Simulation Study of Parameter Estimation and Measurement Planning on Photovoltaics Degradation
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Transcript of Simulation Study of Parameter Estimation and Measurement Planning on Photovoltaics Degradation
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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]
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.
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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
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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
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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 .
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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
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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
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. 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
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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)
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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 .
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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
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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 .
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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)
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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 .
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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
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. 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 .
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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.
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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 .
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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
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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 .
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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
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. 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 .
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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
. 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 .
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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
. 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 .
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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
. 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 .
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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
. 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 .
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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.
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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 .