Solar Irradiance Estimation Using Kalman Filter

Abstract

This work presents a methodology to estimate solar irradiance using Kalman filter for systems with unknown inputs, an approach more adequate to system characteristics than the standard formulation of this tool. A system with photovoltaic panel, dc–dc converter and load was modeled and simulated in order to analyze the proposed methodology in situations of clear, almost clear and cloudy sky days. The proposed estimator and an analytical method are compared with respect to the ability to compute the irradiance and tested against uncertainties in modeling parameters and noise in the voltage and current measurements of the system. The results show that, through a single sensor, the developed methodology allows to estimate and filter not only solar irradiance, but also output current of photovoltaic system and output voltage of converter. This brings benefits in reducing costs with sensors, allows real-time measurements and avoids propagating noisy measures in the management of a solar system.

Appendix: Choice of V, W and \( P^x_{0|0} \)

Two very important parameters of the KFUI are the values of V, W and $P^x_{0|0}$. They are associated with the noises present in the state x and the output y.

To determine appropriate values for the parameters V, W and $P^x_{0|0}$, some simulations were carried out replacing several values in these matrices using genetic algorithm (GA). The following minimization problem was applied

$$\begin{aligned} \mathop {\min }\limits _{{\hat{ {\tilde{x}}}_{k|k}}} J = \sum \limits _{k = 0}^N {\left\| {{{{\tilde{x}}}}_k - {\hat{ {\tilde{x}}}_{k|k}}} \right\| } \end{aligned}$$

(32)

which correspond to the basic idea of a Kalman Filter: minimize the errors of ${\hat{x}}_k$ with respect to a real value ${x}_k$. For this, it was used MATLAB’s genetic algorithm with standard parameters.

Additionally, it is necessary to observe the behavior of unknown input estimation as

$$\begin{aligned} J^z = \sum \limits _{k = 0}^N {\left\| {{{{\tilde{i}}}_{p{h_k}}} - {{\hat{ {\tilde{i}}}_{ph}}_{k|k + 1}}} \right\| }. \end{aligned}$$

(33)

The sum (33) is not included in problem (32) because the tests were performed for both KFUI and SKF. The last one does not estimate the unknown input.

Figure 12 shows the convergences of J and $J^z$ for the irradiance of Day 1. Each individual index is a set of V, W and $P^x_{0|0}$ chosen by GA and evaluated for all values of the real irradiance in Fig. 3. It can be noted that KFUI estimations of ${\hat{ {\tilde{x}}}_{k|k}}$ and ${{\hat{ {\tilde{i}}}_{ph}}_{k|k + 1}}$ and, consequently, irradiance estimation are not affected by variations in the parameters V, W or $P^x_{0|0}$. In Table 6, it can be observed that KFUI presents better estimate than SKF across all generations tested with the GA.

Thus, the parameters V, W and $P^x_{0|0}$ should be chosen more carefully if SKF is used. On the other hand, for this application, any value that satisfies condition $C_5$ can be used in KFUI in order to result in optimal estimates for ${\hat{ {\tilde{x}}}_{k|k}}$ and ${{\hat{ {\tilde{i}}}_{ph}}_{k|k + 1}}$. For simplicity, it was used a KFUI with $V = W = P^x_{0|0} = I$.

Fig. 12

Convergence of J and $J^z$. (Color figue online)

Table 6 Minimum J obtained by the genetic algorithm