# Estimation of Climatic Parameters of a PV System Based on Gradient Method

Authors: Rabiaa Gammoudi; Houda Brahmi; Rachid Dhifaoui
Journal: Complexity
 (2019)

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## Abstract

The behavior of a photovoltaic generator is generally described by its current-voltage characteristic whose appearance depends on the climatic conditions (temperature and solar radiation). The aim of this work is to show the possibility of identifying the values of these two parameters from an experimental curveIpv(Vpv) using numerical method. Subsequently, we will also estimate the series and shunt resistors of the equivalent scheme of a PVG. This estimate is certainly beneficial to properly evaluate its energy balance. The integration of these resistances allows us to obtain a new model. In this work, we will propose a mathematical calculation strategy to ensure the estimation of all the parameters.


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## Body

## 1. Introduction

It is now recognized internationally that the supply of energy, in its various forms, contributes to the sustainable development of the countries. In this context, fossil fuels (oil, gas, uranium, etc.), despite their contributions, have disadvantages in terms of capacity that comes from the deterioration of the world’s reserves and in terms of the impact on the environment. The latter is visible by the rejection of greenhouse gases that causes global warming. These disadvantages give rise to technologies that help the exploitation of inexhaustible clean natural energies. In recent decades, as a result of improved panel performance, photovoltaic, as well as the progression of benefits including subsidies, the profitability of a photovoltaic installation has become justified [1, 2].The output characteristics of a photovoltaic panel depend essentially on the solar radiation to which it is subjected and its temperature, which makes the knowledge of these two parameters is very important. These two quantities can be measured using sensors, also they can be estimed in order to minimize the number of sensors used which is a very important factor in the design of photovoltaic systems.The model of a photovoltaic cell with a single diode is the most used thanks to the simplification of the equivalent electrical circuit which makes it possible to describe the behavior of the cell with the minimum of equations. For constant climatic conditions, this model has five parameters. In the case of variable climatic conditions, these parameters change in consequence. The estimation of the series and shunt resistance of the photovoltaic generator equivalent diagram is certainly beneficial to correctly evaluate its energy balance [3, 4].In this work, in order to treat the characteristics and to analyze the impact of the variations of the quantities intervening from it from a practical point of view, we study the case of the photovoltaic panel TITAN-12-50 of ERCO-INSAT laboratory.In the first part, we will be interested in the mathematical model of a photovoltaic cell, the developed model takes into account climatic variations such as temperature, solar radiation and wind speed. Support simulations are presented to show the dynamic behavior of the system.these simulations show that the shnut and series resistances are influenced by the climatic parameters or the interest of estimating them in real time. In the second part the expression of the two resistances has been developed. The efficiency of the new model has been validated experimentally while using a Dspace1104 card. The third part revolves around the estimation of two main parameters (solar radiation and temperature) through an algebraic strategy from an experimental curveIpv(Vpv).Finally, we did a case study on the TITAN 12-50 panel installed on the roof of the ERCO-INSAT research unit.


## 2. Modeling of Solar Cell and Experimental Database

The equivalent circuit of a photovoltaic cell is given by Figure1.Figure 1
Equivalent circuit of real solar cell.In the sense of this model, we can write the following relation:(1)iph=id+ipv+ish=id+ipv+vpv+rsipvrshwith vpv and ipv being, respectively, the output voltage and the output current of PV cell. rs and rsh are the series and shunt resistors of the cell, ish is the current passing in the rsh resistance, and id is the current of the diode.The current of the diodeid is strongly related to the temperature and the gap energy of the junction. It also depends on the voltage across the diode. It is given by the equation below:(2)id=isexp⁡vpv+rsipvvt-1where is is the reverse saturation current of the diode and vt is the solar cell thermal voltage.The inverse saturation current and the thermal voltage are given by the following equations:(3)is=isTj=isrTjTjr3exp⁡wg1vtr-1vt(4)vt=KIKBqTj,vtr=KIKBqTjrwhere vt1=KIKB/q.We note here that the index (r) used in this paper indicates that the value corresponds to the STC condition (Standard Test Condition:E=Es=1000W/m2, TA=TAr=25°C).w g is the energy of gap, KB Boltzmann constant, KI coefficient of ideality, and q the electron charge, and the main variable of this model is the temperature of the p-n junction of the diode Tj (expressed in °K).The following model was adopted taking into account variations in solar radiation, wind speed, and ambient temperatureTA.(5)Tj=TAC°+273.15+33.75EsfWϑWThe solar radiation Es is expressed inW/m2 and the function fW(ϑW) introduces the effect of the wind speed. (6)fWϑW=exp⁡-γϑW-1ϑW≥1The parameter γ must be positive for the function to vary in the opposite direction of the wind speed. Indeed the increase of the wind speed makes the solar cell cool [5].The current of solar photonsiph is strongly dominated by a variation of solar radiation and very slightly variable under the effect of the temperature. Given these properties, the current of photons is governed by the following model.(7)iph=iphEs,Tj=Esiphr+kTTj-TjrkT is the temperature coefficient that depends a lot on the manufacturing technology and it is basically provided by the manufacturer.Table1 gives the values of the different quantities used in PVG model.Table 1
Quantities related to the physical model of the solar cell.
Notation Designation Numerical values E s r Reference solar radiation (w/m2) in STC 1000 T A r Ambient temperature (°C) in STC 25 T j r Reference junction temperature (°K) in STC 298.15 w g Energy of gap (eV) 1.12 q Electron charge (C) 1.602 × 1 0 - 19 K B Boltzmann constant (J/°K) 1.38065 × 1 0 - 23 K I Constant of ideality 1==>2 k T Temperature coefficient (A/°C) 0.0032All the quantities related to the climatic conditions are defined. The electrical behavior in charge of the solar cell will be described by (8).(8)fvpv,ipv=ipv-iph+is(exp⁡vpv+rsipvvt-1+vpv+rsipvrshThis function is a nonlinear function and involves several variables. In order to resolve it, we need a numerical method; in this work Newton-Raphson method is exploited [6].The nominal powers of solar cells are always limited to a few watts. So the solution in this case is to switch to serial-parallel coupling of several cells. Solar energy is marketed in the form of panels or photovoltaic generator [7–9].A PVG is a configuration comprisingNp rows in parallel; each of the rows is formed byNs panels in series, with each panel being itself formed ofns series cells. So we finally get the following.(9)Ipv=ipv×Np,Vpv=ns×Ns×vpvFor this purpose, we denote, respectively, byIpv andVpv the current and the voltage of PVG.So, finally we get for a photovoltaic generator the following mathematical model (10):(10)FIp,Vp=Ip-Iph+Isexp⁡Vp+RsIpVt-1+Vp+RsIpRsh=0with(11)Rs=NsnsrsNpRsh=NsnsrshIs=NpisIph=NpiphVT=NsnsvT.


### 2.1. Nonlinear CharacteristicIpv=f(Vpv)

The mathematical model of a photovoltaic generator is expressed in the following form.(12)f=Ipv-Iph-Is+Js+Jsh=0To arrange the calculation in the simplest way and therefore as clear as possible, we will define the following quantities.(13)JsVpv,Ipv=Isexp⁡Vpv+RsIpvVT(14)JshVpv,Ipv=Vpv+RsIpvRsh=GshVpv+RsIpvWe must solve the following system with two variables Vpv and Ipv. (15)fVpv,Ipv=Ipv-Iph-Is+Js+Jsh=0This function is nonlinear; the most appropriate method for its resolution is the method of “Newton Raphson”. This technique requires the development of the Jacobian matrix. The elements of this matrix are defined below.(16)J=∂f∂Ipv∂f∂Vpv(17)∂f∂Ipv=1+RsJsVt+RsGsh∂f∂Vpv=JsVt+GshTable2 gives the main data of the solar panel TITAN-12-50.Table 2
Electrical parameters of the solar panel TITAN-12-50.
Name Manufacturer TITAN Description STP-50-01 Cell types Polycrystalline Cell size 100cm2 Cell number 36 Rated power 50Wc Power tolerance ±5% Optimal voltage 17.2V Optimal current 2.9A Open circuit voltage 21V Short circuit current 3.4A Efficiency 11.3% Manufacturer’s warranty 20 years


### 2.2. Influence of Solar Radiation on the Classic Model

To show the influence of solar radiation on theIpv=f(Vpv) characteristic from the previously developed database, the characteristics are measured at the same constant temperature of 26°C and different solar radiation ranging from 200 to 900 W/m2 as shown in Figure 2. It can be seen that the short-circuit current varies in the same direction as the solar radiation while the open circuit voltage varies slightly with the solar radiation.Figure 2
Influence of solar radiation on theIpv=f(Vpv) characteristic.


### 2.3. Influence of Temperature on the Classic Model

For different temperatures and with a constant solar radiation equal to 600W/m2, we acquire the voltage at the terminals of the PVG and the current flow as shown in Figure 3.Figure 3
Influence of temperature on theIpv=f(Vpv) characteristic.We notice that the short-circuit current is not sensitive to the variation of the temperature. On the other hand, the open circuit voltage varies in the opposite direction of the temperature.


## 2.1. Nonlinear CharacteristicIpv=f(Vpv)

The mathematical model of a photovoltaic generator is expressed in the following form.(12)f=Ipv-Iph-Is+Js+Jsh=0To arrange the calculation in the simplest way and therefore as clear as possible, we will define the following quantities.(13)JsVpv,Ipv=Isexp⁡Vpv+RsIpvVT(14)JshVpv,Ipv=Vpv+RsIpvRsh=GshVpv+RsIpvWe must solve the following system with two variables Vpv and Ipv. (15)fVpv,Ipv=Ipv-Iph-Is+Js+Jsh=0This function is nonlinear; the most appropriate method for its resolution is the method of “Newton Raphson”. This technique requires the development of the Jacobian matrix. The elements of this matrix are defined below.(16)J=∂f∂Ipv∂f∂Vpv(17)∂f∂Ipv=1+RsJsVt+RsGsh∂f∂Vpv=JsVt+GshTable2 gives the main data of the solar panel TITAN-12-50.Table 2
Electrical parameters of the solar panel TITAN-12-50.
Name Manufacturer TITAN Description STP-50-01 Cell types Polycrystalline Cell size 100cm2 Cell number 36 Rated power 50Wc Power tolerance ±5% Optimal voltage 17.2V Optimal current 2.9A Open circuit voltage 21V Short circuit current 3.4A Efficiency 11.3% Manufacturer’s warranty 20 years


## 2.2. Influence of Solar Radiation on the Classic Model

To show the influence of solar radiation on theIpv=f(Vpv) characteristic from the previously developed database, the characteristics are measured at the same constant temperature of 26°C and different solar radiation ranging from 200 to 900 W/m2 as shown in Figure 2. It can be seen that the short-circuit current varies in the same direction as the solar radiation while the open circuit voltage varies slightly with the solar radiation.Figure 2
Influence of solar radiation on theIpv=f(Vpv) characteristic.


## 2.3. Influence of Temperature on the Classic Model

For different temperatures and with a constant solar radiation equal to 600W/m2, we acquire the voltage at the terminals of the PVG and the current flow as shown in Figure 3.Figure 3
Influence of temperature on theIpv=f(Vpv) characteristic.We notice that the short-circuit current is not sensitive to the variation of the temperature. On the other hand, the open circuit voltage varies in the opposite direction of the temperature.


## 3. Estimation of Series and Shunt Resistances

The estimation of the series and shunt resistances of the equivalent PVG scheme is certainly beneficial to correctly evaluate its energy balance.


### 3.1. Influence of Series Resistance on the Classic Model

For different values of the series resistance, the behavior of the PVG generator was simulated by plotting theIpv=f(Vpv) characteristics. Figure 4 shows the evolution of the PV current with the PV voltage for different values of series resistance.Figure 4
Influence of series resistance on theIpv=f(Vpv) characteristic.It is noted that the short-circuit current and the open circuit voltage are not affected by the variation of the series resistance. On the other hand, the slope of the characteristic is very sensitive to this variation. The slope variation affects the maximum power as shown in Figure5.Figure 5
Influence of serie resistance on thePpv=f(Vpv) characteristic.So, we can say here that the increase of the series resistance weakens the optimal power. This result is very obvious because a part of the power produced would be lost by Joule effect.


### 3.2. Influence of Shunt Resistance on the Classic Model

We will now set the value of the series resistance to see the influence of the shunt resistor, Figure6.Figure 6
Influence of shunt resistance on theIpv=f(Vpv) characteristic.Like the series resistor, the shunt resistor also does not have an influence on the open circuit voltage and the short-circuit current. The effect of the shunt resistor is very focused on the optimum operating point of the PVG, Figure7.Figure 7
Influence of shunt resistance on thePpv=f(Vpv) characteristic.The increase in shunt resistance slightly weakens the optimal power. The joules losses due to the shunt resistor remain much lower than those due to the series resistance.I phr andIsr are, respectively, the saturation and photons currents in the reference regime. The values of these quantities are usually evaluated from two particular regimes. For the open circuit and short-circuit regimes, the experience confirms that the effect of series and shunt resistances is almost negligible regardless of the weather conditions.In open circuit regime, we have the following.(18)Vpv=VocIpv=0So, we obtain the following. (19)Iph-Id-Ipv-Ish=0Iph-Isexp⁡VocVt-1=0Iph=Isexp⁡VocVt-1In short-circuit regime, we have the following. (20)Vpv=0Ipv=IccSo, we obtain the following.(21)Iph=IccBack to the open circuit regime, we can write the following in reference climatic condition.(22)Isr=Iphrexp⁡Vocr/Vtr-1=Iccrexp⁡-VocrVtr


### 3.3. Estimation of Series Resistance

The internal conductance of the photovoltaic generator in any mode satisfies the relation below.(23)GpvVpv,Ipv=∂Ipv∂Vpv=-GshVT+JsVT+RsJs+VTRsGshTo estimate the value of the series resistor, it is necessary to carry out an open circuit test because the effect of the series resistance is manifested in the vicinity of the open circuit operation, unlike the effect of the shunt resistor, which remains insignificant. The value of the series resistance is estimated by the following relation.(24)Rs=-∂Vpv∂IpvwhenIpv=0andVpv=VocBy applying (23) in (24), we can deduce the following.(25)Rs=-1GpvVoc,0=VtIsexp⁡Voc/Vt=VtIsexp⁡-VocVtUnder Standard Test Conditions (STC), we consider the following peer relationship.(26)Rsr=VtrIsrexp⁡-VocrVtrApplying the expression ofIsr, we obtain the following.(27)Rsr=VtrIccr1-exp⁡-VocrVtr


### 3.4. Estimation of Shunt Resistance

The effect of the shunt resistor is significant around the point of short-circuit operation, so it is necessary to perform a short-circuit test to estimate this resistance.The conductanceGsh has the following expression: (28)Gsh=-∂Ipv∂VpvwhenIpv=IccandVpv=0or, in other terms,(29)Gsh=-Gpv0,Icc=GshVt+JsccVt+RsJscc+VtRsGshJscc=Js0,Icc=Isexp⁡RsIccVtso, finally we get (30).(30)VtRsGsh2+RsIsexp⁡RsIccVtGsh-Isexp⁡RsIccVt=0It is a second-order equation, and its solution passes by the computation of the discriminate, in which the square of the saturation current is involved, which is a very small quantity. We can thus eliminate the second term of this equation and estimate the shunt resistance by the following.(31)Rsh=RsVtIsexp⁡-RsIccVtIn STC, this relation becomes as follows.(32)Rshr=RsrVtrIsrexp⁡-RsrIccrVtrThis relation shows on one side a term proportional to the temperature and on the other side a second term, which decreases exponentially with the temperature.


### 3.5. Validation of the Proposed Model: Checking of the Evolution of Open Circuit Voltage with Temperature

The manufacturer supplies, under STC conditions, a coefficient of sensitivity of the open circuit voltage versus the value temperature. So, we can write the following.(33)Voc=Vocr-0.0798TA-TAr=22.9950-0.0798TAMoreover, in the sense of the model, the open circuit voltage verifies.(34)Voc=VTlog⁡1+IphIsThe curves in Figure 8 correspond, respectively, to formula (33) given by constructor and to formula (34). The difference between the two curves is acceptable.Figure 8
Variation of the open circuit voltage of the TITAN-12-50 panel with temperature.


### 3.6. Experimental Validation

To validate the previously proposed model, the photovoltaic panels installed at the research unit were exploited. The experimental installation comprises a photovoltaic generator whose characteristics are given in Table2, a voltage sensor, a solar radiation sensor, and a temperature sensor. This experimental bench is controlled by a DS1104 dSPACE card.The proposed strategy is implemented and tested in real time with a sampling frequency of 10 kHz. Figure9 shows the complete configuration of the system described above.Figure 9
Configuration of the studied system.Figures10 and 11 show, respectively, the scenario of solar radiation and the comparison between the measured voltage and the voltage estimated by the developed model. It is constant that the tensions converge towards the same value whatever the variation of the solar radiation.Figure 10
Variation of solar radiation.Figure 11
Comparison between measured and estimated open circuit voltage (Voc and Vmes).


## 3.1. Influence of Series Resistance on the Classic Model

For different values of the series resistance, the behavior of the PVG generator was simulated by plotting theIpv=f(Vpv) characteristics. Figure 4 shows the evolution of the PV current with the PV voltage for different values of series resistance.Figure 4
Influence of series resistance on theIpv=f(Vpv) characteristic.It is noted that the short-circuit current and the open circuit voltage are not affected by the variation of the series resistance. On the other hand, the slope of the characteristic is very sensitive to this variation. The slope variation affects the maximum power as shown in Figure5.Figure 5
Influence of serie resistance on thePpv=f(Vpv) characteristic.So, we can say here that the increase of the series resistance weakens the optimal power. This result is very obvious because a part of the power produced would be lost by Joule effect.


## 3.2. Influence of Shunt Resistance on the Classic Model

We will now set the value of the series resistance to see the influence of the shunt resistor, Figure6.Figure 6
Influence of shunt resistance on theIpv=f(Vpv) characteristic.Like the series resistor, the shunt resistor also does not have an influence on the open circuit voltage and the short-circuit current. The effect of the shunt resistor is very focused on the optimum operating point of the PVG, Figure7.Figure 7
Influence of shunt resistance on thePpv=f(Vpv) characteristic.The increase in shunt resistance slightly weakens the optimal power. The joules losses due to the shunt resistor remain much lower than those due to the series resistance.I phr andIsr are, respectively, the saturation and photons currents in the reference regime. The values of these quantities are usually evaluated from two particular regimes. For the open circuit and short-circuit regimes, the experience confirms that the effect of series and shunt resistances is almost negligible regardless of the weather conditions.In open circuit regime, we have the following.(18)Vpv=VocIpv=0So, we obtain the following. (19)Iph-Id-Ipv-Ish=0Iph-Isexp⁡VocVt-1=0Iph=Isexp⁡VocVt-1In short-circuit regime, we have the following. (20)Vpv=0Ipv=IccSo, we obtain the following.(21)Iph=IccBack to the open circuit regime, we can write the following in reference climatic condition.(22)Isr=Iphrexp⁡Vocr/Vtr-1=Iccrexp⁡-VocrVtr


## 3.3. Estimation of Series Resistance

The internal conductance of the photovoltaic generator in any mode satisfies the relation below.(23)GpvVpv,Ipv=∂Ipv∂Vpv=-GshVT+JsVT+RsJs+VTRsGshTo estimate the value of the series resistor, it is necessary to carry out an open circuit test because the effect of the series resistance is manifested in the vicinity of the open circuit operation, unlike the effect of the shunt resistor, which remains insignificant. The value of the series resistance is estimated by the following relation.(24)Rs=-∂Vpv∂IpvwhenIpv=0andVpv=VocBy applying (23) in (24), we can deduce the following.(25)Rs=-1GpvVoc,0=VtIsexp⁡Voc/Vt=VtIsexp⁡-VocVtUnder Standard Test Conditions (STC), we consider the following peer relationship.(26)Rsr=VtrIsrexp⁡-VocrVtrApplying the expression ofIsr, we obtain the following.(27)Rsr=VtrIccr1-exp⁡-VocrVtr


## 3.4. Estimation of Shunt Resistance

The effect of the shunt resistor is significant around the point of short-circuit operation, so it is necessary to perform a short-circuit test to estimate this resistance.The conductanceGsh has the following expression: (28)Gsh=-∂Ipv∂VpvwhenIpv=IccandVpv=0or, in other terms,(29)Gsh=-Gpv0,Icc=GshVt+JsccVt+RsJscc+VtRsGshJscc=Js0,Icc=Isexp⁡RsIccVtso, finally we get (30).(30)VtRsGsh2+RsIsexp⁡RsIccVtGsh-Isexp⁡RsIccVt=0It is a second-order equation, and its solution passes by the computation of the discriminate, in which the square of the saturation current is involved, which is a very small quantity. We can thus eliminate the second term of this equation and estimate the shunt resistance by the following.(31)Rsh=RsVtIsexp⁡-RsIccVtIn STC, this relation becomes as follows.(32)Rshr=RsrVtrIsrexp⁡-RsrIccrVtrThis relation shows on one side a term proportional to the temperature and on the other side a second term, which decreases exponentially with the temperature.


## 3.5. Validation of the Proposed Model: Checking of the Evolution of Open Circuit Voltage with Temperature

The manufacturer supplies, under STC conditions, a coefficient of sensitivity of the open circuit voltage versus the value temperature. So, we can write the following.(33)Voc=Vocr-0.0798TA-TAr=22.9950-0.0798TAMoreover, in the sense of the model, the open circuit voltage verifies.(34)Voc=VTlog⁡1+IphIsThe curves in Figure 8 correspond, respectively, to formula (33) given by constructor and to formula (34). The difference between the two curves is acceptable.Figure 8
Variation of the open circuit voltage of the TITAN-12-50 panel with temperature.


## 3.6. Experimental Validation

To validate the previously proposed model, the photovoltaic panels installed at the research unit were exploited. The experimental installation comprises a photovoltaic generator whose characteristics are given in Table2, a voltage sensor, a solar radiation sensor, and a temperature sensor. This experimental bench is controlled by a DS1104 dSPACE card.The proposed strategy is implemented and tested in real time with a sampling frequency of 10 kHz. Figure9 shows the complete configuration of the system described above.Figure 9
Configuration of the studied system.Figures10 and 11 show, respectively, the scenario of solar radiation and the comparison between the measured voltage and the voltage estimated by the developed model. It is constant that the tensions converge towards the same value whatever the variation of the solar radiation.Figure 10
Variation of solar radiation.Figure 11
Comparison between measured and estimated open circuit voltage (Voc and Vmes).


## 4. Estimation of Solar Radiation and Temperature

To measure the temperature, it is necessary to provide the solar panel with an integrated temperature sensor, which would greatly increase the cost on the one hand and complicate its manufacture on the other hand. A possible indirect solution for the junction temperature evaluation would be to identify it from the measurement results and establish a link with the ambient temperature.With regard to solar radiation, the hardware solution of placing multiple solar radiation sensors would obviously increase the cost of installation [10–13]. The purpose of this section is to show the possibility of identifying the values of solar radiation and temperature from an experimental point of view.Before going on to the method of the estimation of the solar radiation and the temperature for a photovoltaic system, we will present the notions of the adapted gradient method.


### 4.1. Quadratic Minimization by the Conjugate Gradient Method

The optimization, especially numerical optimization, has experienced a significant increase in recent years with the advent of the computer. It is often the last stage of numerical analysis, where we have studied a physical phenomenon, put it into an equation, studied these equations, and shown that one could calculate the solutions with a computer. We begin to optimize the system by changing some parameters to change the solution in a desired direction.Several methods are used for numerical optimization such as Newton’s method, the quasi-Newton method, the conjugate gradient method, linear search, the regions of trust, and Nelder-Mead’s method.In this part, we will use the conjugate gradient method to minimize a nonlinear functionΦ(X_) differentiable at least twice.The development of the Taylor series functionΦ(X_) around an initial condition X0 is expressed as follows.(35)ΦX_0+ΔX_≅ΦX_0+∑i∂Φ∂xiX_0Δxi+12∑i,j∂2Φ∂xi∂xjX_0ΔxiΔxjThis equation can be further rewritten in the following form involving the gradient vector G_ and the Hessian matrix H of the second derivatives evaluated in X_0. The Hessian matrix constitutes the Jacobian matrix of the gradient vector.(36)ΦX_0+ΔX_≅ΦX_0+G_0TΔX_+12ΔX_TH0ΔX_(37)G_0=G_X_0=∂Φ∂X_X_0H0=HX_0=∂G_X_∂X_X_0In expanded form, the matrix H(X_)is symmetric and has the following expression.(38)hij=hji=∂gi∂xj=∂gj∂xi=∂2∂xi∂xj=∂2∂xj∂xiThe solution of minimizing a quadratic problem of the type below using the conjugate gradient method is given by the following system.(39)minX_ΦX_ΦX_=12X_THX_+C_TX_To minimize the function Φ(X_) we could proceed to a local minimization in terms of ΔX_ with G_0 and H0 constant and repeat the same calculation as many times as necessary by rectifying each time the initial condition, the gradient vector, and the Hessian matrix. Obviously, it is admitted here that the Hessian matrix remains positive, which is often the case.


### 4.2. Estimation of Solar Radiation and Temperature: Minimizing the Vp(Ip) Function

In this section, we are interested in the identification ofEs andVt parameters from a given (Vpv,Ipv) operating point.As we have already presented in the first section, the model of a photovoltaic generator is described by the following.(40)f=Ip-Iph-Is+Js+Jsh=0The different terms of this model are defined as follows.(41)VT=ηTjVTr=ηTjrIph=EsIccr+KTrVT-VTrIs=IsrVTVTr3exp⁡WgVTr-WgVTUp=Vp+RsIpJs=Isexp⁡UpVTJsh=GshUpThe main variables of this model are, respectively, the voltage across the generator Vp and the current Ip delivered to the load. These quantities vary, of course, with the load, solar irradiance, and temperature. The solar radiation is denoted here as Es. The junction temperature appears by the magnitude VT, which is proportional to it, provided that the factor of constant ideality is considered. The parameters Rs and Gsh correspond, respectively, to the series resistance and the shunt conductance of the equivalent scheme. The other parameters are constants.The mathematical model of a photovoltaic generator for a given operating point becomes as follows.(42)fVT,Es=0Before exposing the identification technique used, it is necessary to develop the first and second derivatives of the terms of the functionf of the model with respect to these parameters. We establish the following.(43)up=Vpv+RsIpv∂Iph∂Vt=EsKTr∂Iph∂Es=Iccr+KTrVt-Vtr∂Is∂Vt=Is3Vt+WgVt2∂Js∂Vt=Js3Vt+Wg-upVt2∂2Iph∂Es2=∂2Iph∂Vt2=0∂2Is∂Vt2=Is2Vt+Wg2+2Vt2Vt4∂2Js∂Vt2=Js2Vt+Wg-up2+2Vt2Vt4Let us consider an experimental point (Vpv,Ipv) raised to solar radiation and temperature supposed unknown. The injection of this point into the functionf of the mathematical model (40) would lead to a nonzero value because of the error between the real values of the parameters and the values used in the model.By denoting byX_ the vector of the parameters (Vt,Es) to be identified, the function (40) of the model would be denoted accordingly by f(X_). The object is to determine the vector X_ that makes f(X_) tend to zero.In order to take advantage of the available digital quadratic programming tools, we perform the minimization of the following quadratic criterion.(44)ΦX_=12f2X_The Taylor series development expresses Φ(X_) as follows:(45)ΦX_0+x_=ΦX_0+G_0x_+12x_H0x_where G_0 and H0 are, respectively, the gradient vector and the Hessian matrix of Φ(X_) around an initial condition X_0. The vector x_ corresponds to the variation to be made around X_0.The gradient vector and the Hessian matrix are expressed as follows.(46)G_X_=∂Φ∂X_=fX_∂fX_∂X_HX_=∂G_X_∂X_By involving the sensitivity relationships defined above, the gradient vector G_(X_) has for relation (47) the following.(47)ψ1=∂f∂Vt=∂Ipv∂Vt-∂Iph∂Vt-∂Is∂Vt+∂Js∂Vt+∂Jsh∂Vt=-∂Iph∂Vt-∂Is∂Vt+∂Js∂Vt=-EsKtr-Is3Vt+WgVt2+Js3Vt+Wg-upVt2ψ2=∂f∂Es=∂Ipv∂Es-∂Iph∂Es-∂Is∂Es+∂Js∂Es+∂Jsh∂Es=-∂Iph∂Es=-Iccr+KTrVt-VtrG_=ψ1ψ2In order to simplify the calculation without losing too much on precision, the Hessian matrix could be expressed by the approximate Gaussian relation below.(48)HX_≃∂fX_∂X_∂fX_∂X_T=ψ12ψ1ψ2ψ1ψ2ψ22We can express exactly this matrix by the use of the following complete formula.(49)H=ψ1+ψ3fψ1ψ2-KTrfψ1ψ2-KTrfψ22ψ3=∂ψ1∂Vt=∂∂Vt∂f∂Vt=∂∂Vt-∂Iph∂Vt-∂Is∂Vt+∂Js∂Vt=-∂2Is∂Vt2+∂2Js∂Vt2=-Is2Vt+Wg2+2Vt2Vt4+Js2Vt+Wg-up2+2Vt2Vt4The proposed technique is to iteratively apply the conjugate gradient method using H(X_) and G_(X_) as parameters of the quadratic problem. The algorithm consists of the following steps:(i) Give an initial conditionX_0 for the parameter vector X_.(ii) CalculateG_(X_0) and H(X_0) and readjust the limits of the variation vector x_.(iii) To seek the optimal solution of the quadratic problem by the constrained conjugate gradient technique to establish the correctionx_ necessary for the minimization of Φ(X_) around X_0,(50)minx_12x_THx_+G_Tx_x_min≤x_≤x_max.(iv) Correct the parameter vector byX_0=X_0+x_ and return to step (ii).


### 4.3. Estimation ofTj and Es (Application on the Photovoltaic Panel TITAN 12-50)

Just to fix the ideas and check the relations, consider the constant parameters of the Table1 and a function point corresponding to Vp=13.8727V and Ip=2.8545A. Propose as initial conditions X_0=1.31=VTEs0.With this initial condition, we calculate the following.(51)f0=-0.3045G_0=-0.05800.9743H0=0.0362-0.6092-0.609210.2387By completing the algorithm flow with the limits(52)10.8≤X_≤1.61.2we reach the solution: X_=[1.3056,0.9052]V T = 1.3056 and Es=0.9052.We note here that the estimated value of the solar radiation is divided by a ratio of 1000. Still as a test and validation of the developments made, consider the complete curve (Vpv,Ipv) given by Table 3 with the parameters of the TITAN-12-50 module already introduced in Table 2.Table 3
Different experimental values of the characteristics (Vpv, Ipv).
V p v I p v 0 3.300 3.5353 3.2505 5.5500 3.2010 6.7962 3.1680 8.3215 3.1185 9.7205 3.0690 10.9567 3.0195 12.2502 2.9535 13.8727 2.8545 15.2266 2.7555 16.5322 2.5740 17.2528 2.4255 17.9818 2.1945 18.4340 1.9800 18.7883 1.7655 19.1226 1.5345 19.3951 1.3200 19.6467 1.1055 19.8972 0.8745 20.1202 0.6600 20.3465 0.4455 20.5306 0.2145 20.6500 0Figures12 and 13 give, respectively, the variation of solar radiation and the temperature with the load voltageFigure 12
Variation of solar radiation with the load voltage.Figure 13
Variation of temperature with the load voltage.


## 4.1. Quadratic Minimization by the Conjugate Gradient Method

The optimization, especially numerical optimization, has experienced a significant increase in recent years with the advent of the computer. It is often the last stage of numerical analysis, where we have studied a physical phenomenon, put it into an equation, studied these equations, and shown that one could calculate the solutions with a computer. We begin to optimize the system by changing some parameters to change the solution in a desired direction.Several methods are used for numerical optimization such as Newton’s method, the quasi-Newton method, the conjugate gradient method, linear search, the regions of trust, and Nelder-Mead’s method.In this part, we will use the conjugate gradient method to minimize a nonlinear functionΦ(X_) differentiable at least twice.The development of the Taylor series functionΦ(X_) around an initial condition X0 is expressed as follows.(35)ΦX_0+ΔX_≅ΦX_0+∑i∂Φ∂xiX_0Δxi+12∑i,j∂2Φ∂xi∂xjX_0ΔxiΔxjThis equation can be further rewritten in the following form involving the gradient vector G_ and the Hessian matrix H of the second derivatives evaluated in X_0. The Hessian matrix constitutes the Jacobian matrix of the gradient vector.(36)ΦX_0+ΔX_≅ΦX_0+G_0TΔX_+12ΔX_TH0ΔX_(37)G_0=G_X_0=∂Φ∂X_X_0H0=HX_0=∂G_X_∂X_X_0In expanded form, the matrix H(X_)is symmetric and has the following expression.(38)hij=hji=∂gi∂xj=∂gj∂xi=∂2∂xi∂xj=∂2∂xj∂xiThe solution of minimizing a quadratic problem of the type below using the conjugate gradient method is given by the following system.(39)minX_ΦX_ΦX_=12X_THX_+C_TX_To minimize the function Φ(X_) we could proceed to a local minimization in terms of ΔX_ with G_0 and H0 constant and repeat the same calculation as many times as necessary by rectifying each time the initial condition, the gradient vector, and the Hessian matrix. Obviously, it is admitted here that the Hessian matrix remains positive, which is often the case.


## 4.2. Estimation of Solar Radiation and Temperature: Minimizing the Vp(Ip) Function

In this section, we are interested in the identification ofEs andVt parameters from a given (Vpv,Ipv) operating point.As we have already presented in the first section, the model of a photovoltaic generator is described by the following.(40)f=Ip-Iph-Is+Js+Jsh=0The different terms of this model are defined as follows.(41)VT=ηTjVTr=ηTjrIph=EsIccr+KTrVT-VTrIs=IsrVTVTr3exp⁡WgVTr-WgVTUp=Vp+RsIpJs=Isexp⁡UpVTJsh=GshUpThe main variables of this model are, respectively, the voltage across the generator Vp and the current Ip delivered to the load. These quantities vary, of course, with the load, solar irradiance, and temperature. The solar radiation is denoted here as Es. The junction temperature appears by the magnitude VT, which is proportional to it, provided that the factor of constant ideality is considered. The parameters Rs and Gsh correspond, respectively, to the series resistance and the shunt conductance of the equivalent scheme. The other parameters are constants.The mathematical model of a photovoltaic generator for a given operating point becomes as follows.(42)fVT,Es=0Before exposing the identification technique used, it is necessary to develop the first and second derivatives of the terms of the functionf of the model with respect to these parameters. We establish the following.(43)up=Vpv+RsIpv∂Iph∂Vt=EsKTr∂Iph∂Es=Iccr+KTrVt-Vtr∂Is∂Vt=Is3Vt+WgVt2∂Js∂Vt=Js3Vt+Wg-upVt2∂2Iph∂Es2=∂2Iph∂Vt2=0∂2Is∂Vt2=Is2Vt+Wg2+2Vt2Vt4∂2Js∂Vt2=Js2Vt+Wg-up2+2Vt2Vt4Let us consider an experimental point (Vpv,Ipv) raised to solar radiation and temperature supposed unknown. The injection of this point into the functionf of the mathematical model (40) would lead to a nonzero value because of the error between the real values of the parameters and the values used in the model.By denoting byX_ the vector of the parameters (Vt,Es) to be identified, the function (40) of the model would be denoted accordingly by f(X_). The object is to determine the vector X_ that makes f(X_) tend to zero.In order to take advantage of the available digital quadratic programming tools, we perform the minimization of the following quadratic criterion.(44)ΦX_=12f2X_The Taylor series development expresses Φ(X_) as follows:(45)ΦX_0+x_=ΦX_0+G_0x_+12x_H0x_where G_0 and H0 are, respectively, the gradient vector and the Hessian matrix of Φ(X_) around an initial condition X_0. The vector x_ corresponds to the variation to be made around X_0.The gradient vector and the Hessian matrix are expressed as follows.(46)G_X_=∂Φ∂X_=fX_∂fX_∂X_HX_=∂G_X_∂X_By involving the sensitivity relationships defined above, the gradient vector G_(X_) has for relation (47) the following.(47)ψ1=∂f∂Vt=∂Ipv∂Vt-∂Iph∂Vt-∂Is∂Vt+∂Js∂Vt+∂Jsh∂Vt=-∂Iph∂Vt-∂Is∂Vt+∂Js∂Vt=-EsKtr-Is3Vt+WgVt2+Js3Vt+Wg-upVt2ψ2=∂f∂Es=∂Ipv∂Es-∂Iph∂Es-∂Is∂Es+∂Js∂Es+∂Jsh∂Es=-∂Iph∂Es=-Iccr+KTrVt-VtrG_=ψ1ψ2In order to simplify the calculation without losing too much on precision, the Hessian matrix could be expressed by the approximate Gaussian relation below.(48)HX_≃∂fX_∂X_∂fX_∂X_T=ψ12ψ1ψ2ψ1ψ2ψ22We can express exactly this matrix by the use of the following complete formula.(49)H=ψ1+ψ3fψ1ψ2-KTrfψ1ψ2-KTrfψ22ψ3=∂ψ1∂Vt=∂∂Vt∂f∂Vt=∂∂Vt-∂Iph∂Vt-∂Is∂Vt+∂Js∂Vt=-∂2Is∂Vt2+∂2Js∂Vt2=-Is2Vt+Wg2+2Vt2Vt4+Js2Vt+Wg-up2+2Vt2Vt4The proposed technique is to iteratively apply the conjugate gradient method using H(X_) and G_(X_) as parameters of the quadratic problem. The algorithm consists of the following steps:(i) Give an initial conditionX_0 for the parameter vector X_.(ii) CalculateG_(X_0) and H(X_0) and readjust the limits of the variation vector x_.(iii) To seek the optimal solution of the quadratic problem by the constrained conjugate gradient technique to establish the correctionx_ necessary for the minimization of Φ(X_) around X_0,(50)minx_12x_THx_+G_Tx_x_min≤x_≤x_max.(iv) Correct the parameter vector byX_0=X_0+x_ and return to step (ii).


## 4.3. Estimation ofTj and Es (Application on the Photovoltaic Panel TITAN 12-50)

Just to fix the ideas and check the relations, consider the constant parameters of the Table1 and a function point corresponding to Vp=13.8727V and Ip=2.8545A. Propose as initial conditions X_0=1.31=VTEs0.With this initial condition, we calculate the following.(51)f0=-0.3045G_0=-0.05800.9743H0=0.0362-0.6092-0.609210.2387By completing the algorithm flow with the limits(52)10.8≤X_≤1.61.2we reach the solution: X_=[1.3056,0.9052]V T = 1.3056 and Es=0.9052.We note here that the estimated value of the solar radiation is divided by a ratio of 1000. Still as a test and validation of the developments made, consider the complete curve (Vpv,Ipv) given by Table 3 with the parameters of the TITAN-12-50 module already introduced in Table 2.Table 3
Different experimental values of the characteristics (Vpv, Ipv).
V p v I p v 0 3.300 3.5353 3.2505 5.5500 3.2010 6.7962 3.1680 8.3215 3.1185 9.7205 3.0690 10.9567 3.0195 12.2502 2.9535 13.8727 2.8545 15.2266 2.7555 16.5322 2.5740 17.2528 2.4255 17.9818 2.1945 18.4340 1.9800 18.7883 1.7655 19.1226 1.5345 19.3951 1.3200 19.6467 1.1055 19.8972 0.8745 20.1202 0.6600 20.3465 0.4455 20.5306 0.2145 20.6500 0Figures12 and 13 give, respectively, the variation of solar radiation and the temperature with the load voltageFigure 12
Variation of solar radiation with the load voltage.Figure 13
Variation of temperature with the load voltage.


## 5. Conclusion

In this paper, we have proposed a method based on numerical development to estimate through an experimental curve the two main parameters influencing the operation of a PV system. In the same way, we estimated the two series and shunt resistors while showing the influence of the climatic conditions on these resistances, which allows us to develop a new model of a PVG. Finally, we showed the efficiency of this model experimentally using dSPACE 1104.



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*Source: 7385927-2019-02-18.xml*