# On-line Identification of Photovoltaic Arrays’ Dynamic Model Parameters

A. Bobtsov1 F. Mancilla-David2 S. Aranovskiy3 and R. Ortega4

(1Department of Control Systems and Robotics, ITMO University, Kronverkskiy
av. 49, Saint Petersburg, 197101, Russia
<EMAIL_ADDRESS>
2University of Colorado Denver, Denver, Colorado 80204, USA
<EMAIL_ADDRESS>
3IETR–CentaleSupélec, 35576 Cesson-Sévigné, France
<EMAIL_ADDRESS>
4Departamento Académico de Sistemas Digitales, ITAM, México
<EMAIL_ADDRESS>
)

###### Abstract

This paper deals with the problem of on-line identification of the parameters
of a realistic dynamical model of a photovoltaic array connected to a power
system through a power converter. It has been shown in the literature that,
when interacting with switching devices, this model is able to better account
for the PV array operation, as compared to the classical five parameter static
model of the array. While there are many results of identification of the
parameters of the latter model, to the best of our knowledge, no one has
provided a solution for the aforementioned more complex dynamic model since it
concerns the parameter estimation of a nonlinear, underexcited system with
unmeasurable state variables. Achieving such objective is the main
contribution of the paper. We propose a new parameterisation of the dynamic
model, which, combined with the powerful identification technique of dynamic
regressor extension and mixing, ensures a fast and accurate online estimation
of the unknown parameters. Realistic numerical examples via computer
simulations are presented to assess the performance of the proposed
approach—even being able to track the parameter variations when the system
changes operating point.

keywords: Photovoltaic arrays, on-line parameter estimation, identification of
nonlinear systems.

##  1 Introduction

Photovoltaic (PV) arrays are quickly becoming an important source of electric
power around the world. According to the latest solar industry update reported
by the US National Renewable Energy Laboratory [9], 172 GW of PV capacity was
added globally in 2021, bringing cumulative capacity to 939 GW. They also
report that, last year, 5% of global electricity generation came from PV.
Analysts project continued increases in annual global PV installations for the
upcoming years, with estimates suggesting PV could cover a quarter of global
electricity needs by mid-century. Models able to properly capture the
performance of PV arrays for both planning and operation purposes are thus
critical for efficient use of this technology.

The static behavior of a PV array may be captured via a nonlinear
current–voltage ($I$–$V$) characteristic. The single–diode model (SDM) is able
to adequately fit the $I$–$V$ static curve, and is thus widely adopted to
represent the performance of a PV array [13]. This model makes use of five
parameters whose values depend on the solar irradiance ($G$) and the
temperature at the PV junction ($T$). Most of the research has focused on
developing functional forms seeking to capture $G$ and $T$ dependencies,
either considering the physical characteristics of a PV cell [7, 4, 17] or
using functional approximations from discrete measured points of the $I$–$V$
curve [8]. A very large literature is availble for the estimation of the
parameters of this static curve, see [11] for a recent tutorial containing 164
references. Functional forms have been so far unable to fully capture the
behavior of the SDM five parameters over a wide range of operating conditions.
To overcome this defficiency we follow [10] and consider a more realistic
dynamical model of the PV array, which is capable of characterizing a PV array
under any operating point, removing the need for such functional forms. We
take advantage of the fact that in most application PV arrays are interfaced
to a power system through a power converter. Because of the switching, a power
converter imposes a current (or voltage) with ripple to the PV array. Under
these circumstances, the PV array’s parasitic capacitance [12] plays a role on
the synthesis of $I$–$V$ characterization, replacing the static nonlinear
curve by a dynamic nonlinear system with “orbits” around static operating
points—see [18, 10] for a more detailed discussion on this matter. In summary,
we confront in this paper the task of parameter estimation of a nonlinear,
underexcited system with unmeasurable state variables. Providing a solution to
this challenging practically relevant problem is the main contribution of the
paper.

The remainder of the paper is organized as follows. In Section 2 we present
the model of the system and formulate the parameter identification problem.
The key step in the design is the development of a linear regression equation
(LRE) for the system, which is carried out in Section 3. The main result of
the paper is given in Section 4. Simulation results of some examples reported
in the literature are given in Section 5 to illustrate the excellent
performance of the proposed estimator. The paper is wrapped-up with concluding
remarks in Section 6.

##  2 New Dynamic Model and Formulation of the Parameter Identification
Problem

In this section we give the new dynamic model used to describe the behavior of
the system and formulate the parameter identification problem solved in the
paper.

Figure 1: Schematic of PV array connected to a power system through a power
converter.

Fig. 1 shows the circuital representation of the SDM studied in this paper. In
the schematic, $v_{\mathrm{pv}}$ and $i_{\mathrm{pv}}$ represent,
respectively, the voltage and current at the PV array’s physical terminals. As
suggested in the figure, the SDM includes the photoelectric current or
irradiance current generated when the cell is exposed to sunlight, $I_{irr}$,
and three parasitic elements, $R_{\mathrm{p}}$, $R_{\mathrm{s}}$ and $C$,
representing, respectively, the leakage resistance, ohmic losses and the
parasitic capacitance, whose dynamics must be added to the model.

The relation between $I_{d}$ at $V_{d}$ at the PV junction is given by the
well-known Shockley diode equation,

$I_{d}=I_{0}\left(e^{bV_{d}}-1\right),$

where $I_{0}$ is the diode saturation current, and $b$ is the reciprocal of
the modified ideality factor, defined as,

$b=\frac{q}{nkT},$

in terms of the electron’s electric charge, $q=1.602\times 10^{-19}$ C, the
Boltzmann constant, $k=1.3806503\times 10^{-23}$ J/K, and the ideality factor
of the diode, $n$. The dynamic SDM has thus six characterization parameters,
namely $n$, $R_{p},R_{s},C,I_{0}$ and $I_{irr}$. However, the nonideality
factor $n$ is usually assumed to be constant (independent of $G$ and $T$
variations), and may be obtained from manufacturer’s datasheets [17].
Furthermore, sensing $T$ is usually available and changes very slowly compared
to electrical dynamics [6], so it can be assumed the parameter $b$ is known
and constant. That leaves five parameters to be identified:
$R_{p},R_{s},C,I_{0}$ and $I_{irr}$.

It is noted the circuit of Fig. 1 is able to accommodate a PV array of an
arbitrary number of PV cells connected in series/paralell by properly
redefining the various characterization parameters [5]. Fig. 1 also suggests
the PV array is interface to a power system by a generic power converter. In
practice, that converter will be either a boost or buck converter topology. In
the former case, the PV array would be connected to a series inductor followed
by high-frequency switching devices, forcing $i_{\mathrm{pv}}$ to have a dc
value (power component) along with some small ripple (noise component). A buck
converter would require a capacitor to be connected in parallel with the PV
array, forcing $v_{\mathrm{pv}}$ to have a dc value along with some small
ripple [14]. Without loss of generality, we consider a boost–type topological
realization.

Noticing from Fig. 1 that $V_{d}=v_{C}$, the state space model for the system
considered in the paper is given by the following nonlinear differential-
algebraic equations

$\displaystyle C\dot{v}_{C}$ $\displaystyle=-{1\over
R_{\mathrm{p}}}v_{C}-i_{\mathrm{pv}}+I_{irr}-I_{0}\left(e^{bv_{C}}-1\right)$
(1a) $\displaystyle v_{\mathrm{pv}}$
$\displaystyle=v_{C}-R_{s}i_{\mathrm{pv}},$ (1b)

with the following considerations:

  1. (i)

$v_{\mathrm{pv}}(t)\in\mathbb{R}$ is a positive measurable signal;

  2. (ii)

$v_{C}(t)\in\mathbb{R}$ is a positive unknown state variable;

  3. (iii)

$i_{\mathrm{pv}}(t)\in\mathbb{R}$ is a known signal whose first and second
derivative are also known;

  4. (iv)

$C,R_{p},R_{s},I_{0}$ and $I_{irr}$ are positive unknown parameters;

  5. (v)

$b$ is a known positive parameter.

Problem formulation. Consider the system (1) verifying the conditions (i)-(v)
above. From the unique measurements of $v_{\mathrm{pv}}$, $i_{\mathrm{pv}}$,
${d\over dt}{i}_{\mathrm{pv}}$ and ${d^{2}\over dt^{2}}i_{\mathrm{pv}}$
generate globally convergent on-line estimates of the parameters
$R_{p},R_{s},C,I_{0}$ and $I_{irr}$.

All assumptions (i)-(v) are standard and practically reasonable, except
perhaps the assumption of knowledge of ${d\over dt}{i}_{\mathrm{pv}}$ and
${d^{2}\over dt^{2}}i_{\mathrm{pv}}$. However, we follow here the reasoning of
[10, Section 6] regarding some practical considerations pertaining to the
shape of $i_{\mathrm{pv}}$. Namely, that it consists of the sum of a known
mean value current plus a ripple of known frequency. Hence, we can assume that
$i_{\mathrm{pv}}(t)=A(1+\sin{\omega t})$ with known $A$ and $\omega$.

To simplify the reading we rewrite the system (1) using control theory
notation. Towards this end we define the state, input and output signals via
$x:=v_{C},u:=i_{\mathrm{pv}}$ and $y:=v_{\mathrm{pv}}$ and introduce the
constant parameter vector $\eta\in\mathbb{R}^{5}$ as

$\eta:=\operatorname{col}\left({1\over R_{\mathrm{p}}C},{I_{0}\over C},{1\over
C}(I_{irr}+I_{0}),{1\over C},R_{s}\right),$

where $\operatorname{col}(\cdot)$ denotes a column vector. With the
definitions above the system (1) may be rewritten as

$\displaystyle\dot{x}$
$\displaystyle=-\eta_{1}x-\eta_{2}e^{bx}+\eta_{3}-\eta_{4}u$ (2a)
$\displaystyle y$ $\displaystyle=x-\eta_{5}u.$ (2b)

######  Remark 1.

We make the important observation that the system (2) is nonlinear with
unmeasurable state. The task of estimating the parameters $\eta$ is, clearly,
far from obvious and cannot be solved with any of the existing parameter
estimation techniques. Therefore, a radically new technique must be developed
to provide the solution to the estimation problem.

######  Remark 2.

Notice that it is possible to obtain the physical parameters
$C,R_{\mathrm{p}},R_{s},I_{0}$ and $I_{irr}$ from knowledge of $\eta$. More
precisely, there exists a bijective mapping
$\mathcal{M}:\mathbb{R}_{+}^{5}\mapsto\mathbb{R}_{+}^{5}$ such that

$K:=\operatorname{col}(C,R_{p},R_{s},I_{0},I_{irr})=\mathcal{M}(\eta).$ (3)

Clearly, the mapping is defined as

$\mathcal{M}(\eta)=\operatorname{col}\left({1\over\eta_{4}},{\eta_{4}\over\eta_{1}},\eta_{5},{\eta_{2}\over\eta_{4}},{1\over\eta_{4}}(\eta_{3}-\eta_{2})\right).$

######  Remark 3.

The interested reader is referred to [3] where the problem of estimating the
parameters of the windmill power coefficient, which has the additional
difficulty of being nonlinearly parameterized, is solved. Notice that, if the
temperature is not measurable, that is, if $b$ is unknown, we are confronted
with a similar extremely difficult nonlinearly parameterized problem—with the
additional constraint of unmeasurable state.

##  3 Model Reparameterization

The key step in the estimator design is to derive a linear regression equation
(LRE) for the parameter $\eta$, that will be used to estimate them. This
result is given in the proposition below.

######  Proposition 1.

Consider the system (1) verifying the conditions (i)-(v).

  1. C1

The system admits the LRE

$z(t)=\Omega^{\top}(t)\theta+\varepsilon_{t},$ (4)

where $\Omega(t)\in\mathbb{R}^{8}$ and $z(t)\in\mathbb{R}$ are measurable
signals, $\theta\in\mathbb{R}^{8}$ is a vector of unknown parameters and
$\varepsilon_{t}$ is an exponentially decaying term stemming from the filters
initial conditions.111Following standard practice, and without loss of
generality, this term is neglected in the sequel.

  2. C2

There exists a mapping
$\mathcal{F}:\mathbb{R}_{+}^{4}\mapsto\mathbb{R}_{+}^{4}$ such that

$\operatorname{col}(\eta_{1},\eta_{3},\eta_{4},\eta_{5})=\mathcal{F}(\theta_{1},\theta_{2},\theta_{3},\theta_{4}).$

  3. C3

The parameter $\eta_{2}$ verifies the relation

$\eta_{2}=w(\eta_{1},\eta_{3},\eta_{4},\eta_{5},u,y)$ (5)

where $w:\mathbb{R}^{6}\to\mathbb{R}_{+}$ is a known mapping.

###### Proof.

We will establish the proof considering the representation (2) of the system.
Differentiating (2b) with respect to time and using (2a) we get

$\displaystyle\dot{y}$ $\displaystyle=-\eta_{5}\dot{u}+\dot{x}$
$\displaystyle=-\eta_{5}\dot{u}-\eta_{1}x-\eta_{2}e^{bx}+\eta_{3}-\eta_{4}u$
$\displaystyle=-\eta_{5}\dot{u}-\eta_{1}(y+\eta_{5}u)-\eta_{2}e^{b(y+\eta_{5}u)}+\eta_{3}-\eta_{4}u$
$\displaystyle=-\eta_{5}\dot{u}-\eta_{1}y-\eta_{7}u-\eta_{2}e^{by}e^{\eta_{6}u}+\eta_{3}$

where we defined the positive constants

$\eta_{6}:=b\eta_{5},\;\eta_{7}:=\eta_{4}+\eta_{1}\eta_{5}.$

Multiplying by $e^{-by}$ in both sides of the latter equation we get

$\displaystyle e^{-by}\dot{y}=$
$\displaystyle-\eta_{5}\dot{u}e^{-by}-\eta_{1}ye^{-by}-\eta_{7}ue^{-by}$ (6)
$\displaystyle-\eta_{2}e^{\eta_{6}u}+\eta_{3}e^{-by}.$

Defining now the measurable signal

$\psi:=-{1\over b}e^{-by},$ (7)

we can write (6) as

$\dot{\psi}=e^{-by}\dot{y}=\phi^{\top}\mu-\eta_{2}e^{\eta_{6}u},$ (8)

where we defined

$\displaystyle\mu$
$\displaystyle:=\operatorname{col}(\eta_{5},\eta_{1},\eta_{7},\eta_{3})$
$\displaystyle\phi$
$\displaystyle:=\operatorname{col}(-\dot{u}e^{-by},-ye^{-by},-ue^{-by},e^{-by}).$
(9)

Differentiating (8) we get

$\ddot{\psi}=\dot{\phi}^{\top}\mu-\eta_{6}\dot{u}\left(\eta_{2}e^{\eta_{6}u}\right)=\dot{\phi}^{\top}\mu-\eta_{6}\dot{u}\left(\phi^{\top}\mu-\dot{\psi}\right).$

Now, apply to the latter equation an LTI filter of the form
${\lambda^{2}\over(p+\lambda)^{2}}$, where $p:={d\over dt}$ and $\lambda>0$ is
a designer chosen constant, to get

$\displaystyle{\lambda^{2}p^{2}\over(p+\lambda)^{2}}\psi=\mu^{\top}{\lambda^{2}p\over(p+\lambda)^{2}}\phi$
$\displaystyle-\eta_{6}\mu^{\top}{\lambda^{2}\over(p+\lambda)^{2}}(\phi\dot{u})$
$\displaystyle+\eta_{6}{\lambda^{2}\over(p+\lambda)^{2}}(\dot{u}\dot{\psi}).$

Recalling the definitions of $\eta_{6}$ and $\mu$, it follows that
$\eta_{6}=b\mathbf{e}^{\top}_{1}\mu,$ where
$\mathbf{e}_{1}:=\operatorname{col}(1,0,0,0)$. Then the equation above may be
written in the LRE form (4) with the definitions

$\begin{gathered}z:=\frac{\lambda^{2}p^{2}}{(p+\lambda)^{2}}\psi,\
\theta=\begin{bmatrix}\mu\\\ \eta_{5}\mu\end{bmatrix},\\\
\Omega:=\begin{bmatrix}{\lambda^{2}p\over(p+\lambda)^{2}}\phi+\frac{\lambda^{2}}{(p+\lambda)^{2}}(\dot{u}\dot{\psi})b\mathbf{e}_{1}\vspace{3pt}\\\
-b\frac{\lambda^{2}}{(p+\lambda)^{2}}(\phi\dot{u})\end{bmatrix}.\end{gathered}$
(10)

Since $z$ is clearly measurable, to complete the proof of the claim C1 it only
remains to prove that the regressor vector $\Omega$ is also measurable. From
inspection of $\Omega$ we see that the only conflicting term is
${\lambda^{2}\over(p+\lambda)^{2}}(\dot{u}\dot{\psi})$, which involves the
unmeasurable signal $\dot{\psi}$. To prove that this signal is computable
without differentiation we invoke the swapping lemma [16, Lemma 6.3.5] that
ensures the following identity

$\frac{\lambda}{p+\lambda}(\dot{u}\dot{\psi})=\dot{u}\frac{\lambda
p}{p+\lambda}\psi-\frac{1}{p+\lambda}\left(\ddot{u}\frac{\lambda
p}{p+\lambda}\psi\right).$

Using this identity we obtain

${\lambda^{2}\over(p+\lambda)^{2}}(\dot{u}\dot{\psi})={\lambda\over
p+\lambda}\left(\dot{u}\frac{\lambda
p}{p+\lambda}\psi-\frac{1}{p+\lambda}\left(\ddot{u}\frac{\lambda
p}{p+\lambda}\psi\right)\right),$

where, given the knowledge of $\ddot{u}$, the right hand term is computable
without differentiation.

We now proceed to prove the claim C3. For, we apply the filter ${\lambda\over
p+\lambda}$ to the state equation (2a) to get

$\displaystyle{\lambda\over p+\lambda}\dot{x}=$ $\displaystyle-{\lambda\over
p+\lambda}[\eta_{1}(y+\eta_{5}u)]-\eta_{2}\left\\{{\lambda\over
p+\lambda}e^{b(y+\eta_{5}u)}\right\\}$ $\displaystyle+{\lambda\over
p+\lambda}(\eta_{3}-\eta_{4}u)={\lambda p\over p+\lambda}(y+\eta_{5}u),$

where we used (2b) to get the second identity. Noticing that the term in
brackets is bounded away from zero we can rewrite the equation above as

$\displaystyle\eta_{2}=$ $\displaystyle\left({\lambda\over
p+\lambda}e^{b(y+\eta_{5}u)}\right)^{-1}$ $\displaystyle\left({\lambda\over
p+\lambda}(\eta_{3}-\eta_{4}u-\eta_{1}(y+\eta_{5}u))-{\lambda p\over
p+\lambda}(y+\eta_{5}u)\right)$
$\displaystyle=:w(\eta_{1},\eta_{3},\eta_{4},\eta_{5},u,y).$

Finally, the mapping
$\mathcal{F}(\theta_{1},\theta_{2},\theta_{3},\theta_{4})$ is easily derived
as

$\begin{bmatrix}\eta_{1}\\\ \eta_{3}\\\ \eta_{4}\\\
\eta_{5}\end{bmatrix}=\mathcal{F}(\theta_{1},\theta_{2},\theta_{3},\theta_{4})=\begin{bmatrix}\theta_{2}\\\
\theta_{4}\\\ \theta_{3}-\theta_{1}\theta_{2}\\\ \theta_{1}\end{bmatrix},$
(11)

establishing claim C2. ∎

######  Remark 4.

The computation of the estimate of $\eta_{2}$ is done replacing the estimates
of $\eta_{1},\eta_{3},\eta_{4}$ and $\eta_{5}$ in the mapping $w$, that is

$\hat{\eta}_{2}=w(\hat{\eta}_{1},\hat{\eta}_{3},\hat{\eta}_{4},\hat{\eta}_{5},u,y).$

For this reason it is important not to “pull out” the parameter $\eta_{1}$
from the action of the filter.

##  4 Parameter Estimator and Main Result

For parameter estimation, we apply the DREM procedure, see [1, 15]. As it is
shown in Proposition 1, among eight elements of the unknown vector $\theta$,
only the first four are required, see C2 and C3. The DREM procedure transforms
the vector LRE (4) into a set of scalar LREs for each of the unknown
parameters allowing the estimation of $\theta_{i}$ for $i=1,2,3,4$, only.

For the DREM procedure, we introduce the dynamics extension

$\displaystyle\dot{\Phi}_{f}$
$\displaystyle=-a\Phi_{f}+a\Omega\Omega^{\top},\Phi_{f}(0)=0$ (12)
$\displaystyle\dot{Y}_{f}$ $\displaystyle=-aY_{f}+a\Omega z,Y_{f}(0)=0$
$\displaystyle\Phi$ $\displaystyle=c\Phi_{f}+d\Omega\Omega^{\top}$
$\displaystyle Y$ $\displaystyle=cY_{f}+d\Omega z,$

where $z$ and $\Omega$ are defined in (10), $a>0$, $c>0$ and $d>0$ are the
tuning coefficients. The dynamic extension (12) is the combination of
Kreisselmeiers regressor extension scheme that preserves the excitation of the
regressor $\Omega$, see [2], and the feedforward term $d\Omega\Omega^{\top}$
enhancing the excitation, see [15].

After the dynamics extension, the LRE

$Y(t)=\Phi(t)\theta$ (13)

holds. Following the DREM procedure, we define
$\operatorname{adj}\left(\Phi\right)$ to be the adjugate matrix of $\Phi$, and
define the signals

$\Delta:=\det\left(\Phi\right),\;\mathcal{Y}:=\operatorname{adj}\left(\Phi\right)Y.$
(14)

Then, upon multiplication of (13) by $\operatorname{adj}\left(\Phi\right)$, we
obtain the element-wise LREs

$\mathcal{Y}_{i}(t)=\Delta(t)\theta_{i},\;i=1,\ldots,8.$

Then, the required unknown parameters are estimated with the classical
gradient scheme

$\dot{\hat{\theta}}_{i}=-\gamma_{i}\Delta\left(\mathcal{Y}_{i}-\Delta\hat{\theta}_{i}\right),\;i=1,2,3,4$
(15)

where $\gamma_{i}>0$ are tuning coefficients. Note that, thanks to the use of
DREM, we can estimate only the required parameters and not the whole vector
$\theta$. It is straightforward to show that the estimation error
$\tilde{\theta}_{i}:=\hat{\theta}_{i}-\theta_{i}$ obeys

$\dot{\tilde{\theta}}_{i}=-\gamma_{i}\Delta^{2}\tilde{\theta}_{i}.$

Hence, solving these scalar equations yields

$\tilde{\theta}_{i}(t)=e^{-\gamma_{i}\int_{0}^{t}\Delta^{2}(s)ds}\tilde{\theta}_{i}(0).$

Then, the convergence follows under the condition that the integral
$\int_{0}^{t}\Delta^{2}(s)ds$ tends to infinity that is strictly weaker the
the classical condition of persistency of excitation of the regressor
$\Omega$, see [15].

The calculations above provide the proof of our main result given in the
following proposition.

######  Proposition 2.

Consider the system (1) verifying the conditions (i)-(v). Construct the
signals $z$ and $\Omega$ as defined in (10) with the signals $\psi$ and $\phi$
defined in (7) and (9), respectively. Define the DREM parameter estimator
(12), (14) and (15). Use the estimate $\hat{\theta}_{i},\;i=1,\dots,4$, to
reconstruct the estimates $\hat{\eta}$ following C2 and C3 of Proposition 1.
Then, compute the estimate of the physical parameters
$\hat{K}:=\operatorname{col}(\hat{C},\hat{R}_{p},\hat{R}_{s},\hat{I}_{0},\hat{I}_{irr})$
as indicated by (3). Assume the signal $\Delta$ is not square integrable.
Then,

$\liminf_{t\to\infty}\big{|}\hat{K}(t)-K|=0,$

for all values of the system and estimator initial conditions.

######  Remark 5.

As discussed in [1, 15] the assumption that $\Delta$ is not square integrable
is not a stringent one. As shown in the realistic simulations in the next
section it is satisfied in our particular application.

##  5 Illustrative Example

As an example, we consider the 85 W Kyocera KC85TS module. We use the
functional forms available in [17] to compute the “true” values of
characterization parameters at various values of $G$ and $T$. The value of $C$
is generated artificially considering the case studies available in [10, 18].
The point of maximum power at standard test conditions (STC), with $G=1000$
W/m2 and $T=25$ oC, is selected as the main operating point for the
identification illustration. Table 1 summarizes the “true” values of the
parameters at STC.

Table 1: “True” PV array parameters for STC and three random operating points. Description | Symbol | Value
---|---|---
STC | Mode 1 | Mode 2 | Mode 3
Solar irradiance, W/m2 | $G$ | 1000 | 748.9 | 740.4 | 715.8
Junction temperature, K | $T$ | 298.15 | 302.15 | 302.40 | 302.71
Average input current, A | $\bar{i}_{\mathrm{pv}}$ | 4.54 | 3.40 | 3.36 | 3.25
Parasitic capacitance, $\mu$F | $C$ | 0.6 | 0.6 | 0.6 | 0.6
Parallel resistance, $\Omega$ | $R_{p}$ | 112.55 | 150.28 | 152.02 | 157.23
Series resistance, $\Omega$ | $R_{s}$ | 0.2747 | 0.2747 | 0.2747 | 0.2747
Photoelectric current, A | $I_{irr}$ | 5.00 | 3.75 | 3.70 | 3.58
Saturation current, nA | $I_{0}$ | 10.57 | 17.68 | 18.24 | 18.97
Ideality factor, (no units) | $n$ | 1.1287 | 1.1287 | 1.1287 | 1.1287
Exponential coefficient, 1/V | $b$ | 0.958 | 0.945 | 0.944 | 0.943

The input signal $u$ in (2), which is $i_{\mathrm{pv}}$ in (1) is chosen to
have an average value corresponding to the point of maximum power, with a 5%,
20 kHz sinusoidal ripple,

$u(t)=\bar{i}_{\mathrm{pv}}\left(1+0.05\sin(2\pi\times 20\times
10^{3}t)\right),$

where the value $\bar{i}_{\mathrm{pv}}$ corresponds to the (known) average
value of $i_{\mathrm{pv}}$ as given in Table 1.

To implement the estimator we follow Proposition 2. Namely, we construct the
signals $z$ and $\Omega$ as defined in (10) with the signals $\psi$ and $\phi$
defined in (7) and (9), respectively. The filter tuning coefficient is chosen
as $\lambda=6\cdot 10^{5}$. Then we apply the DREM procedure (12), (14) and
(15), where the tuning coefficients are selected as $a=10^{5}$, $c=10^{3}$,
and $d=10^{2}$. The coefficients $\gamma_{i}$ are different for different
simulation scenarios and are provided below. The estimate $\hat{\theta}$ is
further used to construct—applying a certainty equivalence principle—the
estimates $\hat{\eta}$ following C2 and C3 of Proposition 1. Then, estimates
of the physical parameters are constructed as indicated in Remark 2 via (3).
As the parameter reconstruction involves algebraic manipulation and division
by time-varying signals that can be arbitrary close to zero in transients, the
parameter estimates are bounded within the range $[0.1,\ 10]$ times the
nominal value. Note that such saturation is used only for illustrative
purposes; it is applied for the computation of the physical parameters given
in Table 1 and it is not applied to the estimates $\hat{\theta}$. Thus, the
saturation does not affect the convergence analysis of Section 4.

We consider two simulation scenarios. First, the parameter estimator starts
with zero initial conditions and estimates the constant parameters
corresponding to the STC operating point. The coefficients $\gamma_{i}$ are
$\gamma_{1}=\gamma_{2}=20$ and $\gamma_{3}=\gamma_{4}=40$. For this scenario,
we present in Fig. 2 that estimator transients for each of unknown parameters.
Whereas the DREM procedure ensures the monotonicity of the transients for
$\hat{\theta_{i}}$, the physical parameters estimates are obtained via
algebraic manipulations and are thus prone to chattering. Notably, the
estimate $\widehat{I}_{0}$ has the fastest and chattering-free convergence,
the estimate $\widehat{R}_{s}$ is also chattering-free, whereas the estimate
$\widehat{I}_{irr}$ exhibits the strongest chattering. Nevertheless, all
transients converge exponentially in $15$ ms to their true values indicated
with dotted lines in the figure.

(a) Estimate $\widehat{C}$, $\mu$F (b) Estimate $\widehat{R}_{\mathrm{p}}$,
$\Omega$ (c) Estimate $\widehat{R}_{s}$, $\Omega$

(d) Estimate $\widehat{I}_{0}$, nA (e) Estimate $\widehat{I}_{irr}$, A

Figure 2: Parameter estimation in the first scenario. The true values are
given in dashed lines.

To illustrate the ability of the estimator to track parameter variations, we
also include three additional random operating points on Table 1 identified as
Mode 1, Mode 2 and Mode 3, taken from real measurements of $G$ and $T$. We
note the functional forms of [17] assume $R_{s}$ and $n$ to be constant,
independent of $G$ and $T$ variations. Without loss of generality, we assume
$C$ to be constant as well.

So we define a second scenario, where the PV array starts in the Mode 3 with
the correct estimates of the parameters. Then, the PV array switches to Mode
1, then to Mode 2, and then back to Mode 3. The switches occur every $20$ ms;
such a short interval is not practical but is chosen for illustrative
purposes. The coefficients $\gamma$ are $\gamma_{1}=\gamma_{2}=200$ and
$\gamma_{3}=\gamma_{4}=400$. For this scenario, we present only the norm of
the parameter estimation error $\tilde{\theta}$ to illustrate the overall
performance and estimator’s capability of tracking time-varying parameters,
see Fig. 3. After each switch, the estimation error norm jumps to a value
corresponding to the parameter variation between the modes, and then decay
exponentially in approximately $15$ ms.

Figure 3: Parameter estimation error norm $|\tilde{\theta}|$ in the second
scenario.

##  6 Conclusions

In this paper we have provided the first solution to the challenging problem
of on-line estimation of the parameters of a new dynamical model describing
accurately the behavior of a PV array connected to a power system through a
power converter. This problem concerns the parameter estimation of a
nonlinear, underexcited system with unmeasurable state variables. Realistic
numerical examples via computer simulations are presented to assess the
performance of the proposed approach—even been able to track the parameter
variations when the system changes operating point due to its on-line nature.

We are currently working on the practical implementation of the proposed
identification strategy on a physical PV array and we expect to be able to
report our results in the near future.

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