The power output produced by PV arrays is typically influenced by several factors. Irradiance levels and solar rays incidence angle are, of course, two key factors in the determination of available energy which, however, depends also on the cell temperature, on the semiconductor technology (e.g., mono-crystalline, poly-crystalline or amorphous silicon) and on the load impedance.

The relationships between current and voltage values are usually expressed by means of Several I–V characteristic curves. Manufacturers are used to summarize these nonlinear relations by providing in datasheets a given number of electrical parameters, the most typical being those associated to the so-called remarkable points: (i) short circuit point $(V=0,I={I}_{sc})$; (ii) open circuit point $(V={V}_{oc},I=0)$; (iii) maximum power point $(V={V}_{mp},I={I}_{mp})$. These parameters are evaluated under standard test conditions (STC), namely at 1000 W/m^{2} irradiance and 25 °C array temperature.

In this article, we consider several models of PV arrays which have been proposed in the recent scientific literature. These differ in the number and type of components of the equivalent circuit, which are used to model different features of a solar panel. Needless to say, a more accurate model specification needs more details to be taken into account (i.e., more components in the equivalent circuit). However, adding components negatively affects both the parameter extraction procedure and the computational complexity of the model, measured in terms of simulation/emulation time.

#### 2.1. Models Description

In the following, we describe the five models we chose to implement on the emulation board. They are, to the best of our knowledge, representative of a wide range of state-of-the-art models with different features in terms of flexibility, accuracy, and computational requirements for their resolution [

30].

• Two diodes model:

Figure 1 exemplifies a two diodes model with parallel and series resistance. It is currently considered, in literature, a model capable of accurately reproducing a broad spectrum of operating conditions [

19,

30]. It takes into consideration a series of aspects and features related to the production of energy from solar radiation, namely the following:

- (1)
the classic PV effect, by means of current generator ${I}_{PV}$ and diode $D1$;

- (2)
the effect of the recombination current in the depletion region, which particularly affects accuracy in the low-voltage region, by means of diode $D2$;

- (3)
losses due to contact resistance (between silicon and electrodes surfaces) and materials resistance (silicon and electrodes metal), by means of series resistance ${R}_{s}$;

- (4)
sensitivity to temperature variation and the effect of leakage current in the PN junction, by means of parallel resistance ${R}_{p}$.

The current–voltage relation of this equivalent circuit is described in Equation (1).

where

${I}_{PV}$ represents the PV current generated by the light incident on the panel;

${I}_{01}$ is the reverse saturation (or leakage) current associated to diode

$D1$;

${I}_{02}$ is the reverse saturation (or leakage) current associated to diode

$D2$;

${a}_{1}$ is the ideality factor of diode

$D1$;

${a}_{2}$ is the ideality factor of diode

$D2$;

k is the Boltzmann constant (1.3806503 × 10

^{−23} J·K

^{−1}); q is the electron charge (1.60217646 × 10

^{−19} C); T is the temperature of the PN-junction. Introducing the so-called thermal voltage of the array (

${V}_{t}$) as

${V}_{t}=kT/q$ [

19], Equation (1) can be rewritten as:

The model entails, however, the characterization of a significant number of parameters : ${I}_{PV}$, ${I}_{01}$, ${I}_{02}$, ${a}_{1}$, ${a}_{2}$, ${R}_{s}$, ${R}_{p}$.

• Single diode model:

In this model, shown in

Figure 2a, the effect of the recombination current in the depletion region is not taken into account. Indeed, diode

$D2$ is dropped, resulting in the current–voltage relationship described by Equation (3):

This model entails the evaluation of five parameters for its complete characterization: ${I}_{PV}$, ${I}_{0}$, a, ${R}_{s}$, ${R}_{p}$.

• Single diode model without parallel resistance:

Given the usually high values of

${R}_{p}$, some authors have proposed to neglect its effect (i.e., to consider

${R}_{p}=\infty $) to simplify the model [

31,

32,

33,

34,

35,

36]. The equivalent circuit of the single diode model without shunt resistance

${R}_{p}$ is depicted in

Figure 2b. Equation (4) describes the resulting voltage–current relation:

This model entails the evaluation of four parameters ${I}_{PV}$, ${I}_{0}$, a, ${R}_{s}$to complete the characterization.

• Single diode model without series resistance:

In the single diode model, the values of

${R}_{s}$ are typically low in most practical cases. Hence, its effect has been neglected in some works [

37,

38,

39]. The resulting equivalent circuit is shown in

Figure 2c, while the corresponding Several

I–

V relation is described by Equation (5).

There are also four model parameters to be derived in this case, namely: ${I}_{PV}$, ${I}_{0}$, a, and ${R}_{p}$.

• Single diode model without series and parallel resistance:

If neither the series resistance

${R}_{s}$, nor the parallel resistance

${R}_{p}$ are considered (i.e.,

${R}_{s}=0$,

${R}_{p}=\infty $ in the single diode model), the model represented in

Figure 2d is obtained. This is a very simplified model, whose behavior is described by Equation (6).

On one hand, this is an ideal representation of a solar panel and, as such, it can be exploited for reasoning only on some basic, theoretical concepts related to PV arrays. On the other hand, it provides a useful comparison term to be included in our study, because of the low number of parameters to be obtained for its derivation (${I}_{PV}$, ${I}_{0}$, a), and because of the simple structure of the Several I–V equation which allows significant savings in term of execution time. In light of these considerations, it can be taken as a lower bound for computational complexity.

In general, several approaches have been proposed for the derivation of model parameters. A coarse grain classification is usually done among analytical methods and curve fitting/optimization methods [

30].

The former category of methods exploits information contained in components’ datasheets to derive a set of equations which are to be solved for parameters extraction. Given the nature of these equations (usually implicit, transcendental equations) and given that systems of equations relating them are often undetermined (i.e., there are more unknowns than equations) some heuristics are usually applied.

The latter type of solutions cast parameter extraction as a Several I–V curve fitting problem and solves it by means of optimization algorithms. The choice of a proper objective function and of the algorithm used to optimize it leads to various possible alternative methods.

While the focus of this article is on the model execution (in particular on the performance requirements of emulating a given model of small scale solar panels on embedded platforms), the interested reader can refer to a recent, comprehensive review of parameter extraction methods by Chin et al. [

30].

#### 2.2. Numerical Resolution Method

Once all parameters needed to characterize the model of the PV array are derived, emulation/simulation can be carried out. From the functional relationship among current and voltage expressed by Equations (1) and (3)–(6) and given a value of the current

I (or of the voltage

V), the corresponding value of voltage

V (current

I) can be derived from the nonlinear equations by means of numerical methods. We followed [

19] and used the Newton–Raphson algorithm [

40], a standard method to find roots of transcendental equations. In particular, if we formulate each of the Equations (1) and (3)–(6) as

$f(V,I)=0$, the value of

V can be obtained from a given

I by finding the root of the modeling equation. At each iteration, the value of

V is updated as follows:

until the stopping criterion

$|{V}_{i+1}-{V}_{i}|<\u03f5$ is met (

ϵ is an implementation-dependent tolerance value).

To better highlight the computational requirements of the numerical algorithm, we report in the following both the nonlinear modeling function $f(V,I)$ and its partial derivative $\frac{\partial f({V}_{i},I)}{\partial V}$, for each investigated model.

From Equations (8) and (9), it follows that this model entails several arithmetic operations (e.g., four exponentials) to be carried out at run-time by the emulator, which leads to a potential computational bottleneck in real-time applications, as will be discussed in the section devoted to experimental results.

In this case, the evaluation of Equations (10) and (11) requires a lower number of arithmetic operations (e.g., two exponentials instead of four at each iteration of the numerical algorithm) with regards to the two diodes model. This results in an expected improvement of computational performance.

•

Single diode model without parallel resistance:The execution time can be further lowered with respect to that of previous models because of the absence of terms $\frac{V+I{R}_{s}}{{R}_{p}}$ and $\frac{1}{{R}_{p}}$, which enables one to save three additions, a multiplication and two divisions at each of the iterations.

•

Single diode model without series resistance:Neglecting the series resistance enables one to save at each iteration three additions and three multiplications with regards to the single diode complete model. If we take, as a comparison term, the single diode model without parallel resistance, instead, performances are slightly worse: for the single diode model with ${R}_{p}=\infty $, we need, in fact, to compute two more additions and multiplications with regards to the single diode model with ${R}_{s}=0$ which, conversely, needs two more divisions and subtractions.

•

Single diode model without series and parallel resistance:This simplest model enables, in principle, a further speedup in the implementation of the Newton–Raphson algorithm since it allows us to save two additions and two multiplications with regards to the single diode model without shunt resistance.