control objective described above can be written as

maximizeD1,D2

Vo(D1, D2) =

V1(D1, D2) + V2(D1, D2) + V3(D1, D2),

where D1 and D2 are the duty ratios of DPP1 and DPP2,respectively. Then, the maxima of the function Vo(D1, D2)can be found by setting its gradient to zero, as in (2) at thebottom of this page, where D∗

1 and D∗2 are the respective duty

5

ratios of DPP1 and DPP2 that correspond to the maximumpower point. To find D∗

1 and D∗2 , the local controller of each

DPP iteratively adjusts its duty ratio based on (i) local voltagemeasurements, (ii) state variables maintained locally, and (iii)variables maintained by neighboring DPPs. Let k = 1, 2, . . .index the iterations performed by every DPP, and let x1[k],x2[k] be state vectors maintained by the local controllers ofDPP1 and DPP2, respectively. For DPP1, the entries of x1[k]are D1[k]—the actual duty ratio of DPP1, and D1,2[k]—DPP1’s estimate of the duty ratio of DPP2. Similarly, forDPP2, the entries of x2[k] are D2,1[k]—DPP2’s estimate ofthe duty ratio of DPP1, and D2[k]—the actual duty ratioof DPP2. Furthermore, let z1[k] and z2[k] be ancillary statevectors maintained by the local controllers respectively; unlikex1[k] and x2[k], these ancillary states do not correspond to anyphysical variable. For the entire system, we define

x[k] =[︸ ︷︷ ︸

x1[k]

D1[k], D1,2[k],︸ ︷︷ ︸

x2[k]

D2,1[k], D2[k]]T, (3)

z[k] =[︸ ︷︷ ︸

z1[k]

z1,1[k], z1,2[k], ︸ ︷︷ ︸z2[k]

z2,1[k], z2,2[k]]T. (4)

Then, at each iteration, the variables are updated as

x[k + 1] = (I4 − δL)x[k]− δLz[k] + δγu[k], (5)

z[k + 1] = z[k] + δLx[k], (6)

where I4 is the 4 × 4 identity matrix; L = L ⊗ I2, where I2is the 2× 2 identity matrix,

L =

[1 −1−1 0

]

is the Laplacian matrix of the graph representing the exchangeof information between local controllers as depicted in Fig. 3,and “⊗” denotes the Kronecker product of L and I2 (see, e.g.,[21]); δ and γ are parameters that can be used to tune thealgorithm (we will discuss this matter in detail in Section III);and u[k] is defined as in (7) at the bottom of this page, withϕ1 and ϕ2 defined respectively as

ϕ1(D1, D2) := V1(D1, D2) +1

2V2(D1, D2),

ϕ2(D1, D2) :=1

2V2(D1, D2) + V3(D1, D2).

From the above discussion, we see that the update of DPP1’sstates, x1[k + 1] and z1[k + 1], depends on its own states,x1[k] and z1[k], the states of its neighbor, x2[k] and z2[k],and the partial derivatives of the sub-module voltages thatDPP1 is directly attached to, i.e., u1[k]. In this regard, whilethe necessary information that each DPP converter needs forupdating (5) and (6) can be acquired through neighbor-to-neighbor communication, in order to obtain ui[k], each DPPi

must estimate the partial derivatives of the ϕi(·) function.To perform this estimation, at every iteration, each DPPi

alternatively perturbs its duty ratio by a fixed small amount,∆Di, while both local controllers observe the sub-modulevoltages; the timeline of this process is depicted in Fig. 5. Afterboth local controllers have observed the results of each pertur-bation, and upon receiving the necessary information from theneighboring DPP, they can approximate the partial derivativesas ∂ϕi

∂Dj≈ ∆ϕi

∆Dj, i, j = 1, 2, and update their respective state

variables. After k = m iterations, for m sufficiently large,we have that D1[m] ≈ D∗

1 and D2[m] ≈ D∗2 ; thus the string

voltage is maximized and the maximum power is extractedfrom the system at a given string current Io. The fast controlloop continuously tracks the maximum power and once theslow control loop updates Io to a new value, the fast controlloop will maximize the output power for this new condition.The flowchart in Fig. 6 summarizes how the two control loopsoperate and interact. Note that the two control loops are nottime synchronized, but operate independently of one another.

B. Algorithm Formulation for an n-sub-module, (n-1)-DPPSystem

For a system of n sub-modules and n− 1 DPP converters,the communication graph of which is shown in Fig. 7, wedefine the state vector, and the ancillary state vector of theentire system as

x[k] =[x1[k] x2[k] . . . xi[k] . . . xn−1[k]

]T, (7)

z[k] =[z1[k] z2[k] . . . zi[k] . . . zn−1[k]

]T, (8)

where xi[k] and zi[k], i = 1, 2, . . . , n − 1, are state vec-tors maintained by the local controller of DPPi. Each localcontroller maintains an estimate of the duty ratios of allother DPPs in addition to its own; thus, xi[k] consists of

∇V (D∗1 , D

∗2) =

∂V1(D1,D2)∂D1

∣∣∣∣D∗

1 ,D∗2

+ ∂V2(D1,D2)∂D1

∣∣∣∣D∗

1 ,D∗2

+ ∂V3(D1,D2)∂D1

∣∣∣∣D∗

1 ,D∗2

∂V1(D1,D2)∂D2

∣∣∣∣D∗

1 ,D∗2

+ ∂V2(D1,D2)∂D2

∣∣∣∣D∗

1 ,D∗2

+ ∂V3(D1,D2)∂D2

∣∣∣∣D∗

1 ,D∗2

=

[00

](2)

u[k] :=

[

︸ ︷︷ ︸u1[k] = ∂ϕ1(D)

∂D

∂ϕ1

∂D1

∣∣∣∣D1[k],D2[k]

, ∂ϕ1

∂D2

∣∣∣∣D1[k],D2[k]

,

︸ ︷︷ ︸u2[k] = ∂ϕ2(D)

∂D

∂ϕ2

∂D1

∣∣∣∣D1[k],D2[k]

, ∂ϕ2

∂D2

∣∣∣∣D1[k],D2[k]

]T

, (7)

6

time

k k + 1

Converter 1 Perturbs

Converter 2 Perturbs

∆D1

∆ϕ1 ⇒ ∆ϕ1

∆D1≈ ∂ϕ1

∂D1

∆ϕ2 ⇒ ∆ϕ2

∆D1≈ ∂ϕ2

∂D1

observeComputed by converter 1

Computed by converter 2

∆D2

∆ϕ1 ⇒ ∆ϕ1

∆D2≈ ∂ϕ1

∂D2

∆ϕ2 ⇒ ∆ϕ2

∆D2≈ ∂ϕ2

∂D2

observe

Fig. 5: Timeline of distributed perturb and observe for the 3-sub-module, 2-DPP system.

Fig. 6: Simplified flowchart of the proposed algorithm for the 3-sub-module, 2-DPP system.

Di,j [k], i, j = 1, . . . , n− 1, i 6= j and Di[k], i.e.,

xi[k] =[Di,1[k], . . . , Di[k], . . . , Di,n−1[k]

]T,

zi[k] =[zi,1[k], zi,2[k], . . . , zi,n−1[k]

]T.

We also define the input vector as

u[k] =[∂ϕ1(D)

∂D∂ϕ2(D)

∂D . . . ∂ϕn−1(D)∂D

]T, (9)

where∂ϕi(D)

∂D=[∂ϕi(D)∂D1

, ∂ϕi(D)∂D2

, . . . , ∂ϕi(D)∂Dn−1

]T, (10)

with ϕi(D) given by

ϕi(D) =

Vi(D) + 12Vi+1(D), i = 1,

12Vi(D) + 1

2Vi+1(D), 1 < i < n− 1,

12Vi(D) + Vi+1(D), i = n− 1.

(11)

Then, at each iteration, similar to (5) and (6), the variables

are updated as

x[k + 1] = (I(n−1)2 − δL)x[k]− δLz[k] + δγu[k], (12)

z[k + 1] = z[k] + δLx[k], (13)

where I(n−1)2 is the (n − 1) × (n − 1) identity matrix; L =L⊗ In−1, where L is now the Laplacian matrix of the graphin Fig. 7:

L =

1 −1 0 . . . . . . 0−1 2 −1 0 . . . 0

0 −1 2 −1. . .

......

. . . . . . . . . . . ....

.... . . . . . −1 2 −1

0 · · · · · · · · · −1 1

.

By inspecting (12) and (13) for the n-sub-module case, itis easy to see that the state of DPPi at instant k+ 1, xi[k+ 1]and zi[k + 1], depends on its own state at instant k, xi[k]

7

1

x1[k]

z1[k]

2

x2[k]

z2[k]

3

x3[k]

z3[k]

i-1

xi−1[k]

zi−1[k]

i

xi[k]

zi[k]

i+1

xi+1[k]

zi+1[k]

n-2

xn−2[k]

zn−2[k]

n-1

xn−1[k]

zn−1[k]

Fig. 7: Graph of an n-sub-module, (n-1)-DPP system.

and zi[k]; the states of its neighboring DPPs at instant k, i.e.,xi−1[k], zi−1[k], xi+1[k], zi+1[k], all of which can be acquiredthrough the neighbor-to-neighbor communication; and ui[k],which is approximated by DPPi as ∆ϕi(D)

∆D . Similar to the 2-DPP case, after k = m iterations, for m sufficiently large, wehave Di[m] ≈ D∗

i , i = 1, 2, . . . , n− 1.Note that although the algorithm update functions are com-

pactly written in matrix form in (12) and (13), the actual datastorage and computation is distributed among the DPPs. Tounderstand how this algorithm is distributed, consider DPPi

(node i) in Fig. 7 as an example. DPPi locally stores andupdates vector xi[k] and zi[k] that are part of (7) and (8),respectively. For the computation performed by DPPi, only theith row of Laplacian matrix L (representing the informationexchange of DPPi with its closest neighbors) is relevant. If weexpand (12) and (13), and write down only the part needed tocompute xi[k + 1], we obtain that

xi[k + 1] = δxi−1[k] + (1− 2δ)xi[k] + δxi+1[k]+

δzi−1[k]− 2δzi[k] + δzi+1[k] + δγui[k],

zi[k + 1] = zi[k]− δxi−1[k] + 2δxi[k]− δxi+1[k],

which illustrates that the computation of xi[k+ 1] can be per-formed independently by DPPi as long as this DPP converterhas access to the state of its closest neighbors (this is achievedin practice through neighbor-to-neighbor communications).Other states of the system are irrelevant to DPPi’s compu-tation. A similar conclusion applies to other DPP converters;therefore, each DPP converter performs only part of the updatecomputation relevant to itself, and thus the proposed algorithmis distributed.

III. DISCUSSION ON ALGORITHM CHARACTERISTICS

In this section, we first briefly introduces the computer sim-ulation platform of the PV system and the proposed algorithm.Following this, we consider a 6-sub-module, 5-DPP systemas an example, and illustrate via computer simulations thecharacteristics of the proposed algorithm in terms of (i) updatefunction parameter tuning, (ii) communication topology, and(iii) reconfiguration after communication failure.

A. Simulation Platform

We use MATLAB as the platform for all our numericalsimulations. As shown in Fig. 8, the simulation platformconsists of two independent parts: the PV system model blockand the control block. The PV system model contains a setof equations that describes the electrical behaviors of the sub-modules and the DPP converters. The I-V characteristic ofeach PV sub-module is described using the non-linear equa-tions based on the model presented in [22]. The buck-boost

Fig. 8: Components of the simulation platform.

converter are modeled using the steady state average equations[23]. Each converter takes a duty ratio command from thecontrol block and enforces a corresponding voltage ratiobetween two neighboring PV sub-modules, and the Newton-Raphson method is applied to solve the set of equations ofthe PV system model. The control block of Fig. 8 runs thedistributed algorithm as presented in Section II. During eachcontrol algorithm iteration, the control block passes duty ratiocommands to the PV system block, which solves the systemvariables and returns the voltage information to the controlblock. The control block then uses the voltage informationto calculate the update function, after which it starts a newiteration. Parameters in the control algorithm or the irradiancecondition of PV sub-modules can be set before the simulationor changed during the simulation to study various situations.

B. Choice of Tuning Parameter γ and δ

In (5), the parameter γ determines the influence that theinput vector u[k] has on the state vector x[k]. Typically arelatively large value of γ renders a shorter response timebut larger transient overshoot of duty ratios. As an example,Fig. 9a and Fig. 9b shows the duty ratio of a 6-sub-module,5-DPP system evolving over time with different values of γin the update function; the normalized irradiance profile usedin this example is as follows

100%, 100%, 100%, 100%, 100%, 100%t=4 s−→ 100%, 100%, 80%, 80%, 50%, 50%t=6 s−→ 100%, 100%, 100%, 80%, 50%, , 50%t=8 s−→ 100%, 100%, 100%, 100%, 100%, 100%, (14)

with step changes occurring at t = 4 s, 6 s, 8 s as indicated.As shown in Fig. 9a and Fig. 9b, for different values of γ, thealgorithm converges to the same steady-state value after eachirradiance change. Compared to other transients, the duty ratioovershoot is very large at a “cold start”, which corresponds tot = 0 s in Fig. 9b. [At “cold start”, the algorithm is justactivated without receiving any information of the system.]

8

0 2 4 6 8 100.48

0.49

0.5

0.51

0.52

Time [s]

Dut

yra

tio

γ = 0.001

D1 D2 D3D4 D5

(a) γ = 0.001.

0 2 4 6 8 100.48

0.49

0.5

0.51

0.52

Time [s]

Dut

yra

tio

γ = 0.01

D1 D2 D3D4 D5

(b) γ = 0.01.

Fig. 9: Evolution of duty ratios computed by the distributedalgorithm with fixed γ for a 6-sub-module, 5-DPP system.

Although the entries of x[k] can be initialized to 0.5, theentries of z[k] can only be initialized to 0. Since z[k] isintended to balance the input vector u[k] in (5), the state vectorx[k] is very sensitive to u[k] when z[k] is small; in this case, asmaller γ is preferable to suppress the overshoot. Note that theslow convergence resulting from this small γ is not a problemsince, in the field, “cold start” conditions correspond to turningon the DPP system, which typically happens only once perday. Once the system converges and z[k] reaches steady state,a small γ is undesirable due to its slow response, and a largerγ can be used for faster tracking upon irradiance changes (alsoreferred to as “hot start”). Duty ratio overshoot in a “hot start”condition is typically small despite a large γ since the z[k] hasalready reached a steady state value. Therefore, some on-linetuning scheme for γ can be incorporated in the algorithm toimprove its convergence properties; next, we discuss one suchscheme.

To develop a method for smoothly changing the value of γwithout introducing any disturbance to the duty ratio, we firstnote that after the algorithm converges to a steady state value,

0 2 4 6 8 100.48

0.49

0.5

0.51

0.52

Time [s]

Dut

yR

atio

γ = 0.001t=3s−→ γ = 0.01

D1 D2 D3D4 D5

γ adjusted

Fig. 10: Evolution of duty ratios computed by distributedalgorithm with adaptive γ for a 6-sub-module, 5-DPP system.

we have that

x[k + 1] = x[k] = x∗, (15)z[k + 1] = z[k] = z∗. (16)

Substituting (15) and (16) in (12) and (13), respectively, yields

Lz∗ = γu∗, (17)

Lx∗ = 0, (18)

which suggests that scaling γ and z∗ by the same multi-plicative constant does not affect the equilibrium point of thealgorithm. Therefore, the on-line tuning scheme can modifyγ without introducing a disturbance on the duty ratio as longas z[k] is simultaneously changed by the same scaling factor;Fig. 10 illustrates the response of a 5-DPP system after sucha tuning scheme is applied. The system “cold starts” with asmall γ, and modifies the value of γ at t = 3 s without anynoticeable disturbance to the duty ratios. The work in [24]presents the result of long-term measurement illustrating thetransient effects of irradiance changes in solar PV applications,which can be used in the selection of γ for a suitable balancebetween speed and accuracy.

Similar to the choice of γ, the choice of δ in (12) and(13) can also be increased to speed up the convergence ofthe algorithm. While γ controls the update iteration of x[k]to x[k + 1] by scaling the influence of the gradient step asdetermined by u[k], i.e., the larger the value of γ is, the largerthe update of x[k] to x[k+1] will be, δ also scales the influenceof the gradient step as determined by u[k]; but it also scales theinfluence of the auxiliary variable Lz[k]; however, the signs ofthese terms are opposite, so it is not clear a priori which termwill have more influence. On the other hand, δ is also the timeelapsed between two consecutive iterations k and k+ 1; thus,by increasing δ, the iteration time is reduced proportionally.When choosing δ, there is a trade-off between speed andstability of the algorithm. Because the system is highly non-linear, δ can only be determined empirically, and we found δshould be kept smaller than 0.1 to guarantee stability in mostscenarios.

9

1

x1[k]z1[k]

2

x2[k]z2[k]

3

x3[k]z4[k]

4

x4[k]z4[k]

5

x5[k]z5[k]

Fig. 11: Graph of a 6-sub-module, 5-DPP system with redun-dant communications.

0 2 4 6 8 100.48

0.49

0.5

0.51

0.52

Time [s]

Dut

yR

atio

Communication with Closest and Second-closest Neighbors

D1 D2 D3D4 D5

Fig. 12: Evolution of duty ratios computed by distributedalgorithm with redundant communications.

C. Impact of Communication Topology on Convergence Speed

The convergence speed of the algorithm can be improvedby modifying the communication topology; Fig. 11 illustratesa directed graph representing the communication topology ofa 6-sub-module, 5-DPP system. The same irradiance patternin (14) is applied to the 6 sub-modules. By relying onlyon neighbor-to-neighbor communication (as captured by thesolid line in Fig. 11), the evolution of duty ratios is relativelyslow as has already been shown in Fig. 10. If redundancy isintroduced to the communication topology, e.g., each DPP cancommunicate with its second closest neighbor(s) in additionto its direct neighbor(s) (as captured by the dashed line inFig. 11), the convergence speed can be accelerated (as inFig. 12) and the proposed algorithm can be easily adaptedto accommodate this change. We only need to modify theLaplacian matrix L in the update functions in (12) and (13)to reflect the new communication topology, which in this caseis given by

L =

2 −1 −1 0 0−1 3 −1 −1 0−1 −1 4 −1 −10 −1 −1 3 −10 0 −1 −1 2

. (19)

Besides L, no further modification to the algorithm is neces-sary. As illustrated in Fig. 12, the extra communication linkssignificantly improve convergence speed. Further simulationindicates that as we add more communication links, the con-

vergence speed of the algorithm further increases. Althoughit seems that additional communication hardware is requiredin the real system to introduce the modification as discussedabove, this is not really the case. As will be introduced inSection IV, our hardware design, which integrates 3 DPPson one printed circuit board (PCB), readily lends itself to thecommunication topology in which each DPP can communicatewith at least its first, second, and third neighbors in bothdirections without extra hardware.

D. Impact of Communication Topology on Reliability

As shown in [20], the proposed distributed algorithm con-verges to the optimal solution as long as the directed graphdescribing the communication topology remains strongly con-nected. Therefore, when the communication topology hasredundancy, the algorithm can be reconfigured to work undercommunication link failures. Consider the 6-sub-module, 5-DPP system again with the communication topology shown inFig. 13. Each DPP is communicating with its first and secondclosest neighbors. The Laplacian matrix of this graph is givenin (19). If a communication link fails as illustrated by thedashed lines in Fig. 13, and this failure is detected by thelocal controllers of the DPPs on each end of the link, thetwo controllers involved can simply reconfigure the algorithmby modifying the corresponding row of the Laplacian matrixused in their update function to reflect this change in thecommunication topology. Thus, the new Laplacian matrix isgiven by

L =

1 −1 0 0 0−1 3 −1 −1 00 −1 3 −1 −10 −1 −1 3 −10 0 −1 −1 2

; (20)

note that only the first and third row in the Laplacian matrixchange. This change is completely local to the two DPPs onthe ends of the failed communication link, i.e., DPP1 andDPP3, and affects only the update functions of these two DPPs.The update functions for other DPPs remain the same andcontinue to perform MPPT, since other rows of the Laplacianmatrix do not change. If another communication failure occursafter the first one, e.g., as illustrated by the dotted line inFig. 13, the Laplacian matrix can be modified again; thisresults in

L =

1 −1 0 0 0−1 3 −1 −1 00 −1 2 0 −10 −1 0 2 −10 0 −1 −1 2

. (21)

Figure 14 illustrates the simulation result of this reconfigu-ration assuming unchanged irradiance pattern and the afore-mentioned communication link failures happening at t = 4 sand t = 7 s, respectively. As the simulation shows, after eachcommunication failure, our algorithm is able to restore theduty ratio back to the optimal value after a short transient.

10

1

x1[k]z1[k]

2

x2[k]z2[k]

3

x3[k]z4[k]

4

x4[k]z4[k]

5

x5[k]z5[k]

Fig. 13: Graph of a 6-sub-module, 5-DPP system with com-munication failure.

Induced comm. failureInduced comm. failure

0 2 4 6 8 100.48

0.49

0.5

0.51

0.52

Time [s]

Dut

yR

atio

Communication Reconfiguration

D1 D2 D3D4 D5

Fig. 14: Evolution of duty ratios computed by distributedalgorithm with communication failure reconfiguration.

IV. LABORATORY TESTBED

In this section, a brief introduction to the DPP hardwaredesign is provided, followed by a description of the experi-mental setup used to verify the effectiveness of the proposedalgorithm.

A. DPP Hardware Implementation

One of the primary goals of distributed power electronicsfor PV systems, including DPP-based systems, is to achievePV module integration [25]. Nowadays, the cost of separateenclosures for distributed power electronics represents a verysignificant portion of the total system cost, motivating effortsto reduce the converter footprint to fit into the existingweather-resistant junction box of an off-the-shelf PV module.Therefore, for sub-module DPP systems, the goal of thehardware design in this work is to achieve miniaturizationof the DPPs for junction box integration, while maintaininghigh efficiency and capability for the implementation of theproposed distributed MPPT algorithm.

Figure 15 illustrates the wire connection of a DPP system;typically one PV module consists of three sub-modules. There-fore, each junction box needs to integrate three DPPs. Betweenadjacent junction boxes there are three wire connections: oneseries string connection in which the bulk power commonto all sub-modules flows, one differential connection throughwhich the DPPs shuffle the small power mismatch, and oneor more wires for communication. We noted that the bulkand differential power wires can be used for communicationpurpose, but that approach is not used in this work.

Module N

Module N-1

DPP2

DPP3

DPP1

Module N+1

N1

SubModule

N2

SubModule

N3

SubModule

(N+1)1

SubModule

(N-1)3

SubModule

DPP1

DPP2

DPP3

Communication

Comm

Series Wire

Differential Wire

Series Wire

Differential Wire

Junction Box N

Junction Box N+1

Junction Box N-1

DPP3

DPP2

DPP1

DPP3

DPP2

DPP1

Comm

Comm

Fig. 15: PV junction box connection for DPP system.

L

OverlapControl

PWM

Vdrive

Micro-

PWM1

DPP2

VDD

DrMOSmodule 3

Sub-

module 2

Sub-

module 1

Sub-

PWM2

PWM3

Level Shifter

3.3 Vcontroller

5 VVin

Vout

DPP1

DPP3

C

C

Reg

VSS1

VSS

VSS

VSS2

Cshift

Rshift

Fig. 16: Schematic of hardware design (details shown forDPP2 only).

With the design objective described above in mind, weimplement DPPs as bidirectional synchronous buck-boostconverters. As analyzed in [11], the buck-boost topologyallows for the use of low-voltage, fast-switching transistors.By employing a high switching frequency in the order ofhundreds of kHz, the size of the magnetic components, whichtypically dominate the converter size, can be significantlyreduced. Furthermore, while in [11] each DPP has one dedi-cated micro-controller, each micro-controller in our proposedimplementation has enough peripherals to control more thanone DPP. Moreover, since the 3 DPPs in one junction boxare physically very close, they can be integrated on one PCB

11

TABLE III: Main Component ListDevice Model Value Manufacturer

Microcontroller STM32F105 STMicroelectronicsDRMOS SiC780 VishayLinear Regulator UA78L05CPK Texas InstrumentL SER1360-103KL 10 µH CoilcraftC TMK212BBJ106KG-T 10 µF×4 Taiyo YudenDshift 1SS416CT(TPL3) ToshibaCshift 0402 4700 pFRshift 0402 100 kΩI2C Isolator ADuM1250 Analog Devices

and controlled by one micro-controller. This eliminates theneed for communication between DPPs in one junction boxand reduces the control overhead in terms of hardware costand power consumption. In this work, we chose a low-cost32-bit ARM Cortex-M3 micro-controller, which also allowssome flexibility for future expansions, e.g., implementationof fault detection algorithms. Each DPP is constructed witha shielded power inductor and an integrated DRMOS powerstage to achieve high efficiency and a very small footprint onthe PCB; Table III contains a list of the main componentsused.

One unique challenge of controlling 3 DPPs with one micro-controller is sending the control signal across different voltagelevels, as shown in Fig. 16. This figure illustrates the circuitdetails for DPP2 only, while those for DPP1 and DPP3, whichare similar, are omitted. In Fig. 16, while DDP1 can becontrolled directly by the micro-controller, the PWM controlsignals sent to DPP2 and DPP3 must be level shifted becausethey do not have the same voltage reference as the micro-controller. The level shifting circuitry employed in this workis implemented using low-cost passive devices (Rshift, Cshift

and Dshift in Fig. 16). When the micro-controller PWM2

signal is low, the level shifting capacitor gets charged to thevoltage difference between the ground potential of DDP2 andthe ground potential of the micro-controller through the diode.The PWM input pin of DPP2’s DRMOS is thus pulled toDPP2’s ground potential. When the micro-controller PWM2

signal reaches the high-voltage limit (3.3 V), the voltage of theDRMOS’s PWM input pin is pushed to its high-voltage limit(3.3 V) with respect to DPP2’s ground reference, and the diodeturns off. Since the PWM input pin of the DRMOS has a highimpedance, any current that flows through the level shiftingcapacitor is very small, so the input voltage to the DRMOS’sPWM input pin can remain high for a long enough time beforePWM2 goes low again. A resistor is placed in parallel with thediode to prevent over-voltage across the PWM input pin due toa sudden decrease of sub-module 1 voltage, i.e., the voltagedifference between DPP2’s ground reference and the micro-controller’s ground reference. The designer should be awareof certain considerations when selecting the value of level-shifting capacitors and resistors; factors to consider includethe time constant of the level shifter compared to the PWMfrequency, and the driving capability of micro-controller PWMpins. For protection of the micro-controller PWM pin, a smallvalue resistor can be placed in series with the level-shiftingcapacitor to reduce current spike during start-up.

Figure 17a shows an annotated photograph of the front

(a) The front side of the test prototype.

(b) The prototype fitting in a junction box.

Fig. 17: Annotated photograph of the hardware prototype.

TABLE IV: Hardware Prototype Specifications

Sub-module Voltage Range 3− 134 VConverter Power Rating 60 WSwitching Frequency 100 kHzDuty Ratio Resolution (with PWM dithering) 1/3600Converter Peak Efficiency 95%Weight 28 gVolume 8.575 cm3

side of the hardware prototype. Magnetic inductors are onthe back side of the PCB. Note that a large portion of theboard area is consumed by large connectors and ancillarycircuitry to facilitate development and diagnosis, which canbe eliminated in a final product. As shown in Fig. 17a, allessential components of the hardware, including the inductorson the back side of the PCB, only take up a 3.75 cm×3.75 cmarea encompassed by a white rectangle, and, as shown inFig. 17b, can easily fit in a junction box.

A potential limitation of the digital PWM signal generatedby the micro-controller is that with limited clock frequency,a trade-off has to be made between PWM frequency andPWM resolution. The PWM frequency is determined by therequirement of high efficiency and small converter size, so ithas to be maintained at a relatively high value (100 kHz in

12

0 1 2 3 460

70

80

90

100

Output Current [A]

Effi

cien

cy[%

]

DPP1

DPP2

DPP3

Fig. 18: Measured efficiency of the 3 DPPs on one hardwareprototype at 10 V input voltage, 50% duty ratio.

this work), rendering a low PWM resolution given the limitedmicro-controller frequency. In each iteration of the MPPTalgorithm proposed in Section II, the duty ratio updates that arecalculated with high precision must be quantized to within thePWM resolution, impairing the effectiveness of the algorithm.To overcome this limitation, a PWM dithering technique wasadapted from [26]–[28]. By dithering periodically between twoadjacent quantized duty ratio values available from the micro-controller, with proper filtering, the effective PWM resolutionin the hardware prototype is increased by 10 times. Theeffective PWM resolution and other important specificationsof the hardware prototype are listed in Table IV. Note that theDPP converter peak efficiency is above 95%. An efficiencycharacterization of the DPP converter across the entire loadrange is shown in Fig. 18. Note that the relatively low lightload efficiency can be greatly improved using pulse frequencymodulation (PFM) techniques [11].

B. Experimental Setup

To verify the proposed MPPT algorithm, we developed theindoor experimental setup shown in Fig. 19. A power supply(HP 6631A) was connected in parallel with each PV sub-module to replicate the photo-generated current, providing anoutput I-V characteristic similar to the one that would result ifthe PV sub-module were to be illuminated by sunlight. Detailsabout this method to conduct repeatable indoor PV experimentcan be found in [29]. Five DPPs implemented on 2 prototypeboards were connected in parallel with a string of six PV sub-modules. Three of them are from a SunmoduleTM SW 235Poly PV modules, and the other three are from a SolarWorldSunmoduleTM SW 245 Poly PV modules. These two modulesare manufactured in the same process, but binned into twodifferent models by the manufacturer for better characteristicsmatching. In this experiment they are chosen intentionallyto illustrate that PV modules which undergo a less stringentbinning process and thus have more mismatch can benefit fromour proposed solution. Besides, an electronic load (Agilent6060B) acted as a central converter and controlled the string

PVSub-module

PVSub-module

PVSub-module

4

5

6

Agilent 6060B

Electronic Load

Digital Multimeter

Agilent 34401

GPIB

I2C

PCB1

Digital Multimeter

Agilent 34401

Istring

Vstring

PVSub-module

PVSub-module

PVSub-module

1

2

3

DPP

PCB2

DPP

DPP

DPP

DPP

I2C

Fig. 19: Experimental setup.

current. Two digital multimeters (Agilent 34401) were usedto measure the string voltage and current. All power supplies,digital multimeters and the electronic load communicated witha computer through a GPIB interface. Micro-controllers on thetwo prototype boards communicated with each other and withthe same computer through an I2C interface. Note that forthe following tests of the proposed distributed algorithm, thecomputer only received information from the micro-controllersfor data logging and diagnostic purposes. The distributedalgorithm, including data exchange and synchronization ateach iteration, is fully implemented in the micro-controllers.

In the experiments, the current limits of the power supplieswere set to certain percentages of the sub-modules’ nominalshort-circuit current to emulate the corresponding percentageof irradiance (e.g., a combination of 5 A, 4 A and 2.5 A wasused to emulate an irradiance profile of 100%, 80% and 50%normalized value). For a given irradiance mismatch scenario,the string current was kept fixed at a certain value by theelectronic load to emulate the behavior of a central inverterduring its P&O interval. The evolution of duty ratios of allDPPs, along with the string current and voltage after theconvergence of the algorithm, was recorded by the computer.

V. CASE STUDIES

In this section, we verify the effectiveness of the proposeddistributed algorithm through three case studies. We firstpresent and compare simulation and experimental results fromthe simplest 2-DPP system to illustrate its basic features. Then,we focus on the experimental result of a six-sub-module 5-DPP system and analyze its tracking efficiency. Finally, we

13

0 2 4 6 8 10 12 140

2

4

6

Voltage [V]

Cur

rent

[A]

Sub-module1Sub-module2Sub-module3

Fig. 20: I-V characteristics of 3 sub-modules with100%, 80%, 50% normalized irradiance.

0 1 2 3 4 50

50

100

String Current [A]

Stri

ngPo

wer

[W]

Without DPP With DPP

Fig. 21: String P-I characteristic of the a 2-DPP system withirradiance pattern of 100%, 80%, 50%.

demonstrate the scalability of the algorithm by presentingsimulation results for a system containing 31 DPPs and 32sub-modules.

A. 2-DPP system

To study the simplest 2-DPP case, we conducted experi-ments on the setup discussed in Section IV but with only 3 sub-modules and compared the data with a MATLAB simulation.We set the irradiance pattern, string current, and initial dutyratios to be the same in the experiment and simulation, andrecorded the evolution of the states of the algorithm. This testwas performed for different conditions, and we found goodmatching between the simulation and the experimental data.For example, we set up a condition with 100%, 80%, 50%normalized irradiance on sub-modules 1, 2 and 3, respectively(Fig. 20 shows the measured corresponding I-V curves); Weprogrammed the electronic load in Fig. 19 to initially performa reference string current sweep, where the DPP converterswere not operating. Following this, we performed a stringcurrent sweep with the DPP converters operating, executingthe proposed algorithm. The DPP converters were allowed

0 50 100 150 200

0.46

0.48

0.5

0.52

0.54

Iterations, k

Dut

yra

tio

D1[k] D1[k]

D2[k] D2[k]

(a) Duty ratio simulation result.

0 50 100 150 200

0.46

0.48

0.5

0.52

0.54

Iterations, k

Dut

yra

tio

D1[k] D1[k]

D2[k] D2[k]Dref1 Dref2

(b) Duty ratio experimental result.

0 50 100 150 20028

29

30

31

32

Iterations, k

Stri

ngVo

ltage

[V]

String Voltage

(c) String voltage experimental result.

Fig. 22: Evolution of duty ratios computed by distributedalgorithm for a 2-DPP system with irradiance pattern of 100%,80%, 50%, Istring = 3.3A, δ = 0.1 and γ = 0.003.Experimental result of the evolution of the centralized algo-rithm proposed in [11] is also plotted in dotted lines in asa benchmark reference for the final value after convergence.The string voltage change in the experiment as a result of theevolution of the proposed distributed algorithm is also plotted.

time for 200 iterations, which is enough time for the algorithmto converge to maximize the string voltage for that particularstring current. The result of the string current sweeps is shownin Fig. 21. The DPP converters running the proposed algorithmsmoothed out the P-I characteristics of the mismatched PV

14

string and eliminated the local maxima. Therefore, a centralinverter running the current referenced P&O method canconverge to the optimal current given the concave P-I char-acteristics of the PV string with DPP converters. To illustratethe operation of the DPP control algorithm, we consider onesituation in the current sweep as an example. When the stringcurrent is 3.3A (corresponding to the global maximum power),Fig. 22a shows the evolution of the simulated duty ratios forthis condition; Figure 22b shows the evolution of the dutyratios computed using the experimental setup with the samecondition. As the figure shows, the state variables in bothsimulations and experiments quickly converge and roughlyfollow the same trajectory (also note that the duty ratios D1[k]and D2[k] settle to the same value as corresponding dutyratio estimates D1[k] and D2[k]). To verify that the algorithmconverges to the correct value, on the experimental setup, wealso executed the centralized MPPT algorithm in [11], andrecorded its duty ratio evolution as a benchmark (i.e., Dref1

and Dref2 in Fig. 22b). The final duty ratios of the proposedalgorithm after convergence are almost identical to those of thecentralized MPPT algorithm. The string voltage was measuredas the proposed distributed algorithm converged, as shown inFig. 22c, which confirms the increase in string voltage. Testsof other irradiance patterns and string current exhibited similarresults.

B. 5-DPP system

Similar to the 2-DPP case, we tested the proposed algorithmon a 5-DPP system for different conditions. String currentsweeps were performed in the same way as described forthe 2-DPP system. The P-V characteristics are similar tothat of the 2-DPP system and are omitted here. Again, wetake one string current value in the current sweep stepsas an example. Figures 23a and 23b show the simulationand experimental result for the same conditions and initialduty ratios. Note that among all the state variables, onlythe actual duty ratios Di[k], i = 1, 2, . . . , 5, are plotted (thecorresponding Di[k]s are omitted). Along with the distributedMPPT algorithm, the centralized MPPT algorithm proposed in[11] was also executed, and its evolution is plotted in Fig. 23bas a benchmark. As the algorithm converged, an increasein string voltage was experimentally measured. with similarcharacteristics as the 2-DPP case. Note that in both the 2-DPPand the 5-DPP systems, states in simulation and experimentfollow very similar trajectories but do not converge to exactlythe same final values; this is primarily due to two reasons.First, the simulation does not take into account the converterpower losses. Second, as we used two types of PV modules inthe experiment to emulate modules with less stringent binning,there is some variation in the I-V curves of the six sub-modules, which is not captured in the simulation.

At the same time, it is evident that the proposed algorithmand the centralized MPPT algorithm converge to the same finalvalue. However, due to the noise in the voltage measurements,the duty ratio values still have small variations after conver-gence. With a smaller value of γ in (12), the variation aroundthe steady-state value can be significantly reduced as Di[k]

0 100 200 300 400 5000.48

0.49

0.5

0.51

0.52

Iterations, k

Dut

yra

tio

D1[k] D2[k] D3[k]

D4[k] D5[k]

(a) Simulation result, γ = 0.002.

0 100 200 300 400 5000.49

0.5

0.51

0.52

0.53

Iterations, k

Dut

yra

tio

D1[k] D2[k] D3[k]

D4[k] D5[k] Dref1

Dref2 Dref3 Dref4

Dref5

(b) Experimental result, γ = 0.002.

0 100 200 300 400 5000.49

0.5

0.51

0.52

0.53

Iterations, k

Dut

yra

tio

D1[k] D2[k] D3[k]

D4[k] D5[k] Dref1

Dref2 Dref3 Dref4

Dref5

(c) Experimental result, γ = 0.0004.

Fig. 23: Evolution of duty ratios computed by distributedalgorithm for a 5 -DPP system with irradiance pattern of100%, 100%, 80%, 80%, 50%, 50%, Istring = 3.3A, δ = 0.1and different γ. Experimental result of the evolution of thecentralized algorithm proposed in [11] is also plotted indotted lines as a benchmark reference for the final value afterconvergence.

is less dependent on ui[k], but the algorithm will take moreiterations to converge to the steady state value, as shown inFig. 23c, which is consistent with the analysis in Section III-B.In terms of hardware related issues, the noise in the voltagemeasurements can be reduced by averaging multiple reading,

15

TABLE V: Duty ratio comparison and tracking efficiency after algorithm convergenceIrradiance Pattern 100%, 100%, 80%, 80%, 50%, 50% 100%, 100%, 100%, 100%, 100%, 100%

Centralized MPPT Duty Ratio 0.503, 0.503, 0.508, 0.497, 0.503 0.499, 0.500, 0.502, 0.496, 0.501

Distributed MPPT Duty Ratio 0.502, 0.502, 0.510, 0.497, 0.504 0.499, 0.499, 0.502, 0.496, 0.501

Actual Sub-module Power [W] 45.42, N/A, 35.33, 36.59, N/A, 22.34 45.16, N/A, 44.96, 45.99, N/A, 46.54

Maximum Sub-module Power [W] 45.89, 45.63, 35.56, 36.67, 21.43, 22.35 45.89, 45.63, 45.39, 46.06, 45.00, 46.71

Tracking Efficiency 98.98%, N/A, 99.35%, 99.78%, N/A, 99.95% 98.41%, N/A, 99.05%, 99.85%, N/A, 99.63%

or filtering with larger capacitors at the ADC pins, whichmay increase the time needed for each iteration. Therefore,a trade-off between the convergence speed of the algorithmand the ‘noise’ in the steady state values should be carefullyconsidered in practice.

To further verify that the proposed algorithm effectivelytracked the maximum power point, we measured the statictracking efficiency, defined as the ratio of the actual powerextracted to the maximum power of each sub-module for agiven irradiance, when the string current is at the optimalvalue. To acquire the actual power extracted from each sub-module, we used two digital multi-meters (Agilent 34401) toperform synchronized, fast (0.006 NPLC) measurements onthe current and voltage of each sub-module after the algorithmconverged (the algorithm was kept running), so the trackingefficiency losses due to duty ratio perturbations, switchingripples and other transients could be taken into account inthese measurements. The measurements were performed 100times continuously. Sub-module power was calculated fromeach pair of current and voltage measurements, and averagedover the 100 times. On the other hand, to acquire the maximumsub-module power, an output voltage sweep was performed oneach sub-module under the same irradiance, and the maximumoutput power was recorded. The resulting tracking efficiencywas calculated as in in Table V for a few cases. Note that asintroduced in Section IV-B, the six sub-modules used in theexperiment were from two PV modules. Due to the internalwiring of the PV modules (three sub-modules connected inseries in the junction box), the current of the sub-module in themiddle of each PV module was inaccessible, i.e., the currentof sub-module 2 and sub-module 4 could not be directlymeasured when the algorithm was running. Therefore, we onlymeasured the power extracted from the other sub-modules.

Another very important figure of merit is the dynamicMPPT efficiency of the system. It should be noted that thefocus of this work is the control of DPP converters so thatthey can work with an existing central inverter design andcompensate the mismatch between sub-modules in a PV arrayduring partial shading, etc.. While the DPP converters willimprove the dynamic MPPT efficiency of the entire systemby compensating for sub-modules mismatch, the dynamicefficiency of the system is determined and lower-bounded bythe central inverter, which is not the focus of this work. Thedynamic MPPT efficiency will be explored in future field testswhere an existing central inverter will be used.

0 2,000 4,000 6,000 8,0000.4

0.45

0.5

0.55

0.6

Iterations, k

Dut

yra

tio

Fig. 24: Evolution of duty ratios computed by distributedalgorithm for 31-DPP simulation with 100% irradiance on allsub-modules and random initial duty ratios.

C. 31-DPP system

To demonstrate the scalability of the proposed distributedMPPT algorithm, we extended the simulation to model a sys-tem with 32 sub-modules and 31 DPPs. The evolution of theduty ratios found by each of the DPP local controllers using thedistributed algorithm is shown in Fig. 24, for randomly choseninitial conditions, and for an irradiance pattern in which allsub-modules receive 100% irradiance. As Fig. 24 shows, thealgorithm converges to the expected solution wherein the dutyratios are Di = 0.5, i = 1, . . . , n−1. While more iterations arerequired for a larger system to converge from a random initialpoint, some of the terms in ui[k] = ∂ϕi(D)

∂Dj≈ ∆ϕi(D)

∆Dj, as

defined in (10), are effectively zero when i and j are far apart.This allows parallelization of DPP duty ratio perturbations tospeed up the convergence; we plan to explore this idea infuture work.

VI. CONCLUSIONS

In this paper, we addressed the problem of maximizingthe power extracted from a system of series-connected PVmodules outfitted with DPPs. To tackle some of the draw-backs of previously proposed centralized MPP algorithms, weintroduced a distributed perturb and observe algorithm whichrelies on neighbor-to-neighbor communication, and requiresonly local voltage measurements. Hardware suitable for PVjunction box integration was developed. To verify the efficacy

16

of the proposed algorithm, we presented simulation and ex-perimental results for a 2-DPP and a 5-DPP system. We thendemonstrated the scalability of the algorithm by presentingsimulation results for a 31-DPP system.

APPENDIX

In this appendix, a brief introduction to the mathematicalbackground of our proposed algorithm is presented; the readeris referred to [20] for a more detailed account. While [20]derives the algorithm in the continuous-time framework, thealgorithm proposed in this work presents the discrete-timecounterpart.

To derive the algorithm, we will start in the continuous-time framework and discretize the resulting dynamics after-wards. Consider a network with n nodes, i.e., v1, v2, . . . , vn,whose communication topology can be described by a stronglyconnected digraph G, and each node vi, i ∈ 1, . . . , n hasa continuously differentiable and convex function f i that isonly available to this node, then as illustrated in [20], theunconstrained optimization problem

min f(xo) =

n∑

i=1

f i (xo) (22)

can be equivalently formulated as a constrained optimizationproblem

min f(x) =

n∑

i=1

f i (xi) , (23)

subject to Lx = 0n, (24)

where L = L⊗ In; L is the Laplacian Matrix representing G,In is the n×n identity matrix, and “⊗” represents Kroneckerproduct of L and In; 0n is the n-dimensional all-zero vector;xi is node vi’s estimate of the solution to (22), and x is avector containing estimates from all nodes, i.e.,

x = [xT1 , xT2 , . . . , x

Tn ]T . (25)

Then this constrained optimization problem can be solvedusing augmented Lagrangian method, i.e.,

minF (x, z) = f(x) + zT Lx+1

2xT Lx, (26)

where zT is the estimate of the Lagrange multiplier. To solve(26), we need to find the point (x∗,z∗) that sets the gradientof F (x, z) to zero, which we can obtain through the followingdynamics:

x+ Lx+ Lz = −∇f(x), (27)

z = Lx. (28)

As (27) and (28) converge to the equilibrium point (x∗,z∗),x∗ is the solution to (23), and x1 = x2 = · · · = xn in (25) isthe solution to the original problem in (22).

In our specific DPP system setting, the control objective is

maximizeD1,...,Dn

V (D1, . . . , Dn) =

n∑

i=1

Vi (Di) . (29)

This control objective has the same separable structure as (22);thus we can tailor the algorithm in (27) and (28) to our setting.To this end, we discretize (27) and (28) using the forwardrectangular rule, i.e.,

x =x[k + 1]− x[k]

δ, z =

z[k + 1]− z[k]

δ. (30)

Then by defining u = −∇f(x), and substituting it and theexpression in (30) into (27) and (28), we obtain that

x[k + 1] = (I(n−1)2 − δL)x[k]− δLz[k] + δu[k], (31)

z[k + 1] = z[k] + δLx[k]. (32)

After introducing the tuning parameter γ to (31) and (32), wearrive at (12) and (13) as presented in Section II.

REFERENCES

[1] G. R. Walker and P. C. Sernia, “Cascaded dc-dc converter connection ofphotovoltaic modules,” IEEE Transaction on Power Electronics, vol. 19,no. 4, pp. 1130–1139, 2004.

[2] N. Femia, G. Lisi, G. Petrone, G. Spagnuolo, and M. Vitelli, “Distributedmaximum power point tracking of photovoltaic arrays: Novel approachand system analysis,” IEEE Transactions on Industrial Electronics,vol. 55, no. 7, pp. 2610–2621, 2008.

[3] L. Linares, R. Erickson, S. MacAlpine, and M. Brandemuehl, “Improvedenergy capture in series string photovoltaics via smart distributed powerelectronics,” in Proc. of the Applied Power Electronics Conference andExposition, pp. 904–910, 2009.

[4] R. Pilawa-Podgurski and D. Perreault, “Submodule integrated distributedmaximum power point tracking for solar photovoltaic applications,”IEEE Transactions on Power Electronics, vol. 28, no. 6, pp. 2957–2967,2013.

[5] Q. Li and P. Wolfs, “A review of the single phase photovoltaic module in-tegrated converter topologies with three different dc link configurations,”IEEE Transaction on Power Electronics, vol. 23, no. 3, pp. 1320–1333,2008.

[6] R. Erickson and A. Rogers, “A microinverter for building-integratedphotovoltaics,” in Proc. of the Applied Power Electronics Conferenceand Exposition, pp. 911–917, 2009.

[7] A. Trubitsyn, B. Pierquet, A. Hayman, G. Gamache, C. Sullivan, andD. Perreault, “High-efficiency inverter for photovoltaic applications,” inProc. of the Energy Conversion Congress and Exposition, pp. 2803–2810, 2010.

[8] P. Shenoy, K. Kim, B. Johnson, and P. Krein, “Differential power pro-cessing for increased energy production and reliability of photovoltaicsystems,” IEEE Transactions on Power Electronics, vol. 28, no. 6,pp. 2968–2979, 2013.

[9] J. Stauth, M. Seeman, and K. Kesarwani, “Resonant switched-capacitorconverters for sub-module distributed photovoltaic power management,”IEEE Transactions on Power Electronics, vol. 28, no. 3, pp. 1189–1198,2013.

[10] C. Olalla, D. Clement, M. Rodriguez, and D. Maksimovic, “Architec-tures and control of submodule integrated dc-dc converters for photo-voltaic applications,” IEEE Transactions on Power Electronics, vol. 28,no. 6, pp. 2980–2997, 2013.

[11] S. Qin and R. C. Pilawa-Podgurski, “Sub-module differential powerprocessing for photovoltaic applications,” in Proc. of the Applied PowerElectronics Conference and Exposition, pp. 101–108, 2013.

[12] S. Qin, A. J. Morrison, and R. C. Pilawa-Podgurski, “Enhancing micro-inverter energy capture with sub-module differential power processing,”in Proc. of the Applied Power Electronics Conference and Exposition,pp. 621–628, 2014.

[13] C. Schaef and J. T. Stauth, “Multilevel power-point-tracking for variable-conversion-ratio photovoltaic ladder converters,” in Proc. of the Work-shop on Control and Modeling for Power Electronics, pp. 1–7, 2013.

[14] S. Poshtkouhi, A. Biswas, and O. Trescases, “Dc-dc converter forhigh granularity, sub-string mppt in photovoltaic applications using avirtual-parallel connection,” in Proc. of the Applied Power ElectronicsConference and Exposition, pp. 86–92, 2012.

[15] T. Shimizu, M. Hirakata, T. Kamezawa, and H. Watanabe, “Generationcontrol circuit for photovoltaic modules,” IEEE Transactions on PowerElectronics, vol. 16, pp. 293 –300, May 2001.

17

[16] T. Shimizu, O. Hashimoto, and G. Kimura, “A novel high-performanceutility-interactive photovoltaic inverter system,” IEEE Transaction onPower Electronics, vol. 18, no. 2, pp. 704–711, 2003.

[17] H. Bergveld, D. Buthker, C. Castello, T. Doorn, A. de Jong, R. vanOtten, and K. de Waal, “Module-level dc/dc conversion for photovoltaicsystems: The delta-conversion concept,” IEEE Transactions on PowerElectronics, vol. 28, no. 4, pp. 2005–2013, 2013.

[18] H. Field and A. Gabor, “Cell binning method analysis to minimizemismatch losses and performance variation in si-based modules,” inProc. of the Photovoltaic Specialists Conference, pp. 418–421, 2002.

[19] C. E. Chamberlin, M. A. Rocheleau, M. W. Marshall, A. M. Reis,N. T. Coleman, and P. Lehman, “Comparison of pv module performancebefore and after 11 and 20 years of field exposure,” in Proc. of thePhotovoltaic Specialists Conference, pp. 101–105, 2011.

[20] B. Gharesifard and J. Cortes, “Distributed continuous-time convex opti-mization on weight-balanced digraphs,” IEEE Transactions on AutomaticControl, to appear, 2014.

[21] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge UniversityPress, 1990.

[22] M. Villalva, J. Gazoli, and E. Filho, “Comprehensive approach tomodeling and simulation of photovoltaic arrays,” IEEE Transactions onPower Electronics, vol. 24, pp. 1198–1208, May 2009.

[23] R. Erickson and D. Maksimovic, Fundamentals of Power Electronics.Kluwer Academics, 2000.

[24] R. J. Serna, B. J. Pierquet, J. Santiago, and R. Pilawa-Podgurski,“Field measurements of transient effects in photovoltaic panels and itsimportance in the design of maximum power point trackers,” in Proc.of the Applied Power Electronics Conference and Exposition, pp. 3005–3010, 2013.

[25] C. Deline, B. Marion, J. Granata, and S. G. Sigifredo, “A performanceand economic analysis of distributed power electronics in photovoltaicsystems,” Tech. Rep. NREL/TP-5200-50003, January 2011.

[26] A. Peterchev and S. Sanders, “Quantization resolution and limit cyclingin digitally controlled pwm converters,” IEEE Transactions on PowerElectronics, vol. 18, no. 1, pp. 301–308, 2003.

[27] C. Barth and R. Pilawa-Podgurski, “Implementation of dithering dig-ital ripple correlation control,” in Proc. of the Workshop on Control,Modeling and Simulation in Power Electronics, pp. 1–7, 2013.

[28] C. Barth and R. Pilawa-Podgurski, “Dithering digital ripple correlationcontrol with digitally-assisted windowed sensing for solar photovoltaicmppt,” in Proc. of the Applied Power Electronics Conference andExposition, pp. 1738–1746, 2014.

[29] S. Qin, K. Kim, and R. Pilawa-Podgurski, “Laboratory emulation ofa photovoltaic module for controllable insolation and realistic dynamicperformance,” in Proc. of the Power and Energy Conference at Illinois,pp. 23–29, 2013.

Shibin Qin (S’12) received the B.E. degree inElectrical Engineering from Huazhong University ofScience and Technology in 2012 and the M.S. degreein Electrical Engineering from University of Illinoisat Urbana-Champaign in 2014, where he is currentlypursuing a Ph.D. degree. His research is in powerelectronics in photovoltaic applications.

Stanton T. Cady (S’10) received the B.S. degree inElectrical Engineering in 2009 and the M.S. degreein Electrical Engineering in 2011, both from theUniversity of Illinois at Urbana-Champaign, Urbana,IL, USA, where he is currently working toward thePh.D. degree in electrical engineering.

His research interests are include distributed con-trol and its application to electric power systems andpower electronics.

Alejandro D. Domınguez-Garcıa (S’02, M’07) isan Assistant Professor in the Department of Electri-cal and Computer Engineering of the University ofIllinois at Urbana-Champaign, where he is affiliatedwith the Power and Energy Systems area. His re-search interests are in the areas of system reliabilitytheory and control, and their application to electricpower systems, power electronics, and embeddedelectronic systems for safety-critical/fault-tolerantaircraft, aerospace, and automotive applications. Hereceived the Ph.D. degree in Electrical Engineering

and Computer Science from the Massachusetts Institute of Technology, Cam-bridge, MA, in 2007 and the degree of Electrical Engineer from the Universityof Oviedo (Spain) in 2001. After finishing his Ph.D., he spent some timeas a post-doctoral research associate at the Laboratory for Electromagneticand Electronic Systems of the Massachusetts Institute of Technology. Dr.Domınguez-Garcıa received the NSF CAREER Award in 2010, and the YoungEngineer Award from the IEEE Power and Energy Society in 2012. He is alsoa Grainger Associate since 2011. He currently serves as an Associate Editorfor the IEEE Transactions on Power Systems and the IEEE Power EngineeringLetters.

Robert Pilawa-Podgurski (S’06, M’11) was bornin Hedemora, Sweden. He received dual B.S. de-grees in physics, electrical engineering and computerscience in 2005, the M.Eng. degree in electricalengineering and computer science in 2007, and thePh.D. degree in electrical engineering in 2012, allfrom the Massachusetts Institute of Technology.

He is currently an Assistant Professor in theElectrical and Computer Engineering Department atthe University of Illinois, Urbana-Champaign, and isaffiliated with the Power and Energy Systems group.

He performs research in the area of power electronics. His research interestsinclude renewable energy applications, energy harvesting, CMOS powermanagement, and advanced control of power converters. Dr. Pilawa-Podgurskireceived the Chorafas Award for outstanding MIT EECS Master’s thesis, theGoogle Faculty Research Award, and the Richard M. Bass Outstanding YoungPower Electronics Engineer Award of the IEEE Power Electronics Society.He is co-author of two IEEE prize papers.