*3.2. Model Validation*

The analytical model used to predict the clipped power was chosen, since it presents a low modeling error below clipping, presents a low model error compared data sheet stated STC, presents a low computational cost, does not require an optimization stage (neither parameter identification nor training) and is conceptually simple. The model prediction was clipped at the DC power rating of the converter to match the maximum DC power level (clipping level), and then the error between the clipped prediction and the measured DC power was calculated. The technical details applied to model the PV plant are presented in Table 1. PV modules correspond to the Jinko model JKM260-PP.

Figure 3 shows the irradiance, module temperature, DC power model predictions (clipped) and DC power measurements on four different days (18 March 2017, 3 June 2017, 10 August 2017 and 20 September 2017). The top plots in Figure 3a–d show the daily irradiance and temperature measurements, while the lower plots show the DC power (clipped) and measured DC power. The analytical model display an adequate tracking of the measured DC power; to present a complete analysis, some error metrics and other characteristics of the models were considered.


**Table 1.** PV plant and PV modules parameters.

The error metrics applied to validate the PV plant model were Normalized Root Mean Squared Error (*NRMSE*), Normalized Mean Absolute Error (*NMAE*), Pearson linear correlation factor (*Pearson*) and Normalized Root Mean Squared Error Fitness (*NRMSEF*). For the first two error metrics, *NRMSE* and *NMAE*, the optimal value is 0%, while, in the second pair of error metrics, *Pearson* and *NRMSEF*, the optimum is 100%. Equations (2)–(5) correspond to the mathematical description of the metrics applied to analyze the error between the DC power measurement (*Xi*) and the clipped DC output power predicted by each model (*X* ˜ *i*). Variables *μ<sup>X</sup>*, *μX*˜ , *σX* and *<sup>σ</sup>X*˜ in Equation (4) correspond, respectively, to the mean of *Xi* and *X* ˜ *i*, and the standard deviation of *Xi* and *X* ˜ *i*. *N* corresponds to the number of samples. Normalized metrics were measured respect to the clipping power level (*P*clip).

$$NRMSE = \frac{\sqrt{\frac{1}{N} \cdot \sum\_{i=1}^{N} (X\_i - \mathcal{X}\_i)^2}}{P\_{\text{clip}}} \cdot 100\tag{2}$$

$$NMAE = \frac{\frac{1}{N} \cdot \sum\_{i=1}^{N} |X\_i - \bar{X}\_i|}{P\_{\text{clip}}} \cdot 100 \tag{3}$$

$$Pearson = \frac{\sum\_{i=1}^{N} \left[ \left( \frac{X\_i - \mu\_{\bar{X}}}{\sigma\_{\bar{X}}} \right) \cdot \left( \frac{\overline{X}\_i - \mu\_{\bar{X}}}{\sigma\_{\bar{X}}} \right) \right]}{N - 1} \cdot 100 \tag{4}$$

$$NRMSEF = \left(1 - \frac{||X - \tilde{X}||}{||X - \mu\_X||}\right) \cdot 100\tag{5}$$

**Figure 3.** Daily analysis of measured DC power versus models, top plot module temperature and irradiance daily measurements, bottom plot clipping level (*P*clip), measured DC power (*P*pv) and analytical model estimated power (*P*mpp): (**a**) 18 March 2017; (**b**) 3 June 2017; (**c**) 10 August 2017; and (**d**) 20 September 2017.

Table 2 summarizes the error metrics applied to the model. The analytical model presents a low calculation time, a simple approach and small errors metrics. Therefore, this method was chosen to predict PV clipped power. It is also useful for predicting the clipped power in real time.

Aging and soiling effects were not considered in the previously described analytical model, nonetheless the model enables a straight forward update by fitting the STC efficiency term (*η*) in Equation (1).


