Dynamic characteristics and improved MPPT control of PV generator

Abstract

This paper presents a mathematical model of photovoltaic (PV) module and gives a strategy to calculate online the maximum power point (MPP). The variation of series and shunt resistor are taken into account in the model and are dynamically identified using the Newton-Raphson algorithm. The effectiveness of the proposed model is verified by laboratory experiments obtained by implementing the model on the dSPACE DS1104 board.

Keywords

modeling of photovoltaic (PV) generator / maximum power point tracking (MPPT) / estimation parameters / real time controller

Introduction

Energy technologies play a significant role in economic and social development. Indeed, the production of energy is closely linked to environmental contamination, economic development and quality of life [1,2]. Today’s renewable energy technologies, such as photovoltaic energy, are considered a good alternative for the production of electricity since they are environmentally friendly and sustainable forms of energy [3-5].

Models are important in the control of systems because they present the dynamic process of underlying systems; surely a system cannot be controlled without having defined a specific model taking into account different parameters.

This paper aims to give a dynamical model of PV generator and a new strategy to establish the maximum power point (MPP). Besides, it also presents the control methodology to implant the model on the dSPACE DS1104 board.

First, it proposes the strategy for the photovoltaic generator model taking into account all the dynamic variations of internal parameters such as series and shunt resistor. Then, the proposed maximum power point tracking (MPPT) method is given. Next, simulation supports are put forward, giving the effect of the variation of solar radiation and temperature on series and shunt resistors. These simulations are performed with the data of a photovoltaic generator installed at the RME Research Unit. Finally, experimental results are provided to illustrate the validity of the adopted strategy.

PV source electrical model

The traditional equivalent circuit of photovoltaic cell represented by a current source in parallel with one diode [6,7] is shown in Fig. 1.

The definitions of PV parameters are given in Table 1.

The current i_{ph} is the current emitted by the cell under the effect of solar radiation. It is related to the intensity of the illumination and depends only on the temperature variation. The diode current i_d is related to the temperature and the energy gap of the junction [8,9]. In sense of this equivalent circuit, the current of the photovoltaic cell is given by

(1)i_{ph}=i_d+i_p+i_{sh}=i_d+i_p+{\displaystyle \frac{v_p+r_s{i}_p}{r_{sh}}}.

The current i_d is expressed by Eq. (2)

(2)i_d=i_s\left[\exp \left({\displaystyle \frac{v_p+r_s{i}_p}{v_T}}\right)-\mathrm{1}\right],

where i_s is the diode saturation current and v_T is the thermal voltage. These variables are defined respectively by Eqs. (3) and (4)

(3)i_s=i_s(T_j)=i_{sr}{\left({\displaystyle \frac{T_j}{T_{jr}}}\right)}^{\mathrm{3}}\exp \left(w_g\left({\displaystyle \frac{\mathrm{1}}{v_{Tr}}}-{\displaystyle \frac{\mathrm{1}}{v_T}}\right)\right),

(4)v_T={\displaystyle \frac{{}_I{}_B{T}_j}{q},}{v}_{Tr}={\displaystyle \frac{{}_I{}_B{T}_{jr}}{q}}\mathrm{.}

The index (r) used in this paper indicates that the value corresponds to the STC regime defined by an incident irradiance E_sr = 1000 W/m² and junction temperature T_jr = 25°C.

Considering the effect of the irradiance and the temperature on photo current, this current can be approximated as

(5)i_{ph}=i_{ph}(E_s,T_j)={\displaystyle \frac{E_s}{E_{sr}}}[i_{phr}+K_T(T_j-T_{jr})].

It is noted that the temperature variation is indirectly affected by the variation of wind speed. Equation (6) gives the expression of the temperature including the effect of the irradiance and wind speed [10,11].

(6)T_j=T_A(°C)+\mathrm{273.15}+\mathrm{33.75}{E}_s{f}_W({\vartheta }_W),

where f_W({\vartheta }_W) is the function introducing the effect of wind speed. Here, Eq. 7, which is the flowing expression, is considered.

(7)\{\begin{array}{l}{f}_W({\vartheta }_W)=\exp (-\gamma ({\vartheta }_W-\mathrm{1})),\hfill \\ {\vartheta }_W\ge \mathrm{1.}\hfill \end{array}

The parameter γ, that must be positive, varies in the opposite direction of wind speed. Indeed, when wind speed increases the temperature of the junction decreases.

The electrical behavior on charge of the solar cell can be evaluated by Eq. (8) which describes the evolution of the current delivered by the celli_p and the voltage at its terminal v_p.

(8)f(i_p,v_p)=i_p-i_{ph}+i_s\left[\exp \left({\displaystyle \frac{v_p+r_s{i}_p}{v_T}}\right)-\mathrm{1}\right]+{\displaystyle \frac{v_p+r_s{i}_p}{r_{sh}}}=\mathrm{0},

f(i_p,v_p) is a non-linear function which involves several variables. In order to resolve it, a numerical method is needed. In this work, the Newton-Raphson iterative method is exploited because of its safe and fast convergence.

For a photovoltaic generator having N_p cells in parallel and N_s cells in series, the shunt and series resistors are calculated by Eq. (9):

(9)\{\begin{array}{l}{R}_s={\displaystyle \frac{N_s{n}_s{r}_s}{N_p}},\hfill \\ R_{\mathrm{sh}}=N_s{n}_s{r}_{\mathrm{sh}}.\hfill \end{array}

Based on Eqs. (3)-(8) and the determined parameters, the formulated electrical model of photovoltaic modules is obtained by Eqs. (11)-(15).

\{\begin{array}{ll}{T}_j=T_A+\mathrm{273.15}+\mathrm{0.55}{f}_W({\vartheta }_W)E_s,\hfill & (\mathrm{11})\hfill \\ V_T=\left({\displaystyle \frac{N_s{n}_s{}_I{}_B}{}}\right)T_j,\hfill & (\mathrm{12})\hfill \\ I_s=I_{sr}{\left({\displaystyle \frac{V_T}{V_{Tr}}}\right)}^{\mathrm{3}}\exp \left({\displaystyle \frac{W_g}{V_{Tr}}}-{\displaystyle \frac{W_g}{V_T}}\right),\hfill & (\mathrm{13})\hfill \\ I_{ph}=E_s[I_{phr}+{}_T(V_T-V_{Tr})],\hfill & (\mathrm{14})\hfill \\ I_p-I_{ph}+I_s\left[\exp \left({\displaystyle \frac{V_p+R_s{I}_p}{V_T}}\right)-\mathrm{1}\right]+{\displaystyle \frac{V_p+R_s{I}_p}{R_{sh}}}=\mathrm{0},\hfill & (\mathrm{15})\hfill \end{array}

where

(16)\{\begin{array}{l}{W}_g=N_s{n}_s{w}_g,\hfill \\ V_{\mathrm{Tr}}=\mathrm{298.15}\left({\displaystyle \frac{N_s{n}_s{}_I{}_B}{Q}}\right),\hfill \\ {}_T={\displaystyle \frac{N_p{k}_{T}Q}{N_s{n}_s{}_I{}_B}}.\hfill \end{array}

This model involves two parameters I_sr and I_phr which correspond to the saturation current and photon current at the reference regime. The values ​​of these quantities are evaluated respectively in open circuit and short circuit conditions.

Expression of the internal conductance

Estimating series resistance and shunt resistance of the equivalent circuit of the PV generator is beneficial to properly assess its energy balance. In this sense, it is useful to introduce some sensitivity coefficients. First, the two quantities should be defined:

(17){}_s(V_p,I_p)=I_s\exp \left({\displaystyle \frac{V_p+R_s{I}_p}{V_T}}\right),

(18){}_{\mathrm{sh}}(V_p,I_p)={\displaystyle \frac{V_p+R_s{I}_p}{R_{\mathrm{sh}}}}=G_{\mathrm{sh}}(V_p+R_s{I}_p).

The derivatives of these quantities over V_p and I_p are expressed by

(19){\displaystyle \frac{\partial {}_s}{\partial I_p}}={\displaystyle \frac{R_s{}_s}{V_T}},{\displaystyle \frac{\partial {}_s}{\partial V_p}}={\displaystyle \frac{{}_s}{V_T}};

(20){\displaystyle \frac{\partial {}_{\mathrm{sh}}}{\partial I_p}}=R_s{G}_{\mathrm{sh}},{\displaystyle \frac{\partial {}_{\mathrm{sh}}}{\partial V_p}}=G_{\mathrm{sh}}.

For the purposes of solving Eqs. (19) and (20), the mathematical model described by Eq. (1) can be expressed as

(21)I_{\mathrm{p}}-I_{\mathrm{p}\mathrm{h}}-I_{\mathrm{s}}+J_{\mathrm{s}}{J}_{\mathrm{s}\mathrm{h}}=\mathrm{0.}

The internal conductance of the photovoltaic panel verifies Eq. (22):

(22)G_p(V_p,I_p)={\displaystyle \frac{\partial I_p}{\partial V_p}}=-{\displaystyle \frac{G_{\mathrm{sh}}{V}_T+{}_s}{V_T+R_s{}_s+V_T{R}_s{G}_{\mathrm{sh}}}}.

Characterization of the optimal operating point

To extract the maximum available power from the PV modules, it is necessary to operate the PV modules at their maximum power point (MPP) [12,13,14].

The operating point at maximum power is characterized by the following equivalent conditions:

(23){\displaystyle \frac{\partial P_v}{\partial V_p}}=\mathrm{0}\mathrm{o}\mathrm{r}{I}_p+V_p{\displaystyle \frac{\partial I_p}{\partial V_p}}=\mathrm{0.}

Involving relation (Eq. (5)), Eq. (24) can be obtained:

(24)I_p(V_T+R_s{}_p+V_T{G}_{\mathrm{sh}}{R}_s)-V_p{}_p-V_T{G}_{\mathrm{sh}}=\mathrm{0.}

To calculate the optimal regime, the optimality condition (Eq. (24)) above must be addressed in conjunction with Eq. (21) of the mathematical model of the generator. In equivalent terms, the following system must be solved in two variables V_p and I_p. Thus the maximum power point is characterized by the following system:

\{\begin{array}{l}{f}_{\mathrm{1}}(I_{\mathrm{p}},V_{\mathrm{p}})=I_{\mathrm{p}}(V_{\mathrm{T}}+R_{\mathrm{s}}{J}_{\mathrm{p}}+V_{\mathrm{T}}{G}_{\mathrm{s}\mathrm{h}}{R}_{\mathrm{s}})-V_{\mathrm{p}}{J}_{\mathrm{p}}-V_{\mathrm{T}}{G}_{\mathrm{s}\mathrm{h}}=\mathrm{0},(\mathrm{25})\hfill \\ f_{\mathrm{2}}(I_{\mathrm{p}},V_{\mathrm{p}})=I_{\mathrm{p}}-I_{\mathrm{p}\mathrm{h}}-I_{\mathrm{s}}+J_{\mathrm{p}}+U_{\mathrm{p}}=\mathrm{0.}(\mathrm{26})\hfill \end{array}

This system is non linear, the most appropriate method for its resolution is the Newton-Raphson method. This technique requires the development of the Jacobian matrix. The elements of this matrix are defined as

(27)\left[\begin{array}{cc}{\displaystyle \frac{\partial f_1}{\partial I_p}}& {\displaystyle \frac{\partial f_1}{\partial V_p}}\\ {\displaystyle \frac{\partial f_2}{\partial I_p}}& {\displaystyle \frac{\partial f_2}{\partial V_p}}\end{array}\right]=\left[\begin{array}{cc}{J}_{11}& J_{12}\\ J_{21}& J_{22}\end{array}\right],

where

(28)\{\begin{array}{l}{J}_{11}=(1+R_s{G}_{\mathrm{sh}})V_T+\left(1-{\displaystyle \frac{V_p-R_s{I}_p}{V_T}}\right)R_s{}_p,\hfill \\ J_{12}=-\left(1+{\displaystyle \frac{V_p-R_s{I}_p}{V_T}}\right){}_p,\hfill \\ J_{21}=1+{\displaystyle \frac{R_s{}_p}{V_T}}+R_s{G}_{\mathrm{sh}},\hfill \\ J_{22}={\displaystyle \frac{{}_p}{V_T}}+G_{\mathrm{sh}}.\hfill \end{array}

The algorithm of the Newton-Raphson method is summarized by the flowchart illustrated in Fig. 2.

Estimation of resistance parameters

Estimation of series resistance

The effect of the series resistance is more significant in open circuit conditions [15]; besides, the effect of the shunt resistor is insignificant. Based on this physical reality, it is proposed in this paper to estimate the series resistance around the open circuit regime while ignoring the effect of the shunt resistor.

(29)R_s=-{\displaystyle \frac{\partial V_p}{\partial I_p}}\mathrm{when}{I}_p=\mathrm{0}\mathrm{and}{V}_p=V_{\mathrm{oc}}.

By applying Eq. (26), Eq. (30) can be deduced:

(30)R_s=-{\displaystyle \frac{\mathrm{1}}{G_p(V_{\mathrm{oc}},\mathrm{0})}}={\displaystyle \frac{V_T}{I_s\exp (V_{\mathrm{oc}}/V_T)}}.

In the STC, the following relation can be obtained.

(31)R_{\mathrm{sr}}={\displaystyle \frac{V_{\mathrm{Tr}}}{I_{\mathrm{sr}}}}\exp \left(-{\displaystyle \frac{V_{\mathrm{ocr}}}{V_{\mathrm{Tr}}}}\right).

Estimation of shunt resistance

The shunt resistance acts near the maximum power point (MPP), which is the reason to approximate it in short circuit regime.

(32)G_{\mathrm{sh}}=-{\displaystyle \frac{\partial I_p}{\partial V_p}}\mathrm{when}{I}_p=I_{\mathrm{cc}}\mathrm{and}{V}_p=\mathrm{0.}

Under Eq. (22), Eqs. (33)-(35) can be obtained:

(33)G_{\mathrm{sh}}=-G_p(\mathrm{0},I_{\mathrm{cc}})={\displaystyle \frac{G_{\mathrm{sh}}{V}_T+{}_p(\mathrm{0},I_{\mathrm{cc}})}{V_T+R_s{}_p(\mathrm{0},I_{\mathrm{cc}})+V_T{R}_s{G}_{\mathrm{sh}}}},

(34){}_p(\mathrm{0},I_{\mathrm{cc}})=I_s\exp \left({\displaystyle \frac{R_s{I}_{\mathrm{cc}}}{V_T}}\right),

(35)V_T{R}_s{G}_{\mathrm{sh}}^2+\left(R_s{I}_s\exp \left({\displaystyle \frac{R_s{I}_{\mathrm{cc}}}{V_T}}\right)\right)G_{\mathrm{sh}}-I_s\exp \left({\displaystyle \frac{R_s{I}_{\mathrm{cc}}}{V_T}}\right)=\mathrm{0.}

Making the necessary arrangements, the estimation of the shunt resistance is given by

(36)R_{\mathrm{sh}}=\sqrt{{\displaystyle \frac{R_s{V}_T}{I_s}}\exp \left(-{\displaystyle \frac{R_s{I}_{\mathrm{cc}}}{V_T}}\right)}.

In the STC, Eq. (37) can be obtained:

(37)R_{\mathrm{shr}}=\sqrt{{\displaystyle \frac{R_{\mathrm{sr}}{V}_{\mathrm{Tr}}}{I_{\mathrm{sr}}}}\exp \left(-{\displaystyle \frac{R_{\mathrm{sr}}{I}_{\mathrm{ccr}}}{V_{\mathrm{Tr}}}}\right)}.

Adopted strategy

The ability to assess with sufficient precision the series resistance and shunt resistance to any regime of solar radiation and cell temperature have been shown, which, therefore, offers the opportunity to integrate the calculation of these resistances in the whole model. In this case, a completely electronic climate model has been obtained, avoiding the possible intervention to modify the values of R_s ​​and R_{sh} on regime. On the contrary, the integration of these resistances in the model provides a self-adjusting of current regime.

The model is from Eq. (38) to Eq. (47). The main purpose of this model is obviously to determine the operating point of the PV generator from three climate variables: solar radiation, ambient temperature and wind speed.

(38)f_W({\vartheta }_W)=\exp (-\gamma ({\vartheta }_W-\mathrm{1})){\vartheta }_W\ge \mathrm{1},

(39)T_j=T_A+\mathrm{273.15}+\mathrm{33.75}{f}_W({\vartheta }_W)E_s,

(40)V_T={\displaystyle \frac{N_s{n}_s{}_I{}_B}{Q}}{T}_j,

(41)I_s=I_{\mathrm{sr}}{\left({\displaystyle \frac{V_T}{V_{\mathrm{Tr}}}}\right)}^{\mathrm{3}}\exp \left({\displaystyle \frac{W_g}{V_{\mathrm{Tr}}}}-{\displaystyle \frac{W_g}{V_T}}\right),

(42)I_{\mathrm{ph}}=E_s[I_{\mathrm{phr}}+{}_T(V_T-V_{\mathrm{Tr}})],

(43)V_{\mathrm{oc}}=V_T\log \left(\mathrm{1}+{\displaystyle \frac{I_{\mathrm{ph}}}{I_s}}\right),

(44)R_s={\displaystyle \frac{V_T}{I_s}}\exp \left(-{\displaystyle \frac{V_{\mathrm{oc}}}{V_T}}\right),

(45)R_{\mathrm{sh}}=\sqrt{{\displaystyle \frac{R_s{V}_T}{I_s}}\exp \left(-{\displaystyle \frac{R_s{I}_{\mathrm{ph}}}{V_T}}\right)},

(46)I_p-I_{\mathrm{ph}}+I_s\left[\exp \left({\displaystyle \frac{V_p+R_s{I}_p}{V_T}}\right)-\mathrm{1}\right]+G_{\mathrm{sh}}(V_p+R_s{I}_p)=\mathrm{0},

(47)I_p\left(V_T+R_s{I}_s\exp \left({\displaystyle \frac{V_p+R_s{I}_p}{V_T}}\right)+V_T{G}_{\mathrm{sh}}{R}_s\right)-V_p\left[G_{\mathrm{sh}}{V}_T+I_s\exp \left({\displaystyle \frac{V_p+R_s{I}_p}{V_T}}\right)\right]=\mathrm{0.}

This model requires knowledge of numerical values ​​of six parameters identifiable from the following data:

1) The data related to the panel: the ideality factor, the number of cells in series, the open circuit voltage and the short-circuit current on STC regime, the sensitivity of the short-circuit current when the temperature varies;

2) The data related to the generator: number of panels in series and number of panels in parallel;

3) The general data: the gap energy, the Boltzmann constant and the electron charge;

4) The modeling data: the parameter of the descriptive function of the influence of the wind speed on the junction temperature.

When these data are present, the numerical values ​​of the parameters are deduced as listed below:

(48)\{\begin{array}{l}{W}_g=N_s{n}_s{w}_g\hfill \\ V_{Tr}=\mathrm{298.15}{\displaystyle \frac{N_s{n}_s{\kappa }_I{\kappa }_B}{\ell }}\hfill \\ {\kappa }_T={\displaystyle \frac{N_p{k}_T}{V_{T\mathrm{1}}}}\hfill \\ I_{phr}=N_p{I}_{ccr}\hfill \\ I_{sr}={\displaystyle \frac{N_p{I}_{ccr}}{\exp \left({\displaystyle \frac{N_s{V}_{ocr}}{V_{Tr}}}\right)-\mathrm{1}}}\hfill \end{array}

Once these data are ready, Eq. (46) is solved using Newton RAphson method. If it is of interesting to calculate the Maximum Power point, it is necessary to add Eq. (46) to the optimality condition Eq. (47). One of the aspects of utility of this model is to locate on (V_p, I_p) the locus of points giving the maximum power to the margins of large variations in climatic conditions.

Illustrative simulation

The photovoltaic panels considered are installed in the laboratory of the RME Research Unit at INSAT, as demonstrated in Fig. 3. This solar system is composed of ten identical panels, having the characteristics indicated in Table 2. In nominal conditions (solar radiation, temperature, humidity, etc.), a maximum power of 500 W can be extracted. The current and voltage of the system naturally depend on the mode of association series/parallel panels, the matching stage, the load resistance, and etc.

For STC regime, the following values have be calculated and resumed in Table 3.

Influence of temperature on series and shunt resistance

Equation (30) shows that in case of a constant solar radiation, the series resistance is positively proportional to the junction temperature. As the latter is referenced relative to 0 K, this implies that the temperature increase is small compared to this reference value. Figure 4 shows the dynamic variation of the series resistance with the variation of the temperature.

Equation (36) of the shunt resistor shows, on the one hand, a term proportional to the temperature and, on the other hand, a term which decreases exponentially with temperature. This results in the decrease in the overall behavior indicated by the curve in Fig. 5.

Influence of solar radiation on series and shunt resistance

From Eq. (30) it can be deduced that at constant temperature, the series resistance is marked by the inverse of short-circuit current which increases proportionally with the solar radiation. Figure 6 confirms this and shows that the series resistor varies in opposite of the solar radiation.

Equation (36) shows a term that decreases with the short-circuit current. Moreover, the denominator of Eq. (36) is proportional to the saturation current and is very low. Figure 7 shows the variation of shunt resistor with solar radiation.

The ability to assess with sufficient precision the series resistance and shunt resistance to any of solar radiation and cell temperature has been shown. This, therefore, offers the opportunity to integrate the calculation of these resistances in the model. Thereby the integration of these resistances in the model provides a self-adjusting current regime.

One of the aspects of utility of this model is to locate the points giving the maximum power to the margins of large variations in climatic conditions. Figure 8 reports the maximum power points for the photovoltaic panel TITAN-12-50 incorporating the effect of the series resistance and shunt resistance. This result corresponds to 1000 climatic points generated randomly. It is particularly useful to guide the user to locate the optimum power by a real-time control structure. Moreover, the delimitation of the optimal domain can easily be done by assigning formulas to determine the final left and right boundaries of research as shown in Fig. 3. The same kind of idea is also plotted in Fig. 9, which shows the dominance of the solar radiation on the optimal regime.

Experimental setup and results

To validate the previously proposed model, the photovoltaic panel installed in the RME research unit has been exploited, as shown in Fig. 1. The experimental setup consists of a photovoltaic generator described in Section 7, a voltage sensor, a solar radiation sensor, a temperature sensor and a wind sensor. The experimental bank is controlled by a dSPACE DS1104 card.

The different sensors are made in a didactic way and they are properly developed as part of the activities of the RME Research Unit. The illumination sensor is made ​​on the basis of a Pasan Cell Sorter 801 photovoltaic cell. This sensor is installed adjacent to the plane of photovoltaic power. The temperature sensor is analog and is located just near the panels; this sensor provides a voltage proportional to temperature. The proposed strategy is implemented and tested in real time with a sampling frequency equal to 10 kHz.

Figure 10 presents the complete configuration of the system described above.

To test the effectiveness of the model, the open circuit voltage measured directly from the panel and the open circuit voltage estimated from the model developed has been compared. Figure 11 shows the comparison between the measured open voltage (1:1) and the estimated one (1:2), where a good agreement between the two values can be observed.

Conclusions

In this paper a method has been presented for modeling and control of the PV system through the MPPT using a new method based on a numerical algorithm. The ability has also been shown to evaluate online the series and shunt resistor. Results illustrate the effectiveness of the method of identification and control adopted in this system. This study is a starting point for investigating photovoltaic systems and improving their efficiency.

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