�Abstract-- Recent interest in distributed generation (DG) due

to the opening of the electricity market and the need foralternatives to conventional fossil fuel-based electricitygeneration has revived interest in grid-connected photovoltaic(PV) systems. Studies need to be performed at the power systemlevel to examine the impacts of grid-connected PV systems andseveral models for PV arrays have been proposed in theliterature for this purpose. However, the complexities of thesemodels and the difficulties of implementing them in softwareprograms can be deterrents to their use. This paper proposes asimple, robust and flexible piecewise linear PV device model fordynamic and transient power system studies. The circuit-basednature of the model is beneficial because it facilitatesunderstanding of the PV device and its behavior in the connectedcircuit. Software implementation is straightforward and it caneven be constructed using standard software library components,as demonstrated using PSCAD/EMTDC.

Index Terms—circuit modeling, circuit simulation, nonlinearcircuits, photovoltaic cells, photovoltaic power systems, piecewiselinear approximation, power system simulation, semiconductordevice modeling.

I. INTRODUCTION

HE opening of the electricity market and the need foralternatives to conventional fossil fuel based electricity

generation has brought renewed interest in grid-connectedphotovoltaic (PV) systems. New studies are being performedat the power system level to examine the impacts of greaterpenetrations of this form of distributed generation (DG), but itis not uncommon to find that the PV array models used areoversimplified.

The output of a PV array is highly non-linear, and tosimplify the array to a constant voltage source or a current-controlled voltage source is often not appropriate. Severalmodels for PV arrays have been proposed in the literature [1]-[4]. However, in some cases, the models are not fullyexplained and in all cases, their complexity, when coupledwith the difficulties of implementing them in various softwareprograms, can be a deterrent to their use. Furthermore, thelevel of detail inherent in these models may not be necessary

This work was supported by the Power System and Computer ApplicationLab (PSCAL), Department of Electronic and Electrical Engineering,Sungkyunkwan University, Suwon, S. Korea.

R. C. Campbell is currently with the Department of Electrical Engineering,University of Washington, Seattle, WA 98195 USA (e-mail:ryanchasecampbell@hotmail.com).

for system level studies, and it is helpful to have a flexiblemodel that can be based either on theory or on measuredmodule/array I-V characteristics. In regards to the latter, PVmodule manufacturers often provide measured data points fortheir products, available either on their Websites or uponrequest. There also exists a useful empirical model with adatabase of well over one-hundred commercially available PVmodules and several installed arrays [5].

This paper proposes a PV device model suitable for usewith power electronic inverters in dynamic and transientpower system studies, and reflects the PV array's non-linearDC characteristics. The model is robust, flexible, and itsimplementation in software is straightforward; it can even beconstructed using standard software library components as isdemonstrated in this paper using PSCAD/EMTDC.

II. PV MODEL DEVELOPMENT

A. Background

There are several ways to model a PV cell, module, orarray for simulation. Custom components are often coded byadvanced software users (though the researchers who do thisrarely reveal their code). However, mathematical and softwareintegration complexities involving software numericalintegration methods and modeling non-linearities at the smalltime-steps needed for simulation with power electronicconverters can make this a challenging task. The resultingmodel may not be as robust or reliable as desired; trouble-shooting and debugging can also potentially be time intensive.Furthermore, modeling the continuous non-linear I-V curvebased on (1) (see Section B) using the compensating currentsource method can result in excessive computations whichconsiderably slow simulation speed [6].

To overcome these problems, this paper presents a PVarray model based on the single-diode model described below,and implemented directly as an electrical circuit in a way thatavoids the mathematical complexities of the implicit and non-linear (1). This circuit-based method is advantageous becauseit allows the electrical engineer to visualize and betterunderstand the PV device, and its behavior in the connectedcircuit, than if it were simply a black box or in the form of acomplex equation.

A Circuit-based Photovoltaic Array Model forPower System Studies

Ryan C. Campbell, Student Member, IEEE

T

97

978-1-4244-1726-1/07/$25.00 c© 2007 IEEE

B. Theoretical Circuit Equivalent

The ideal solar cell can theoretically be modeled as acurrent source anti-parallel with a diode, as shown connectedby the solid lines in Fig. 1. The DC current, I'sc, generatedwhen the cell is exposed to light, varies linearly with solarirradiance. Refinement of the PV cell model includes theeffects of series and shunt resistance, as shown connected bythe dashed lines. A second diode can also be included [7], [8],[1], as it provides an even more accurate I-V curve thataccounts for the difference in current flow at low currentvalues due to charge recombination in the semiconductor'sdepletion region. Note that in this paper, only the single diodemodel is used.

Fig. 1. PV Cell Equivalent Circuit Model

Since a PV module is composed primarily of series-connected cells, and a PV array is composed of series- andparallel-connected modules, the single cell circuit cantheoretically be scaled up to represent any series/parallelcombination.

C. Theoretical Equation

The equation for the single-diode model, including seriesand shunt resistances, is given by (1) (adapted from [7]):

sq(V+IR )skT

sc o1sh

(V + I RI = I' I (e 1)

R

)� � � (1)

whereI'sc is the light-generated current (short-circuit value

assuming no series/shunt resistance)Io1 is the dark saturation currentq is the charge of an electron (coul)k is the Boltzman constant (j/K)T is the cell temperature (K)I, V, Rs, Rsh are cell current (A), voltage (V), series and

shunt resistance (Ohms) as indicated in Fig. 1.

Fig. 2 illustrates the I-V characteristics of the single diodemodel, showing the ideal curve and illustrating the effects ofseries and shunt resistances.

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Voltage (V)

Cur

rent

(A)

IdealShunt R

Series R

Fig. 2. I-V Curves for DC PV Cell Models (Single Diode Model)

D. Piecewise Linear (PWL) Model

It is challenging to obtain a good fit to the non-linear curveof Fig. 2 with power system simulation programs usingstandard software library components due to the simplificationof diode representation: in most circuit simulators that utilizethe trapezoidal integration solution method, diodes aremodeled piecewise linear (PWL) as voltage controlledresistors with two states: on and off resulting in low and highresistance values, respectively. Using a PWL diode, the circuitmodel for the non-linear response is shown in Fig. 3 alongwith the resulting two-segment I-V curve. Since manysoftware programs provide parameters for the diode's turn-onvoltage and on-resistance, a PV cell, module, or even an entirearray can be modeled with this simple circuit by setting thediode voltage near the cell/module/array open-circuit voltage,setting the diode on-resistance to give the appropriate slope,and adjusting the current source to the desired value of light-dependent short-circuit current.

0.00 0.20 0.40 0.60 0.80-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50 +y

-y

-x +x

PWL

Ideal

Voltage, (V)

Cur

rent

,(A

)

Fig. 3. Two-segment PWL Circuit & I-V Curve

While this approximation may be sufficient for somestudies, it is not a very good match to the ideal curve and willclearly over-estimate the maximum power point (MPP) of thePV array, which is located at the knee of the curve. It will alsobe problematic for MPP tracking controls, since the slope of

98 2007 39th North American Power Symposium (NAPS 2007)

the curve changes so abruptly - in reality, the MPP is notnearly so sharp so it is far easier for the power electronics tocontrol the operating point. Some improvement could be madein these respects by aligning the MPP's, but this would requireexaggerated series and shunt resistances, which would reducethe overall curve fitting accuracy.

A much better fit to the non-linear curve can be achievedusing three series diodes and two low valued bypass resistors,as shown in Fig. 4. In fact, the curve can be fit to any degreeof desired accuracy in this way by simply adding more bypassresistor and diode pairs. Methods for determining theappropriate values for the resistors and diode voltages will bediscussed in the next section.

0.00 0.20 0.40 0.60 0.80-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50 +y

-y

-x +x

Ideal

PWL

Voltage, (V)

Cur

rent

,(A

)

4

3

21

Fig. 4. Four-segment PWL Circuit & I-V Curve

Since a PV device's operating point is determined by theequivalent resistance of its load, I-V curves such as thoseabove can be generated using a variable resistor that sweepsfrom zero to infinite resistance. Operation of the circuit in Fig.4 can thus be explained as follows:

1) When the load resistance is small, all diodes are off, andthe bypass resistors provide a path for voltage to buildup across the bottom diode, resulting in the first segmentof the curve in the constant-current region.

2) As the load resistance increases, voltage builds up acrossthe bottom diode until its turn-on voltage is reached, atwhich point more current is drawn through the branch(though limited by the bypass resistors) resulting in thesecond PWL segment.

3) If the value of the middle resistor is greater than that ofthe top resistor, then the middle diode will turn on next,resulting in operation in the third PWL segment.

4) Finally, the top diode switches on and the deviceoperates in the (nearly) constant voltage region of thePWL curve. The slope of this segment is determined bythe equivalent resistance of the branch comprised of 3diode on-resistances and 2 bypass resistors.

E. Calculation of PWL Model Parameters

Methods for determining the vertex points and calculatingthe values of the bypass resistors and diode turn-on voltagesare thus needed, and several are presented below:

- The values of bypass resistors and diode turn-onvoltages can be selected iteratively by trial-and-error tofit any reference curve available (such as the ideal curveof Fig. 2).

- PV module manufacturers often provide measured datafor their products, either on their Websites, or uponrequest. Depending on the data quality, an appropriatePWL-type curve fitting technique, such as the MDL(minimum descriptor length) method in [9] or [10], canbe used to fit the data. Low quality data sets may benefitfrom pre-fitting using a PV-specific method (e.g., [11]).

- Another option is to use the Sandia NationalLaboratories model and database [5], which is based onreal module and array data. The Sandia model can beused to generate the vertices of the PWL curve based onthe five key points of the I-V curve shown in Fig. 5.Note that compared to the previous figures, which gavecurves for individual cells, Fig. 5 gives a curve for amodule composed of series-connected cells. Thismethod is also suitable for entire arrays, and the Sandiamodel does provide data for several arrays.

Fig. 5. PWL Vertex Determination via the Sandia Model [5]

The Sandia model or any curve for which the appropriatedata points are available (i.e., Ix and Ixx as shown in Fig. 5)can then be used to determine the resistor and diode voltagevalues of the proposed model according to the followingprocedure:

1) Choose diode on resistances to be very small (e.g., 0.001Ohm), depending on the software limitations;

2) Choose diode off resistances to be very large (e.g., 1E6Ohm);

3) Calculate Rsh and Rs from (2) and (3), which give theslopes of the first and forth segments, respectively, asdefined by Fig. 4 or Fig. 5:

2007 39th North American Power Symposium (NAPS 2007) 99

oc scsh

sc x

0.5V VR R1

I I

�� �

�(2)

oc oc mps

xx oc

V 0.5(V + V )R = R4 =

(I I )

�

�(3)

4) Calculate the top and middle bypass resistor values (Rtop

and Rmid, respectively) from the simultaneous solution of(4) and (5), in which the values of diode on and offresistances suggested in steps 1 and 2 are used forclarity:

topmidsh

mid tops

topmidsh

mid top

1E6 R1E6 RR 0.001+ +

1E6 + R 1E6 + RR2 = R +

1E6 R1E6 RR + +

1E6 + R 1E6 + R

� �� � (4)

topmidsh

mid tops

topmidsh

mid top

1E6 R0.001 RR 0.001+ +

0.001 + R 1E6 + RR3 = R +

1E6 R0.001 RR + +

0.001 + R 1E6 + R

� �� � (5)

where R2 and R3 are given by (6) and (7) as the slopes ofthe second and third segments, respectively, of Fig. 4 orFig. 5:

mp oc

x mp

V 0.5VR2

I I

��

�(6)

oc mp mp

mp xx

0.5(V V ) VR3

I I

� ��

�(7)

5) Finally, calculate the values of diode forward voltagedrop from Equations 8, 9 and 10, in which the values ofdiode on and off resistances suggested in steps 1 and 2are again used for clarity:

oc sc x sbot

top mid

(0.5V V ) + I RV =

R + R1+

1E6

�(8)

� topmid mp oc s mp x

top mid

R 0.001V V 0.5V R (I I ) 1

R R 0.001

�� � � � �

� �

� �� �

(9)

top oc bot midV = V V V� � (10)

F. Accounting for Changes in Cell Temperature or IncidentIrradiance

As mentioned in Section B, PV device I-V characteristicsare temperature dependent; however, for many studies, suchas for transient analyses, changes in cell temperature are not ofmuch interest and can often be assumed zero. Solar celltemperature varies with incident sunlight, ambient temperatureand wind, but these effects can not change the temperature ofa PV module very quickly, thus the assumption of constanttemperature is often reasonable. However, in some cases, itmay be necessary to examine a scenario at one temperatureand then change the cell temperature and re-run thesimulation. It may also be necessary to account for the smallvariation in open-circuit voltage produced by changes inincident sunlight. Rather than re-calculating the diode voltagesand bypass resistors based on a curve at another temperatureor photo-current, an easy modification of the proposed modelis to insert an inverted ideal DC voltage-source into the diodebranch below the bottom diode, as shown in Fig. 6.

Fig. 6. Addition of Voltage Source to Account for Open-circuit VoltageVariation

The value of the offset voltage needed to account for thevoltage dependence at the new value of temperature or photo-current can be calculated as follows:

1) since output current equals zero at open-circuit, the IRs

term in (1) is zero and (1) can be solved for the newopen-circuit voltage, Voc,new as given by (11):

ocqV

kToc,new sh sc oV = R I' I ( e 1)

� �� �� �

(11)

2) The difference between the original array voltage, Voc,old

and Voc,new is then equal to the DC voltage offset, Voc,adj,as given by (12):

oc,adj oc,old oc,newV = V V� (12)

100 2007 39th North American Power Symposium (NAPS 2007)

III. CONCLUSIONS

This paper proposed a PV device model suitable for usewith power electronic converters in dynamic and transientpower system studies. The model accounts for the importantnon-linear DC response of PV devices, and is simple, robustand flexible. For power system level studies, existing modelssuffer from unnecessary complexities and fromimplementation difficulties in various software programs. Theproposed model, however, is simple and implementation isstraightforward: it can even be constructed using standardsoftware library components as demonstrated in this paperusing PSCAD/EMTDC. Furthermore, compared to black-boxPV device models, the model’s circuit-based nature enablesthe electrical engineer to better visualize and understand thePV device and its behavior in the connected circuit.

IV. ACKNOWLEDGMENT

The author gratefully acknowledges the advice andencouragement of Professor Myong-Chul Shin of theDepartment of Electronic and Electrical Engineering,Sungkyunkwan University, Suwon, S. Korea and Dr. Hak-man Kim of the Korea Electrotechnology Research Institute(KERI), Uiwang City, Gyeong-gi Province, S. Korea. Theauthor also wishes to thank Mr. Chang-Dae Yoon and themembers of the Power System and Computer Application Lab(PSCAL) for their assistance and support.

V. REFERENCES

[1] J.A. Gow, C.D. Manning, "Development of a photovoltaic array modelfor use in power-electronics simulation studies," IEE Proceeding onElectric Power Applications, Vol. 146, No. 2, 1999.

[2] M.W. Park, and I.K. Yu, "A Novel Real-Time Simulation Technique ofPhotovoltaic Generation Systems Using RTDS," IEEE Transactions onEnergy Conversion, Vol.19, No.1, 2004.

[3] Y.T. Tan, D.S. Kirschen, and N. Jenkins, "A Model of PV GenerationSuitable for Stability Analysis," IEEE Trans. on Energy Conv., V19.4,2004.

[4] S.K. Kim, J.H. Jeon, and E.S. Kim, "PSCAD/EMTDC Based Modelingand Simulation Analysis of a Grid-Connected Photovoltaic GenerationSystem", Transactions of KIEE, Vol. 54A, No. 3, 2005.

[5] D.L. King, W.E. Boyson, and J.A. Kratochvil, "Photovoltaic ArrayPerformance Model," Sandia National Lab., SAND2004-3535, Aug.2004.

[6] Manitoba HVDC Research Center, PSCAD v4.2 On-line Help,"Interpolation and Switching," and "Switching and Non-linearElements," Nov. 2005.

[7] A. Luque, and S. Hegedus (Eds.), Handbook of Photovoltaic Scienceand Engineering, John Wiley & Sons Ltd., 2003.

[8] T. Markvart, and L. Castaner (Eds.), Practical Handbook ofPhotovoltaics: Fundamentals and Applications, Elsevier Ltd., 2003.

[9] S. Banerjee, W. Niblack and M. Flickner, "A Minimum DescriptionLength Polygon Approximation method," IBM Research report RJ10007 (89096) pp.19, Dec. 1996.

[10] L. Allison (1999), Curve and Polygon Fitting, Faculty of InformationTechnology (Clayton), Monash University, Australia. [Online].Available:www.csse.monash.edu.au/~lloyd/tildeProgLang/Java/PolyFit/

[11] M. Haouari-Merbaha, M. Belhamelb, I. Tobiasa, and J.M. Ruiza,"Extraction and analysis of solar cell parameters from the illuminatedcurrent-voltage curve," Solar Energy Materials and Solar Cells, Vol. 87,pp. 225-233, Elsevier B.V., 2005.

VI. BIOGRAPHY

Ryan C. Campbell (SM’2006) was born in Ft.Collins, Colorado, USA, where he earned his B.S. inengineering science with a concentration inenvironmental engineering from Colorado StateUniversity (CSU) in 2001. He then followed aninternational path through the Baha’i World Centre inHaifa, Israel, and ended up back in academia atSungkyunkwan University (SKKU) in Seoul/Suwon,S. Korea, where he earned his M.S. degree inelectrical engineering in 2006. He is now working on

his PhD. in electrical engineering at the University of Washington (UW) inSeattle, Washington, USA, where he been awarded a second year of graduatefellowship support under the Grainger Power Engineering Program.

His employment experience includes management in the mid-level retailsector, and engineering/technician work in the commercial sector. In additionto his studies, he now works as the lead student at the University ofWashington’s new Department of Energy (DOE) sponsored IndustrialAssessment Center (IAC).

2007 39th North American Power Symposium (NAPS 2007) 101