Simulation Study of Parameter Estimation and Measurement Planning on Photovoltaics Degradation

8/17/2019 Simulation Study of Parameter Estimation and Measurement Planning on Photovoltaics Degradation

1/16

International Journal of Energy and StatisticsVol. 3, No. 3 (2015) 1550013 (16 pages)c Institute for International Energy Studies

DOI: 10.1142/S2335680415500131

Simulation study of parameter estimation and measurement

planning on photovoltaics degradation

Dazhi Yang

Singapore Institute of Manufacturing Technology (SIMTech)Agency for Science, Technology and Research (A∗STAR)

71 Nanyang Drive, Singapore 638075, Singapore [email protected]

[email protected]

Received 19 July 2015Revised 28 August 2015

Accepted 31 August 2015Published 30 September 2015

Photovoltaics degradation is one of the key parameters in PV performance evaluation.Units under a degradation study can be either modules or systems. As a single set of degradation measurements based on one unit cannot represent the population nor beused to estimate true degradation of a particular PV technology, repeated measuresthrough multiple units are essential. Linear mixed effects model is a suitable tool foranalyzing longitudinal data. In this paper, I use LME model to explain the degrada-tions in PV modules/systems which are installed at a shared location with modulesof same technology. The degradation parameters including degradation rate can thenbe estimated using maximum likelihood estimation. Beside the degradation rate, otherparameters of interest, e.g., the degradation distribution quantiles, are also derived toprovide valuable information for PV manufacturers and system owners.

Two types of measurements describe PV degradation, namely, a regression-basedlow-accuracy measurement through monitoring data (such as solar irradiance, moduletemperatures and various electrical parameters) and the flash test which can be consid-ered as a high-accuracy measurement. Given the underlying true degradation of a setof units, the two methods differ mainly in measurement accuracy. The error differencesbetween the low- and high-accuracy experiments are analyzed through simulation.

Keywords: Maximum likelihood estimation; linear mixed model; PV degradation.

Nomenclature

BVN : Bivariate normal distribution.

HE : High-accuracy Experiments.

LE : Low-accuracy Experiments.

LME : Linear Mixed Effects.

ML : Maximum Likelihood.

MLE : Maximum Likelihood Estimation.

1550013-1

I n t

. J . E n e r g y S t a t . 2 0 1 5 . 0 3 . D o w n l o a

d e d f r o m w w w . w o r l d s c i e n t i f i c . c o m

b y m r s w i s s e p r a b a w a t i o n 1 0 / 1 1 / 1 5 . F o r p e r s o n a l u s e o n l y .

http://dx.doi.org/10.1142/S2335680415500131http://dx.doi.org/10.1142/S2335680415500131


2/16

D. Yang

MVN : Multivariate normal distribution.

pdf : Probability density function.

PR : Performance Ratio.PV : Photovoltaics.

STC : Standard Test Condition.

se : Standard errors.

1. Introduction

The degradation rate of photovoltaic modules and systems is a key parameter in

PV performance evaluation and reliability analyses. It is also an important param-

eter in projecting the long-term power generation of PV systems. The degradationrates reported in the literature can be directly linked to PV manufacturer war-

ranty, especially when the reported time period became longer with more reliable

estimations in the past decades. The typical module manufacturer power output

warranty increased from 5 years to 25 five years since 1985 [1] owing to the fact

that module durability has increased through the years. As degradation rate is

receiving more attention, many researchers have reported degradation rates based

on available data. A comprehensive review of published degradation rates can be

found in Ref. [1]. Reference [2] reviewed some of the mechanisms which cause PV

degradation.

Degradation in PV can be quantified at the module level [3, 4] and at the sys-

tem level [5, 6]. Based on the study by Jordan and Kurtz [1], degradation rates

of modules and systems differ by only small margins, despite their distinct degra-

dation mechanisms. In this simulation study, I do not differ module degradation

from system degradation, as the methodology herein used aims at quantifying the

degradation rate and standard errors other than degradation mechanisms. In con-

sideration of this, other factors such as climate/weather conditions which affect the

degradation [7] can be relaxed.Also of interest is the study of PV degradation across different technologies

[8–11]. Five mainstream technologies are often seen in the literature, namely, amor-

phous silicon, cadmium telluride, copper indium gallium selenide, mono-crystalline

silicon and multi-crystalline silicon. Among these technologies, crystalline silicon

received the most attention at the reported time [1]. Crystalline silicon is found to

have a smaller degradation rate as compared to thin-film technologies. It is also

found that the spread (variance) of the thin-film degradation rates is much larger

than silicon technologies. Furthermore, the degradation rates observed in the first

year of operation may be higher due to the light induced degradation (especiallyfor thin-film technologies) and other early degradation mechanisms [12]. Therefore

in the later analyses, without loss of generality, degradation model parameters are

set based on crystalline silicon technology with early degradation effects removed.

The nameplate power measured at laboratory condition is a commonly used

parameter to describe the expected module energy output. However, two modules

1550013-2

I n t





3/16

Simulation study on photovoltaics degradation

●

● ●

●●

●

● ●

● ●

●

● ●

●● ●

●●

●

●

● ●

●●

●

● ●

● ● ●

●

●

●

●●

● ●●

●

●●

● ●

●●

● ●

● ● ●

● ●●

● ●

●

● ●

●

●●

● ●

● ●

● ●

●●

●● ●

●●

●

● ● ●

●●

●●

●

● ● ●

●● ●

● ●

●

●●

●●

●●

●●

●

●●

●

●

● ●●

● ● ●

●

●● ●

●

●●

●

●

● ●

● ●

●

●

● ●●

●●

●●

● ● ●

● ●●

● ●●

● ●

● ●

●●

●●

●

●● ● ●

● ●

●

●●

●

● ●●

●●

● ●● ●

●●

● ● ●

●

● ●

●●

● ●●

● ●

●●

●

● ●●

●

● ●

● ●

● ●

●

●

●

● ●

●●

●

● ●

●● ● ●

●

●● ●

●●

●●

●●

● ●●

● ●

● ●●

● ●

●

●

●

●●

● ● ●

●●

●●

●●

●

● ●

●

●

●●

●● ●

●●

● ● ●●

●

● ●

●● ●

●

●●

●

● ●

●

● ●

●

●●

●

●

●

●

●

●●

● ●

● ●

●●

●

●●

●

●●

●

70

80

90

100

0 5 10 15 20 25Years in operation

%

o f n a m e p l a t e p o w e r a

t S T C

unit

●

●

●

●

●

●

●

●

●

●

●

●

1

2

3

4

5

6

7

8

9

10

11

12

Fig. 1. Simulated degradation curves for 12 crystalline silicon modules installed at a shared loca-tion. Simulation is performed based on Eq. (7), see below.

with same nameplate power may have very different energy production [12]. As the

degradation rate calculated by using a single set of measurements cannot repre-

sent the population, repeated measures are essential. Figure 1 shows the simulated

degradation curves for 12 crystalline silicon modules based on Eq. (7). It is assumed

that these modules have identical nameplate information. It is also assumed that

these modules are installed at a shared climate condition. The details of this simu-

lation can be found in later sections.

If the early degradation effects are removed, it is reasonable to assume a linear

degradation model. Although some publications use an exponential degradation

model [13]; it is shown that for a typical starting degradation rate, the models do

not differ significantly up to 25 years [14]. I introduce linear mixed effects model [15]

together with some useful statistics in Sec. 2.By setting up the LME model with certain assumptions, PV degradation rate

can be found through maximum likelihood estimation. As the degradation rate

introduced in this paper is based on repeated measures, it reflects a more robust

estimation on the true degradation of a certain technology in a specific environment.

However, the point estimation of degradation rate is often not sufficient for manufac-

turing decision making. For example, knowledge of the quantile of the degradation

distribution is essential for warranty setting. I therefore show the method for degra-

dation parameter estimation and quantile estimation in Sec. 3 through simulation

examples.

2. Models and Methods

Repeated measures are defined if an outcome is measured repeatedly through a set

of units [15]. The data is called longitudinal data if these repeated measures are

1550013-3

I n t





4/16

D. Yang

taken sequentially in time [16]. When we consider PV modules/systems as units,

the outcome is the degradation of a particular technology (e.g., multicrystalline

silicon) at a specific outdoor condition.

2.1. Degradation model

By defining unit i, where i = 1, . . . , n, we can have mi measurements of degradation

of that unit. Let yij be the measured degradation of unit i at time tij , where

j = 1, . . . , mi. The linear degradation model is given by:

yij = b0,i + b1,itij + εij, (1)

where b0,i and b1,i denote the intercept and the gradient of the linear model forunit i; εij denotes a random effect. Suppose there are many identical units, i.e.,

PV modules/systems with same technology and under the same environmental and

climate conditions, the intercept and the gradient can be modeled using a bivariate

normal distribution, (b0, b1) ∼ BVN(β, V) with mean vector

β = (β 0, β 1) (2)

and covariance matrix

V = σ2b0 ρσb0σb1ρσb0σb1 σ2b1 . (3)While b0 and b1 are correlated (with correlation ρ) random variables, b0,i and

b1,i are a particular pair of realizations of these random variables. The probability

density function of this bivariate normal distribution is:

f (b0, b1;β, V) = 1

2πσb0σb1

1 − ρ2 exp− κ

2(1 − ρ2)

, (4)

where

κ =b1 − β 1

σb0

2+ b2 − β 2

σb1

2 − 2ρb1 − β 1σb0

b2 − β 2σb1

. (5)

2.2. Linear mixed model

It is convenient to consider Eq. (1) as a linear mixed model [15, 17]:

yij = (β 0 + b∗

0,i) + (β 1 + b∗

1,i)tij + εij . (6)

Write the above equation into matrix form:

Yi = Xiβ + Zib∗

i + εi, (7)

where Yi = (yi1, . . . , yimi); εi = (εi1, . . . , εimi)

; b∗i = (b∗0,i, b

∗1,i)

; (b∗0, b∗1) ∼

BVN(0, V);

εi ∼ MVN(0, σ2Ii); (8)

1550013-4

I n t





5/16


V(b∗i , εi) = 0; (9)

Xi = Zi = 1 ti1... ...1 timi

(10)and Ii is an mi by mi identity matrix. Equation (8) implies that εi is independent

and normally distributed. Equation (9) reveals that b∗i and εi are independent.

Under these settings, Yi has a multivariate normal distribution with mean vector

Xiβ and covariance

Σi = V(Yi) = ZiVZ

i + σ2Ii, (11)

which is a special case of the repeated-measured models in Ref. [ 18], i.e. Yi ∼MVN(Xiβ, Σi). The multivariate normal random vector Yi has pdf:

f (yi; Xiβ, Σi) = 1

(√

2π)mi |Σi|1/2exp

−1

2(yi − Xiβ)Σ−1i (yi − Xiβ)

, (12)

where |Σi| is the determinant of Σi.

2.3. Parameter estimation

Given the measured degradation data, the linear mixed model parameters can be

estimated using statistical procedures. The multivariate normal distributions give

convenience to many parameter estimation methods such as maximum likelihood

estimation where the results have been derived. In the aforementioned model, there

are six parameters to be estimated:

θ = (β 0, β 1, σb0 , σb1 , ρ , σ). (13)

Following the notations in Eq. (12), the log-likelihood for unit i is

i(θ) = −12

(yi − Xiβ)Σ−1i (yi − Xiβ) − 12 log |Σi|; (14)the total log-likelihood for n units is:

(θ) =n

i=1

i(θ) = −12

ni=1

(yi − Xiβ)Σ−1i (yi − Xiβ) − 1

2

ni=1

log |Σi|. (15)

The ML estimates of parameters, θ, can thus be estimated by setting the deriva-tive of (θ) to zero. Statistical software R [19] is used throughout the paper. The

MLE routine implementation is from the nlme package.

2.4. Degradation quantiles and Fisher information

Maximum likelihood estimates of the degradation rate would be sufficient for many

applications. However our discussion on the methodology does not stop at the point

estimation. Point estimation provides a single “best guess” of some quantity of

1550013-5

I n t





6/16

D. Yang

interest [20], in this case, the degradation rate. Other statistics such as degradation

distribution and the covariance matrix of the ML estimators are also important,

especially in degradation measurement planning. In this paper, the planning isstudied based on the standard errors of the estimated degradation quantiles.

2.4.1. Degradation quantiles

Consider the degradation model in Sec. 2.1, let the true degradation at time t be

D = b0 + b1t. From the definition, I note that the true degradation D refers to thequantity on the y-axis. In Fig. 1, D is the percentage of nameplate power at STC.Since b0 and b1 have a bivariate normal distribution, the mean and variance of the

true degradation are

E(D) = E(b0 + b1t) = β 0 + β 1t (16)and

V(D) = V(b0 + b1t) = σ2b0 + t2σ2b1 + 2tρσb0σb1 (17)respectively. The p quantile of the degradation distribution at time t is:

d p(t) = E(

D) + V(D)Φ−1( p), (18)

where Φ−1( p) is the inverse standard normal CDF.

2.4.2. Fisher information

The reason to discuss Fisher information here is to derive the large-sample approxi-

mated covariance matrix of the ML estimators; the reason to discuss the covariance

matrix is to derive the standard error for degradation quantiles; so that the degra-

dation measurements can be effectively and efficiently planned.

The Fisher information I (θ) of some parameter θ is defined as:

I (θ) = −E

∂ 2(θ)

∂ θ2

. (19)

When parameter θ is written into θ = (β 0, β 1, σb0 , σb1 , ρ , σ) = (β,ϑ), the

Fisher information of unit i can be noted using the Hessian matrix:

I i(θ) = −E(Hi) = −E

Hββ,i Hβϑ,i

Hϑβ,i Hϑϑ,i

= −E

∂ 2i/(∂ β∂ β) ∂ 2i/(∂ β∂ ϑ)

∂ 2i/(∂ ϑ∂ β) ∂ 2i/(∂ ϑ∂ ϑ)

=

Xi Σ

−1i Xi 0

0 Mi

, (20)

1550013-6

I n t





7/16


where the element on row r and column s of the symmetrical 4 by 4 matrix Mi is:

M i,rs = 1

2tr(Σ−1i Σ̇irΣ

−1i Σ̇is), (21)

r, s = 1, . . . , 4 and the explicit representations of Σ̇ir or Σ̇is are obtained by differ-

entiating Eq. (11) with respect to each parameter in ϑ:

Σ̇i1 = ∂ Σi∂ϑ1

= ∂ Σi∂σb0

= Zi

2σb0 ρσb1

ρσb1 0

Zi ; (22)

Σ̇i2 = ∂ Σi∂ϑ2

= ∂ Σi∂σb1

= Zi

0 ρσb0

ρσb0 2σb1

Zi ; (23)

Σ̇i3 = ∂ Σi∂ϑ3

= ∂ Σi

∂ρ = Zi

0 σb0σb1

σb0σb1 0

Zi ; (24)

Σ̇i4 = ∂ Σi∂ϑ4

= ∂ Σi

∂σ = 2σIi. (25)

The Fisher information for all n units is the sum of the Fisher information for each

unit:

I (θ) =n

i=1 I i(θ). (26)

The Fisher information matrix can be used to obtain the standard errors of ML

estimates. Wasserman [20] states the following theorem:

Theorem 1. (Asymptotic Normality of the MLE ) Let se =

V( θ). Under appro-

priate regularity conditions , the following hold :

(1) se ≈

1/ I (θ) and

( θ − θ)se

N(0, 1). (27)

(2) Let se = 1/ I ( θ). Then ,( θ − θ) se N(0, 1). (28)

Symbol denotes convergence in distribution.

If we extend the theorem to multiparameter cases, we have:

V( θ) = [ I (θ)]−1, (29)where V(·) denotes the approximated variance-covariance matrix of the ML esti-mators. In other words, the se2 of each parameter is given by the correspondingdiagonal term of [ I (θ)]−1; the covariance between the parameters are given theoff-diagonal terms of [ I (θ)]−1. An estimate of V(·) at the ML estimates is V( θ).

1550013-7

I n t





8/16

D. Yang

2.5. Standard error and confidence interval of the degradation

quantiles

The variance–covariance matrix of the ML estimators can be obtained through theinverse of the Fisher information matrix. With this information, together with the

degradation quantile d p evaluated at the ML estimates (denoted by d p or equiv-alently d p( θ)), the standard error of the quantile can be estimated through the“so-called” delta-method. Wasserman [20] states the following theorem:

Theorem 2. (Multiparameter delta method ) Suppose that ∇g evaluated at θ is not 0. Let

τ = g(

θ). Then

( τ − τ ) se( τ ) N(0, 1), (30)where

se( τ ) = ( ∇g) V( θ)( ∇g), (31) ∇g is ∇g evaluated at θ = θ.

In our case, ∇g is the vector of partial derivatives of d p with respect to theparameters. The elements of this vector are:

∂d p/∂β 0 = 1; (32)

∂d p/∂β 1 = t; (33)

∂d p/∂σb0 = ζ (2σb0 + 2tρσb1); (34)

∂d p/∂σb1 = ζ (2t2σb1 + 2tρσb0); (35)

∂d p/∂ρ = ζ (2tσb0σb1 ); (36)

∂d p/∂σ = 0, (37)where

ζ = Φ−1( p)

2

σ2b0 + t2σb1 + 2tρσb0σb1

. (38)

The estimated standard error of the quantile of degradation distribution at the

ML estimates is thus given by:

se( d p) = c V( θ) c, (39)where c is the vector of partial derivatives of d p. The 1−α% confidence interval of the estimated quantile is thus given by: d p ± zα/2 se( d p), (40)under the normal-based interval.

1550013-8

I n t





9/16


3. Simulation Study of PV Degradation

The simulated degradation curves of 12 crystalline silicon modules are shown in

section 1. There are two reasons for using simulation instead of using empirical data:(1) I am interested in the parameters of the LME model, hence using the simulated

data facilitates the analyses and benchmarking; (2) I do not possess long enough

dataset (20+ years for example) to demonstrate the complete set of statistical

analyses involved in this paper. Before I set the parameters for the simulation, two

types of degradation measurements are described.

3.1. Low- and high-accuracy degradation measures

Two types of PV degradations experiments are commonly used, namely, the

regression-based low-accuracy experiments through outdoor measurements and the

high-accuracy experiments through indoor flash tests. Similar to many other real-

world problems, the LE are easier to obtain as compared to the HE. Furthermore,

there may be more than one low-accuracy experiment which is available. To fully

utilize the results from LE, the outcomes from the HE are often used to benchmark

various LE to determine the corresponding accuracy [21]. However, the limitation

of the LE is obvious during the decision making process of the manufacturers, for

example, setting the degradation warranty based on inaccurate degradation ratesleads to financial risks [22].

Degradation rates determined using the outdoor measured data depend on

different regressands (explained variable), with time being the usual regressor

(explanatory variable). Jordan and Kurtz [21] compared four methods including

the DC/GPOA method [23], PR method, PR method with temperature correc-

tion [25] and PVUSA method [24]. The core idea of these LE is to use the drops in

certain performance indicators (such as PR) to represent degradation in PV mod-

ules/systems through the years. It is therefore important to consider various type

of correction and data filtering.In mid-latitude locations, PR varies in a year with winter showing a relatively

higher PR than summer. Module temperature is commonly used to adjust this

seasonal variation. In a recently proposed method [26], PR is normalized further

by removing the weather dependency. Conventional PR is given by:

PR =

t EN AC,t

t

P STC

GPOA,tGSTC

, (41)while the weather-corrected performance ratio, P Rcorr, is:

PRcorr =

t EN AC,t

t

P STC

GPOA,tGSTC

[1 − γ (T mod typ avg − T mod,t)]

, (42)with γ being the temperature coefficient for power, with a typically value of

−0.4%/◦C; EN AC being the measured AC power generation (in kW); P STC being

1550013-9

I n t





10/16

D. Yang

the nameplate power (in kW); GPOA being the in-plane irradiance (in kW/m2);

GSTC being the irradiance at standard test condition (1 kW/m2); T mod being the

module temperature (in ◦

C) and T mod typ avg being the average cell temperaturecomputed from a typical meteorological year. The summation,

t (not to be con-

fused with covariance Σi), in the above equations can be calculated over any defined

period of time, may it be days, weeks, months or years. It is shown that the seasonal

cycles in the PR can be effectively removed using this weather correction regardless

if monthly or daily PR is used [26]. In the simulation below, a weather-corrected

PR is assumed. Beside the corrections in PR, data filtering is also commonly used

to remove certain data points. For example, an irradiance filter can be applied

to remove data points far from STC; a module temperature filter can be used to

remove data points which deviate largely from the T mod typ avg. In addition, outlierfilters and stability filters are also frequently involved in the data quality control

process [21]. I will again assume that in the LE example presented below, the data

are filtered accordingly.

It is mentioned earlier that many other factors may affect the degradation rates

determined by the LE such as climate condition and soiling. It is therefore reason-

able to assume that the measurement error through the LE is high. On the other

hand, although the flash testing systems may have certain error originated from

the spectrum of the artificial light [27], calibration using a reference module can beused. It is therefore assumed that the power measurements at STC through a flash

test have a small error variance.

3.2. Parameter estimation using MLE

In Sec. 2, six parameters are identified to be estimated, namely, β 0, β 1, σb0 , σb1 ,

ρ, σ. PV modules experience early degradation such as light induced degradation

during the first year of operation. To simulate the approximate 3% drop in the first

year, the intercept of true degradation curve β 0 is set at 97. It was reported thatsome crystalline modules may have more than 4% power loss after the first weeks

of operation [12], σb0 = 0.5 is used to represent the variations of early degradation

among the sample modules. This means that the PV modules under simulation

preserve 97% of nameplate power at STC at time t = 0 with a standard deviation

of 0.5. I note that t = 0 denotes the beginning of the simulation, one can consider

this to be the beginning of the second actual operating year. Only the simulation

time reference t will be used hereafter.

In Ref. [1], a rich literature review is presented on the degradation rate of crys-

talline silicon modules. It was found that the average degradation rate of siliconmodules is 0.7%/year, i.e., β 1 = 0.7. Further to that, σb1 = 0.1 is interpreted from

Ref. [1] to denote the variation in the degradation rate distribution, see Fig. 5 from

Ref. [1] for this interpretation.

One of our model assumptions is that the intercept and the gradient of the

degradation can be modeled using a bivariate normal distribution with correlation ρ.

1550013-10

I n t





11/16


In PV degradation, this parameter does not carry significant physical implication.

However, it is reasonable to assume a small positive correlation between b0 and b1,

which means a module with a higher starting β 0 degrades slower. I use ρ = 0.3 inthe simulation.

In our degradation model, εi ∼ MVN(0, σ2Ii) is the error term. In HE simula-tion, it can be assumed that the error is small, so σ = 0.5 is set to explain the year-

to-year variations in the performance index. In LE simulation, higher experimental

errors are expected so that the corresponding σ is also higher. As an example, I set

the LE σ to be 2. With expected life time of the PV modules being 25 years, we

simulate 24 years (excluding 1 burn-in year) of HE data as shown in Fig. 1 using

Eq. (7). For each of the 12 units, the specific degradation curves are draw from

the bivariate normal distribution (parametered by Eqs. (2) and (3)). Noise term isthen added to these curves using random numbers drawn from a normal distribu-

tion (parametered by Eq. (8)). Using Eq. (15), the estimated HE parameters are: β 0 = 96.982, β 1 = −0.706, σb0 = 0.481, σb1 = 0.087, ρ = 0.443 and σ = 0.516.The ML estimates for LE parameters are β 0 = 96.858, β 1 = −0.709, σb0 = 0.405, σb1 = 0.086, ρ = 0.631 and σ = 2.062. It is shown that the MLE produces preciseestimates.

At this stage, I have demonstrated using the LME model to produce point esti-

mates of degradation rate. Very often, additional information about the degradation

distribution is required when manufacturers are setting the warranty policy.

3.3. Degradation quantiles evaluated at ML estimates

In the above examples, the true degradation D is the “percentage of nameplatepower at STC”, as shown on the y-axis of Fig. 2. Since the linear transformation

of normal random variables is also normal, Eqs. (16) and (17) give the mean and

the variance of D. Based on the ML estimates obtained earlier for the LE, at eachtime instance t, the degradation distribution can be plotted. Figure 2 shows thedegradation distribution f (D) at t = 0, 3, . . . , 24. The evolution of the probabilitydensity function of D is apparent. Following Eq. (18), the p quantile at any instancet can then be calculated based on the inverse standard normal CDF. Figure 2 shows

the 0.05 (dotted line on x–y plane), 0.50 (solid line), 0.999 (dashed line) degradation

quantiles at ML estimates of the LE.

Further to this, the standard error and confidence interval of the p quantile at

any instance t can be calculated through Eqs. (39) and (40) respectively. Equa-

tion (39) relies on two pieces of information, namely, V( θ) and c. As the estimationof V( θ) depends on the fisher information, thus depends on time matrix Zi or Xi. Tovisualize this effect, the 95% confidence intervals of 0.50 degradation quantile using

5 and 15 years of data are plotted in Fig. 3 respectively. In Fig. 3(a), fisher informa-

tion and the ML estimates are evaluated based on first 5 years of the LE data. The

quantiles at t > 5 are extrapolated using the linear degradation model. In Fig. 3(b),

the calculations are based on data up to 15 years. It is evident that the confidence

1550013-11

I n t





12/16

D. Yang

Y e a r s i n

o p e r a t i o n

0

5

10

15

2025

% o f n a m e p

l a t e p o w e r a t S T C

70

75

80

85

9095

100

P r

o b a b i l i t

y d e n s i t

y

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 2. Evaluations of the 0.05 (dotted line on x–y plane), 0.50 (solid line), 0.999 (dashed line)degradation quantiles at ML estimates of the LE.

(a) After 5 years (b) After 15 years

70

80

90

100

0 5 10 15 20 25 0 5 10 15 20 25Years in operation

% o f n a m e p l a t e p o w e r a t S T C

Fig. 3. The 95% confidence intervals of 0.50 degradation quantile based on 5 and 15 years of data.The estimated 0.5 quantiles are shown as the solid black line. The shaded regions denote theconfidence intervals.

of 0.5 quantile estimates increases significantly when time period increases. We can

also perform similar analysis on arbitrary quantiles, similar results are expected.

3.4. Degradation measurement planning using a simple test plan

Up to this point, all quantile related information is derived using the LE data. It was

shown that by monitoring the PV performance continuously, the degradation can

1550013-12

I n t





13/16


be estimated with high confidence as more data become available, e.g., surveillance

up to 15 years. To set up a LE in real-life operation, fixed cost is the dominant

cost. In other words, once the monitoring system is set up, streaming data will beavailable as long as the monitoring system is maintained. The composition of the

HE cost is however different. Recall that the HE in PV degradation is the flash

test. Once the modules/systems are deployed, it becomes difficult to access the

flash test especially when the installation is remote. The cost of the HE is related

to the number of measurements and the number of units under study. It is therefore

important to consider HE planning in PV degradation. I am interested in the trade-

off between the number of measurements and the number of test units in terms of

standard errors of degradation quantile.

As the estimated degradation quantile standard error se( d p) is a function of time t as shown in Fig. 3, t is fixed in this section. Suppose the HE degradation

study is expected to run for 15 years, the degradation quantiles d p at the end of

the experiment is of interest. Standard error is used as the metric to measure the

goodness of a particular experiment. To demonstrate the planning strategy, I set

p = 0.50, Fig. 4(a) shows the contour plot of the estimated standard error of the

estimated 0.50 quantile,

se( d0.50), at the end of evaluation period.

The contour plot is interpreted here. For n = 3 and m = 3, the case corresponds

to situation where 3 units measured 3 times each during the course of 15 years at

t = 0, t = 7.5 and t = 15 respectively. The estimation is se( d0.50) = 0.95, reflectedby the contour line at the bottom left corner of Fig. 4(a). Similarly, se( d0.50) =0.5, a smaller standard error, is found for setup with n = 11 units and m = 3

measurements. An important conclusion drawn from the HE simulation is: a trade-

off can be made by using fewer indoor flash tests without losing much on precision.

Further to that, to improve the estimation precision, more units should be used. In

comparison with HE setup, the LE contour plot of

se(

d0.50) is shown in Fig. 4(b).

0. 5

0. 5 5

0. 6

0. 6 5

0. 7 0. 7

5 0

. 8 0.

8 5 0. 9

0. 9 5

0. 8

0. 9

1 1. 1

1. 2 1. 3 1

. 4

0. 7

0. 6

(a) High−accuracy experiment (b) Low−accuracy experiment

3

4

5

6

7

8

9

10

3 4 5 6 7 8 9 10 11 12 3 4 5 6 7 8 9 10 11 12Number of units

N u m

b e r o f m e a s u r e m e n t s

Fig. 4. Contour plot of b se( b d0.50) at the end of evaluation period using different number of unitsand different number of measurements.

1550013-13

I n t





14/16

D. Yang

(a) High−accuracy experiment (b) Low−accuracy experiment

1

2

3

0 5 10 15 20 25 0 5 10 15 20 25Years in operation 0

. 5 0 q u a n t i l e s t a n d a r d

e r r o r

After 5 years

After 10 yearsAfter 15 years

Analysis

Extrapolation

Fig. 5. b se( b d0.50) as functions of time for the HE and LE.

Evaluation period for the LE is also 15 years. It is clear from the LE contours that

the estimation precision is more affected by the number of measurements over time.

Under the same n and m, the standard error for the LE is also higher than that of

the HE owing to the higher measurement uncertainty.

Beside the choice for n and m, the expected runtime of the experiment is also

important in degradation studies. While Fig. 3 demonstrates the 95% confidenceintervals of 0.50 degradation quantile for the LE experiment, the standard error

is considered in Fig. 5(b). In addition, the standard error for the HE experiment

under the same conditions is shown in Fig. 5(a). Simulated data shown in Fig. 1

are used here. Three different analysis periods are shown, namely, using the first

5, 10, and 15 years of data. The estimates of se( d0.50) at the remaining years foreach case are found through extrapolation using the degradation model. Based on

the simulated data, it is found that the standard error from the LE is comparable

to that of the HE when the monitoring period is long enough, such as a period of 15 years. However, the trade-off is present when the monitoring period is short.

The above simple test plan enables PV module manufacturer to plan the degra-

dation studies effectively. The particular choice of experiment and setup can be

decided by experts based on some specific tolerable upper bound of the standard

error. Together with the above mentioned cost constraints for HE and LE, the prob-

lem can be considered as a multi-objective optimization task. However, the solution

to this task is not within the scope of this work.

4. Conclusion

A practical PV degradation model is introduced. The six-parameter model enables

flexible simulation and design exercises for photovoltaic degradation. Instead of

using the conventional regression based methods for gradient estimation, maxi-

mum likelihood estimation is used to identify the degradation rate together with

1550013-14

I n t





15/16


other parameters simultaneously. Degradation quantile is considered with detailed

formulation. This facilitates the PV manufacturers in setting up warranty policies.

Degradation measurement planning is also discussed. Several design parametersneed to be evaluated and optimized. These parameters include:

• Type of experiment: HE versus LE;• Evaluation period;• Number of measurements made throughout the evaluation period;• Number of units;• Cost considerations.

References

[1] Jordan, D. C. and Kurtz, S. R. (2013). Photovoltaic degradation rates — An analyt-ical review. Progress in Photovoltaics : Research and Applications , 21(1), 12–29.

[2] Ndiaye, A., Charki, A., Kobi, A., Kèbé, C. M. F., Ndiaye, P. A. and Sambou,V. (2013). Degradations of silicon photovoltaic modules: A literature review. Solar Energy , 96(0), 140–151.

[3] Skoczek, A., Sample, T. and Dunlop, E. D. (2009). The results of performance mea-surements of field-aged crystalline silicon photovoltaic modules. Progress in Photo-voltaics : Research and Applications , 17(1), 227–240.

[4] Dunlop, E. D. and Halton, D. (2006). The performance of crystalline silicon pho-tovoltaic solar modules after 22 years of continuous outdoor exposure. Progress in Photovoltaics : Research and Applications , 14(1), 53–64.

[5] So, J. H., Jung, J. S., Yu, G. J., Choi, J. Y. and Choi, J. H. (2007). Performanceresults and analysis of 3-kW grid-connected PV systems. Renewable Energy , 32(11),1858–1872.

[6] Alawaji, S. H. (2001). Evaluation of solar energy research and its applications inSaudi Arabia-20 years of experience. Renewable and Sustainable Energy Reviews ,5(1), 59–67.

[7] Kempe, M. D. and Wohlgemuth, J. H. (2013). Evaluation of temperature and humid-ity on PV module component degradation. In Photovoltaic Specialists Conference (PVSC), 2013 IEEE 39th , June 2013, 120–125.

[8] Ishii, T., Takashima, T. and Otani, K. (2011). Long-term performance degradation of various kinds of photovoltaic modules under moderate climatic conditions. Progress in Photovoltaics : Research and Applications , 19(2), 170–179.

[9] Makrides, G., Zinsser, B., Georghiou, G. E., Schubert, M. and Werner, J. H. (2010).Degradation of different photovoltaic technologies under field conditions. In 35th IEEE Photovoltaic Specialists Conference (PVSC), June 2010, 2332–2337.

[10] Makrides, G., Zinsser, B., Georghiou, G. E., Schubert, M. and Werner, J. H. (2010).Evaluation of grid-connected photovoltaic system performance losses in Cyprus. In Power Generation , Transmission , Distribution and Energy Conversion (MedPower

2010), 7th Mediterranean Conference and Exhibition on , November 2010, 1–7.[11] Marion, B., Adelstein, J., Boyle, K., Hayden, H., Hammond, B., Fletcher, T., Canada,

B., Narang, D., Kimber, A., Mitchell, L., Rich, G. and Townsend, T. (2005). Per-formance parameters for grid- connected PV systems. In 31st IEEE Photovoltaic Specialists Conference (PVSC), 03–07 January 2005, 1601–1606.

[12] Cereghetti, N., Bura, E., Chianese, D., Friesen, G., Realini, A. and Rezzonico, A.(2003). Power and energy production of PV modules statistical considerations of 10

1550013-15

I n t





16/16

D. Yang

years activity. In 3rd World Conference on Photovoltaic Energy Conversion , May2003, 1919–1922, Vol.2.

[13] Chuang, S., Ishibashi, A., Kijima, S., Nakayama, N., Ukita, M. and Taniguchi, S.(1997). Kinetic model for degradation of light-emitting diodes. Quantum Electronics ,IEEE Journal of , 33(6), 970–979.

[14] Vázquez, M. and Rey-Stolle, I. (2008). Photovoltaic module reliability model basedon field degradation studies. Progress in Photovoltaics : Research and Applications ,16(5), 419–433.

[15] Verbeke G., Molenberghs G. and Rizopoulos D. (2010). Random effects models forlongitudinal data. In Longitudinal research with latent variables , ed. by van Montfort,K., Oud, J. H. L. and Satorra, A., 37–96. Springer, Berlin.

[16] Faraway, J. J. (2006). Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models , Chapman and Hall/CRC.

[17] Weaver, B. P., Meeker, W. Q., Escobar, L. A. and Wendelberger, J. (2013). Methodsfor Planning Repeated Measures Degradation Studies. Technometrics , 55(2), 122–134.

[18] Jennrich, R. I. and Schluchter, M. D. (1986). Unbalanced Repeated-Measures Modelswith Structured Covariance Matrices. Biometrics , 42(4), 805–820.

[19] R Core Team (2014). R: A Language and Environment for Statistical Computing. RFoundation for Statistical Computing. Vienna, Austria. http://www.R-project.org/.

[20] Wasserman, L. (2003). All of Statistics : A Concise Course in Statistical Inference ,Springer, USA.

[21] Jordan, D. C. and Kurtz, S. R. (2014). The Dark Horse of Evaluating Long-Term

Field Performance–Data Filtering. Photovoltaics, IEEE Journal of , 4(1), 317–323.[22] Jordan, D. C. (2011). NREL PV Module Reliability Workshop, Golden, Colorado.

http://www.nrel.gov/pv/pvmrw.html.[23] Jordan, D. C. and Kurtz, S. R. (2012). PV degradation Risk. In World Renewable

Energy Forum , May 2012.[24] Jennings, C. (1988). PV module performance at PG&E. In 12th IEEE Photovoltaic

Specialists Conference (PVSC), 1988, 1225–1229, Vol.2.[25] Haeberlin, H. and Beutler, C. H. (1995). Normalized representation of energy and

power for analysis of performance and on-line error detection in PV systems. In 13th Eur. Photovoltaic Sol. Energy Conf., 1995.

[26] Dierauf, T., Growitz, A., Kurtz, S., Becerra Cruz, J. L., Riley, E. andHansen, C. (2013). Weather-Corrected Performance Ratio, NREL Techical Report.http://www.nrel.gov/docs/fy13osti/57991.pdf.

[27] Munoz, M. A., Alonso-Garćıa, M. C., Vela, N. and Chenlo, F. (2011). Early degrada-tion of silicon PV modules and guaranty conditions. Solar Energy , 85(9), 2264–2274.

I n t




Documents

Simulation Study of Parameter Estimation and Measurement Planning on Photovoltaics Degradation