Keywords

1 Introduction

The evaluation of energy conversion efficiency and hence the operation of inverters for photovoltaic (PV) systems connected to the grid are based on electrical parameters (voltage input and output, DC and AC power circuits, etc.) and thermal equipment. Several mathematical models of energy conversion efficiency inverters for such systems have been developed over time [1–4]. For sizing PV systems, designers need a reliable mathematical model and simple application that performs computer simulations and consequently generates estimates of energy production of PV systems with a lower margin of uncertainty in order to provide better financial return and investment in the project.

Here, we studied two energy conversion efficiency inverter mathematical models for grid-connected PV systems proposed by [1, 2]. They were chosen because they are simple to operate and because each one is based on data from the manufacturer’s inverter datasheets and can also use operational data (from the field) of inverters that are coupled to small PV systems in real operation conditions. Each one was implement in a mathematical computational tool and validated, under different climatic conditions, by operating data from the Xantrex GT2.8-NA-240/280 UL-05 inverter and the Enphase® M215-60-2LL-S22-IG microinverter and data from the manufacturer’s datasheets [5, 6]. In the following sections we present the cited models, their development, validation and results analysis using statistical indicators.

2 Methodology

The models proposed by [1, 2] were chosen and implemented using MATLAB® software R2013a. Each one was validated using datasheets from the inverter manufacturers [5, 6], and field data from two small PV systems operating under different environmental conditions. The Xantrex model (rated at 2.8 kW) is coupled to a grid-connected PV system with 3.36 kWp installed power, with m-Si modules Kyocera KC 120. The Enphase model with 215 W nominal power is coupled to a 245 Wp PV module (m-Si Aleo S19G245) in a bus interconnected to a 220 V AC grid. Both systems were located in the test area of the Group of Studies and Development of Alternatives Energy (GEDAE) at the Federal University of Pará (UFPA) in Belém city in the north of Brazil. A detailed description of these small PV systems, their instrumentation, and monitoring and data acquisition are described in [7, 8]. The data generated by the inverters were first treated and then applied to the models. A comparison of datasheets and field data was carried out using two figures of merit commonly used in statistical literature [9, 10]—MBEFootnote 1 and RMSEFootnote 2 errors and a parallel of the models was performed. Equations (9.1) and (9.2) show the calculations of average errors where ‘K’ is a general variable and ‘n’ is the number of datasets used.

$$RMSE\,=\,{{\left(\frac{1}{n}*\underset{i\,=\,1}{\overset{n}{\mathop \sum}}\,{{\left( {{k}_{\text{calculated}}}-{{k}_{\text{measured}}}\right)}^{2}} \right)}^{\frac{1}{2}}}$$
(9.1)
$$ MBE\,=\,\frac{1}{n}*\underset{i\,=\,1}{\overset{n}{\mathop \sum }}\,({{k}_{\text{calculated}}}-{{k}_{\text{measured}}}) $$
(9.2)

3 Energy Conversion Efficiency Inverter Mathematical Models for Grid-Connected PV Systems

3.1 Jantsch Model

The Jantsch model, developed by [1] proposes the calculation of the losses of the inverter depending on their power output, not taking into account other possible variations that may occur. The model considers the losses as a second-degree function and seeks to connect the terms of each order to the actual losses in the inverters. Equation. (9.3) shows total losses.

$$ {{P}_{\text{losses}}}\,=\,{{k}_{0}}\,+\,{{k}_{1}}*{{P}_{\text{output}}}\,+\,{{k}_{2}}*P_{\text{output}}^{2} $$
(9.3)

The input power and efficiency are defined as in Eqs. (9.4) and (9.5).

$$ {{P}_{\text{input}}}\,=\,{{P}_{\text{output}}}\,+\,{{P}_{\text{losses}}}\,=\,{{P}_{\text{output}}}\,+\,{{k}_{0}}\,+\,{{k}_{1}}\text{*}{{P}_{\text{output}}}\,+\,{{k}_{2}}\text{*}P_{\text{output}}^{2} $$
(9.4)
$$ \eta \,=\,\frac{{{P}_{\text{output}}}}{{{P}_{\text{input}}}}\,=\,\frac{{{P}_{\text{output}}}}{{{P}_{\text{output}}}\,+\,{{k}_{0}}\,+\,{{k}_{1}}\text{*}{{P}_{\text{output}}}\,+\,{{k}_{2}}\text{*}P_{\text{output}}^{2}}$$
(9.5)

Having knowledge of the parameters k0, k1 and k2, can set losses throughout the range of operation of the inverter, and the energy conversion efficiency. These characteristic parameters are defined by Eqs. (9.6)–(9.8).

$$ {{k}_{0}}\,=\,\frac{1}{9}*\frac{1}{{{\eta }_{\text{inv}}}1}-\frac{1}{4}*\frac{1}{{{\eta }_{\text{inv}}}0.5}\,+\,\frac{5}{36}*\frac{1}{{{\eta }_{\text{inv}}}0.1} $$
(9.6)
$$ {{k}_{1}}\,=\,\frac{-4}{93}*\frac{1}{{{\eta }_{\text{inv}}}1}\,+\,\frac{33}{12}*\frac{1}{{{\eta }_{\text{inv}}}0.5}-\frac{5}{12}*\frac{1}{{{\eta }_{\text{inv}}}0.1}-1 $$
(9.7)
$$ {{k}_{2}}\,=\,\frac{20}{9}*\frac{1}{{{\eta }_{\text{inv}}}1}-\frac{5}{2}*\frac{1}{{{\eta }_{\text{inv}}}0.5}\,+\,\frac{5}{18}*\frac{1}{{{\eta }_{\text{inv}}}0.1} $$
(9.8)

where η inv1, η inv0.5 and η inv0.1 refer to the instant efficiencies at 100, 50 and 10 % of rated output power, respectively. K0 is the loss of self-consumption, without depending on power output. K1 refers to the linear losses in power output (e.g. voltage drops in semiconductor devices, diodes and switches, IGBT). Finally, K2 is the quadratic loss with output power, for example, the ohmic losses.

3.2 Sandia Model

The mathematical model of the energy conversion efficiency for PV inverters proposed by [2], also called the Sandia/King model, is an empirical model which has good accuracy in results and good versatility, having been successfully applied to large and small PV systems. The model consists of an expression that relates AC power inverter output as a function of input power and DC bus voltage; these being the two variables needed to estimate the behavior of the conversion efficiency for an inverter. The Sandia model is described by the power AC expression, Eq. (9.9), which is complemented by Eqs. (9.10)–(9.12).

$$ {{P}_{\text{ac}}}\,=\,\left\{\left(\frac{{{P}_{\text{aco}}}}{A-B} \right)-C*(A-B) \right\}*({{P}_{\text{dc}}}-B )\,+\,C*{{({{P}_{\text{dc}}}-B)}^{2}}$$
(9.9)
$$ A\,=\,{{P}_{\text{dco}}}*\left\{1\,+\,{{C}_{1}}*({{V}_{\text{dc}}}-{{V}_{\text{dco}}}) \right\} $$
(9.10)
$$ B\,=\,{{P}_{\text{so}}}*\left\{1\,+\,{{C}_{2}}*({{V}_{\text{dc}}}-{{V}_{\text{dco}}}) \right\} $$
(9.11)
$$ C\,=\,{{C}_{0}}*\left\{1\,+\,{{C}_{3}}*({{V}_{\text{dc}}}-{{V}_{\text{dco}}}) \right\} $$
(9.12)

where P ac is the power output of the inverter, P dc is the input power of the inverter, P aco is rated output; P dco is the input power to the inverter-rated power supply output, P so is own consumption or minimum power for start-up of the inverter, V dco is nominal voltage, and V dc is input voltage.

4 Validation of Jantsch and Sandia Models

4.1 Analysis of Results for the Xantrex GT2.8-NA-240/208 UL-05 Inverter

Table 9.1 shows the performance of the Jantsch and Sandia models by comparing the results of MBE and RMSE using data from the manufacturer’s datasheet and field data measurements.

Table 9.1 Results obtained for the Jantsch and Sandia models from field data and datasheet data
Full size table

According to Table 9.1, for output power and also efficiency (through RMSE% error), the Jantsch model showed better results compared to the Sandia model, when used datasheet data. However, the (Δdatasheet) difference between the numerical results for the RMSE% error is sparingly small between models, for example, 0.05 % (for output power) and 0.07 % (for efficiency). Compared to field data, the Sandia model showed better results (for the output power) and the Δfield difference is equal to 0.02 %; however, for efficiency, the Jantsch model was slightly better but the Δfield difference is equal to 0.03 %, which is still very low. Therefore, both models showed good quality outcomes, mean errors, and are very similar. Therefore, both models represented the inverter manufacturer Xantrex model GT 2.8 kW well.

Regarding the indicator MBE% error, for output power in Table 9.1, the Sandia model tended to underestimate (− 0.17 %) but the Jantsch model tended to overestimate (0.11 %) the datasheet data (∆datasheet) showing a value equal to 0.28 %, i.e., a small variation in the end result. For the field data, both models underestimated, but datasheet data (Δdatasheet) showed a very low result (0.03 %) between the models. Regarding MBE% for the efficiency using datasheet data, the Sandia model had a tendency to underestimate (− 0.70 %) as opposed to the Jantsch model which overestimated their perspective (0.15 %); the (Δdatasheet) between models was 0.85 %. Regarding efficiency using field data, again both models tended to underestimate but (Δdatasheet) between models was very low; therefore, the results achieved were almost similar between the two models in this regard.

Finally, Table 9.1 shows good results (RMSE% and MBE% errors) for efficiency and output power, good equivalence and small variations in values (as mentioned above), showing the high quality of the Sandia and Jantsch models for representing a model of energy conversion efficiency to the Xantrex GT 2.8 kW inverter. Figure 9.1 shows the efficiency curves for the Jantsch model for the manufacturer’s datasheet (black curve), for field measurements (red curve) and the curve of the inverter under real operation conditions (green) and Fig. 9.2 shows the efficiency curves for the Sandia models for the Xantrex GT 2.8 kW inverter. The red curve is from the datasheet, the black curve is from field measurements and the green curve is the actual curve of the inverter in real operation conditions.

Fig. 9.1
figure 1

Efficiency curves of the Jantsch model for the Xantrex GT 2.8 kW inverter using datasheet data and field data

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Fig. 9.2
figure 2

Efficiency curves of the Sandia model for the Xantrex GT 2.8 kW inverter using datasheet data and field data

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4.2 Analysis of Results for the Enphase Energy 215-60-2LL-IG Microinverter

Table 9.2 shows the performance of the Jantsch and Sandia models compared with the results of MBE% and RMSE% errors using data from the manufacturer’s datasheet and from field measurements.

Table 9.2 Results for the Jantsch and Sandia models from field measurements and datasheet data
Full size table

According to Table 9.2, for the output power and also efficiency (through RMSE% error), the Jantsch model showed worse results in comparison with the Sandia model, when using datasheet data. However, the (Δdatasheet) difference between the numerical results for the RMSE% error is very small between models, for example, 3 % (for output power). For efficiency, when using datasheet data, the RMSE% error was quite high for both models with values > 10 %. Compared with field data, the Sandia model showed worse results (for the output power) and the difference (Δfield)with a value equal to 0.25 %; however, for efficiency, the Jantsch model was slightly better with a (Δfield) difference equal to 0.04 %, which was very low. Therefore, for both models, the RMSE% error output power showed good quality for datasheet data and field data with small variations; however, for efficiency, the RMSE% error for the Sandia and Jantsch models using datasheet data was high, showing poor quality for both models. On the other hand, for efficiency, the RMSE% error was very similar (4.30 %) and therefore both models represented the microinverter Enphase Energy 215 W for field data well.

Regarding the indicator MBE% error to the power output in Table 9.2, the Sandia model tended to underestimate (− 0.17 %) but the Jantsch model tended to overestimate (3.98 %) using datasheet data with Δdatasheet equal to 3.81 %. The same happened for field data for both models. The Sandia model tended to underestimate (− 0.06 %) while the Jantsch model tended to overestimate (0.27 %) with Δfield equal to 0.33 %. Regarding MBE% error for efficiency with datasheet data, both models had a tendency to overestimate (0.35 % for Sandia and 15.20 % for Jantsch) with Δdatasheet equal to 14.85 %. Regarding efficiency with field data, again both models tended to overestimate but with small values (0.57 % for Sandia and 1.37 % for Jantsch) with Δfield equal to 0.8 %, showing that the difference between the models is very low; therefore, the result achieved were almost similar between the two models in this regard.

Finally, Table 9.2 shows good results; the RMSE% and MBE% errors for efficiency and output power showed good equivalence and small variations in their values (as mentioned above), showing the high quality of the Sandia and Jantsch models for representing a model of energy conversion efficiency to the Enphase Energy 215 W inverter for field measurements. Figure 9.3 shows the efficiency curves for the Jantsch model for the manufacturer’s datasheet (green curve), for field measurements (red curve) and the curve of the inverter for real operation conditions (dots blue). Fig. 9.4 shows the efficiency curves for the Sandia model for the Enphase Energy 215 W inverter. The red curve is from datasheet data, the black curve is from field measurements and the green curve (dots) is the actual curve of the inverter in real operation conditions.

Fig. 9.3
figure 3

Efficiency curves of the Jantsch model for the Enphase Energy model M215 microconverter using datasheet data and field data

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Fig. 9.4
figure 4

Efficiency curves of the Sandia model for the Enphase Energy model M215 microinverter using datasheet data and field data

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5 Conclusion

Regarding the validation of the Jantsch and Sandia models using MBE% and RMSE% errors, both showed good results on average errors, good equivalence between models and small variations between their values (as shown in Sect. 4.1). This confirmed the high quality of the Sandia and Jantsch models for representing the energy conversion efficiency for the Xantrex model GT 2.8 inverter.

For the Enphase Energy model M215-60-2LL-S22-IG microinverter, the validation of the Jantsch and Sandia models was good for power output and efficiency for data from field measurements with slight variations as shown in Table 9.2. This demonstrates the quality of both models for representing the energy conversion efficiency for the inverter. On the other hand, for datasheet data, the Sandia and Jantsch models presented very high values for RMSE% errors for efficiency. The explanation for this refers to datasheet data from the manufacturer that provides values for the efficiency curve beginning at 10 % of rated output power disregarding the values below this loading; therefore, increasing the average error and the quality of the models. Note, for MBE% errors, the Sandia model results were better than the Jantsch model. Finally, the Sandia and Jantsch mathematical models are a good and reliable for representing the energy conversion efficiency for an inverter for computer simulations and consequently to generate estimates of energy production of PV systems under different climatic conditions.

6 Special Thanks

This work has been supported by the Brazilian agency CAPES, CNPq, FIP PUC Minas, GEDAE/UFPA, PPGEE/UFMG and developed under the Strategic Program Nº13 Call R & D Technological Electricity Sector regulated by ANEEL.