Likelihood Methods for Single Diode Model ...

Likelihood Methods for Single Diode Model Parameter Estimation from I-V Curve Data with Noise Brian Zaharatos∗ , Mark Campanelli† , Clifford Hansen‡ , Keith Emery† , Luis Tenorio∗ ∗ Colorado

School of Mines, Golden, CO, 80401, USA Renewable Energy Laboratory, Golden, CO, 80401, USA ‡ Sandia National Laboratories, Albuquerque, NM, 87185, USA

† National

Abstract—We develop a likelihood function for estimation of the five parameters of a single diode model for the electrical performance of photovoltaic devices. The likelihood function gives the probability of observing a given I-V curve dataset with a particular noise structure, conditional upon a choice for the model parameters. The maximum likelihood estimator (MLE) method searches for model parameters that maximize the probability of observing the given data. With independent and identically distributed (i.i.d) normal noise in the data, the MLE is equal to the least squares estimate, but MLEs have quantifiable statistical properties such as confidence intervals for uncertainty quantification. Performance parameter estimates (e.g., Pmax0 ) are derived from the MLE for the model parameters. Key challenges are the identifiability of the model with respect to the given data and effective computation of the MLE. Likelihood functions can be extended to more realistic noise models for measurements over a full operating range of irradiance and temperature and to Bayesian inference with uncertainty quantification. Index Terms—likelihood function, maximum likelihood estimator, noise model, parameter estimation, single diode model.

I. I NTRODUCTION We present a single diode model formulation amenable to model parameter estimation by a maximum likelihood estimator (MLE). The likelihood function for the single diode model is developed, incorporating a noise model for the points in one or more measured I-V curves. The MLE computation is fast, but issues such as identifiability require special attention. II. S INGLE D IODE M ODEL For non-negative terminal voltages, the following singlediode lumped-parameter equivalent-circuit model adequately describes the current-voltage (I-V) performance of many series-wired photovoltaic (PV) devices at a fixed temperature:  V +IRS  0 V + IRS0 NS n0 Vth 0 − 1 I = IL − IS0 e − , (1) RP0 where the device’s photocurrent IL is substituted by  E ISC RS  0 0 E ISC0 RS0 NS n0 Vth 0 IL := E ISC0 + IS0 e −1 + , RP0

(2)

the thermal voltage at standard reporting conditions (SRC) is Vth0 := kB T0 /q, and where I V E ISC0

device terminal current (A), device terminal voltage (V), effective irradiance ratio (unit-less), short-circuit current of device at SRC (A),

IS0 NS n0 RS0 RP0 kB T0 q

diode reverse saturation current of device at SRC (A), number of cells connected in series (unit-less), ideality factor of device at SRC (unit-less), series resistance of device at SRC (Ω), parallel (shunt) resistance of device at SRC (Ω), Boltzmann constant (1.3806504 × 10−23 J/K), junction temperature of device at SRC (298.15 K), electron charge (1.602176487 × 1019 C).

(I, V, E) are observed triples, some/all components of which have significant measurement noise or uncertainty. ISC0 , IS0 , n0 , RS0 , and RP0 are the five positive model parameters to be inferred (with quantified uncertainty) from the observations of (I, V, E). For brevity, we abbreviate these five parameters with the vector θ := (ISC0 , IS0 , n0 , RS0 , RP0 ), and we assume throughout that T = T0 and that NS has been specified. The effective irradiance ratio E is a unit-less measure of the “effective” irradiance illuminating the device relative to 1-sun at SRC. E is defined as the ratio of the device’s short circuit current at the prevailing irradiance, ISC , to the device’s short circuit current at SRC, ISC0 . This ratio is readily measured using a calibrated reference device. From the definition of the unit-less spectral correction factor M [1], which compensates for spectral response differences between the device and the reference device, it follows that E :=

ISC ISC,R = , ISC0 M ISC,R0

(3)

where ISC,R is the reference device’s short circuit current at the prevailing irradiance and ISC,R0 is the reference device’s short circuit current at SRC (typically a calibration value). Auxiliary equation (2) describes the device’s photocurrent generation as a function of E and the model parameters. This equation is derived by setting V = 0 and I = ISC in (1), using definition (3) to substitute ISC = E ISC0 , and then solving for IL . We choose to infer the short circuit current at SRC, ISC0 , instead of the photocurrent at SRC, IL0 , because ISC0 is typically of primary interest for performance evaluation. IL0 can be derived directly from ISC0 using (2) with E = 1. Our initial investigation into likelihood methods for inferring the five single diode model parameters centers around I-V curve measurements taken very close to 1-sun equivalent irradiance in well-controlled laboratory conditions, i.e., E ≈ 1 and T ≈ T0 = 25◦ C. Thus, we do not consider the effects that temperature or irradiance have on several of the model parameters. For example, IS0 or RP0 would typically change

if SRC were defined at a different temperature or irradiance, respectively. Including these dependencies typically requires additional equations auxiliary to (1) that capture the deviation of the diode model parameters from their SRC-values as temperature and irradiance change [2]. Additional auxiliary equations typically introduce additional model parameters that likely depend upon the PV material system. Such model extensions and comparisons are a subject of current investigation. A. Solving the Single Diode Model Given θ, solutions to the single diode model (1) define a performance surface in (V, I, E)-space. This surface implicitly defines three functions: I = gI (V, E; θ), V = gV (I, E; θ), and E = gE (V, I; θ). The Lambert W function can be employed for the efficient computation of any of these three functions [3]. The PV LIB Toolbox for MATLAB provides vectorized computation of Lambert W via the function wapr_vec [4]. Computation of I-V curves in the first quadrant is verified against an explicit computational scheme using junction voltages (detailed in [5]), over a very large parameter space for θ and for a range of practical values for NS . This verification ensures that the MLE search algorithm operates correctly for a wide variety of PV devices, as described later. III. T HE L IKELIHOOD F UNCTION The likelihood function gives the probability of observing a given I-V curve dataset with a particular noise structure, given some choice for the model parameter vector θ. Here we assume that the single diode model (1)–(2) is a sufficiently accurate representation of the PV device’s performance near 1 sun.1 The I-V curve dataset specifies the observables, which in general are N measured (V, I, E) triples, denoted (Vn , In , En ), for n = 1, . . . , N . In order to define a likelihood function, one must specifiy a noise model for the observables. For the present work, the (V, I, E) triples correspond values from an I-V curve swept under conditions such that E ≈ 1. For this initial investigation, we assume that the noise in the measurement of V and I is negligible as compared to the measurement noise in E. Thus, conditional upon the value of θ, we consider an observed value En to be a generated by the single diode model (1)–(2) through the corresponding function gE of Vn and In . Given a value for θ, the statistical model for the value of the nth observable En is thus given by En = gE (Vn , In ; θ) + εn . Here, the additive noise εn is assumed to be i.i.d. normally 2 distributed
with zero mean and variance σE , i.e., εn ∼ 2 N 0, σE . Although we specify a distribution family for the 2 noise model, we do not specify σE and instead infer it also from the observed data. Given a value for θ and σE , the ˜n , is thus given by distribution of En , denoted E
2 ˜n ∼ N gE (Vn , In ; θ), σE E . 1 Uncertainty due to so-called model discrepancy is not considered here. Also, measurement calibration uncertainty and uncertainty due to spatial nonuniformity of solar simulators are issues for future investigation.

In terms of the normal probability density function (PDF) for ˜n , one can write E (En −gE (Vn ,In ;θ))2 q − 0 2σ 2 2, E 2 π σE fE˜n(En ; θ ) = e using the extended parameter vector defined by θ 0 := (ISC0 , IS0 , n0 , RS0 , RP0 , σE ). The likelihood function for the nth datum is defined by fixing En to an actual observed value, and instead viewing the PDF as a function of θ 0 . For the vector E = (E1 , . . . , EN ) of all N observed data points, the independent noise assumption gives the following overall likelihood function by taking products of the individual likelihoods: PN (En −gE (Vn ,In ;θ))2 
N − n=1 0 2 2 2σ 2 E 2 π σE L(θ ; E) = e . (4) The logarithm of (4) is often employed in analysis and computational work. IV. M AXIMUM L IKELIHOOD E STIMATION Given observed data E, a maximum likelihood estimator (MLE) is a value θˆ0 of the model parameter vector θ 0 that maximizes the likelihood function L(θ 0 ; E). Put another way, an MLE is a value θˆ0 that maximizes the probability of observing the data at hand [6, Ch. 8]. An MLE is not guaranteed to exist nor is an MLE guaranteed to be unique if it does exist. For a unique MLE to exist, it is necessary for the parameters of a model to be identifiable, i.e., two distinct parameter vectors should not give rise to the same distribution of observables. Equivalently, an infinite number of observations should allow one to uniquely identify a single value for the true parameter vector θ 0 [7]. Parameter identifiability is important in practice, because a non-identifiable model could admit multiple parameter sets that explain the data equally well. This situation could lead to confusion from (potentially large) disagreements between parameter values computed by different groups using the same data. Subsequently, it should not be surprising when different model parameters give numerically equivalent values for the derived performance parameters at SRC (e.g., ISC0 , Pmax0 , and VOC0 ). Even for identifiable models with a unique MLE, the computation of the maximizer θˆ0 can be challenging. Complex likelihood functions (e.g., multiple local maxima) and strong dependencies between model parameters, such as IS0 and n0 , can create the computational semblance of non-identifiability.2 By taking partial derivatives of (4) with respect to the components of θ 0 , setting them to zero, and considering the resulting system of equations for θˆ0 , the MLE is seen to be equivalent to a nonlinear least squares estimator for θ. In addition, the MLE approach provides an estimate of the noise level σE and generalizes in a straightforward way to other noise models. Extensions to the basic MLE computation 2 This situation is analogous to computational difficulties with parameter estimation via nonlinear least squares.

TABLE 1 SRC Performance Param’s Pmax0 (W) Vmax0 (V)

RP0 (Ω)

Noise Level σE

0.04831 — 0.04838

0.4537 — 0.4532

0.1138 0.1138 0.1139

— 1 × 10−10 4.704 × 10−10

— 1 1.0802

— 0.1 0.03165

— 1000 737.8

— 0.01 0.001077

17.94 — 17.96

13.80 — 13.91

1.414 1.414 1.412

— 1 × 10−7 1.822 × 10−7

— 1.5 1.439

— 1.54 0.6460

— 721 1162

— 0.01 0.002747

A. Computing the MLE In general, assessing the identifiability of a statistical model with several parameters is difficult. Often, estimates of parameters are calculated under the assumption that the model is, in fact, identifiable [7]. In order to assess the identifiability of L(θ 0 ; E), we are investigating a method called data cloning developed in [7]. Data cloning utilizes a Bayesian framework, with Markov chain Monte Carlo (MCMC) methods, to assess identifiability. One advantage of data cloning is that it provides an MLE of θ 0 over the domain on which θ 0 is identifiable. For the present work, we assume identifiability and compute the MLE using the bounded-region maximization available in the mle function in MATLAB’s Statistics Toolbox [8]. Such iterative, optimization-based solution methods can be sensitive to the choice of initial condition for the parameter θ 0 . For the present work, an initial condition close to the presumed ∗ ∗ MLE, denoted θ 0 , is selected manually by setting σE = 0.01 ∗ (a middle value) and adjusting the elements of θ until the I-V curve measured (nominally) at 1-sun is in approximate visual agreement with the I-V curve points computed as In∗ = gI (Vn , E = 1; θ ∗ ). B. Numerical Examples Table 1 and Fig. 1 summarize the parameter fits for two devices: (i) a 2 cm x 2 cm mono-Si cell, and (ii) a small mono-Si module with 30 cells in series. Each fit computation took less than two seconds using the given starting value. The performance parameters at SRC for the fit curve are compared to the measured performance parameters derived from the I-V data points in which the current values have been corrected to 1-sun by linear scaling. V. C ONCLUSION The MLE approach emphasizes the statistical inference behind nonlinear least squares, and it can be extended to quantify parameter uncertainty. MLEs appear to be no harder to compute than nonlinear least squares for a single diode model with simple noise, and they can be easily adapted to other physical and noise models. Parameter identifiability and certain computational issues require further investigation.

Corrected and Fit I−V Curves at SRC for 2cm x 2cm Mono−Si Cell 0.12 Terminal Current (A)

can quantify uncertainty in the estimator via confidence intervals (CIs) for the parameters under conditions of asymptotic normality. We are currently investigating the accuracy of such CIs using Monte Carlo simulation and bootstrap resampling.

0.1 0.08 0.06 0.04 Current−Corrected Data Points Single Diode Mode Fit

0.02 0

0

0.1

0.2

0.3 Terminal Voltage (V)

0.4

0.5

0.6

Corrected and Fit I−V Curves at SRC for Small Mono−Si Module with 30 Cells in Series 1.5 Terminal Current (A)

Mono-Si Cell Measured Starting Value Fit Value Mono-Si Module Measured Starting Value Fit Value

Parameter Estimation Summary Single Diode Model Parameters, θ ISC0 (A) IS0 (A) n0 RS0 (Ω)

1

0.5 Current−Corrected Data Points Single Diode Mode Fit 0

0

2

4

6

8 10 Terminal Voltage (V)

12

14

16

18

Fig. 1. MLE parameter estimation for the single diode model (1)–(2). The fit curve at E = 1 is compared to the measured I-V data points in which the current values have been corrected to 1-sun by linear scaling. Top: A 2 cm x 2 cm Mono-Si cell. Bottom: A small Mono-Si module with 30 cells in series.

ACKNOWLEDGMENT This work was supported by the U.S. Department of Energy under Contract No. DE-AC36-08-GO28308 with the National Renewable Energy Laboratory (NREL). The lead author is supported by NREL through a National Center for Photovoltaics (NCPV) Graduate Fellowship.

R EFERENCES [1] C. R. Osterwald, “Translation of device performance measurements to reference conditions,” Solar Cells, vol. 18, pp. 269–279, 1986. [2] W. De Soto, S. A. Klein, and W. A. Beckman, “Improvement and validation of a model for photovoltaic array performance,” Solar Energy, vol. 80, pp. 78–88, 2006. [3] A. Jain and A. Kapoor, “Exact analytical solutions of the parameters of real solar cells using Lambert W-function,” Solar Energy Materials & Solar Cells, vol. 81, pp. 269–277, 2004. [4] J. S. Stein, D. Riley, and C. W. Hansen, PV LIB Toolbox (Version 1.1). Albuquerque, NM, USA: Sandia National Laboratories, 2014, pvpmc.org/pv-lib/. [5] J. W. Bishop, “Computer simulation of the effects of electrical mismatches in photovoltaic cell interconnection circuits,” Solar Cells, vol. 25, pp. 73– 89, 1988. [6] G. A. Young and R. L. Smith, Essentials of Statistical Inference. New York, NY, USA: Cambridge University Press, 2005. [7] S. R. Lele, K. Nadeem, and B. Schmuland, “Estimability and likelihood inference for generalized linear mixed models using data cloning,” Journal of the American Statistical Association, vol. 105, no. 492, pp. 1617–1625, 2010. [8] MATLAB Statistics Toolbox, Version 8.3 in MATLAB R2013b (8.2.0.701). Natick, MA, USA: The MathWorks, Inc., 2013, www.mathworks.com.