In modern photovoltaic (PV) power stations, geographical constraints and optimization for energy yield increasingly lead to complex installation layouts. Solar panels within the same plant may have vastly different tilt angles ($\beta_s$) and azimuth angles ($\gamma_s$). This complexity poses a significant challenge for traditional fault detection schemes, which often assume uniform installation conditions or rely heavily on precise environmental sensors. This article presents a novel, data-driven fault detection methodology specifically designed for large-scale PV plants with such heterogeneous installation parameters. The core idea is to perform a comparative analysis of the operational states between different solar panels by extracting a characteristic value that is independent of their physical orientation.
The performance of a solar panel is fundamentally governed by its current-voltage (I-V) characteristics under given environmental conditions. While detailed physical models like the single-diode model exist, they often require intricate parameter identification and are sensitive to measurement noise. For the purpose of comparative fault analysis across many solar panels, a model that balances accuracy with analytical simplicity is preferable. We adopt a linearized form derived from the Sandia PV Array Performance Model (SPAM). The maximum power output $P_{mp}$ of a solar panel can be expressed as a function of the effective irradiance $E_e$ on its plane:
$$ P_{mp} = C_o \cdot V_{mpo} \cdot I_{mpo} \cdot E_e $$
where $C_o$ is a current conversion coefficient, and $V_{mpo}$, $I_{mpo}$ are reference voltage and current. The effective irradiance $E_e$ itself is a combination of beam (direct), diffuse, and ground-reflected components incident on the tilted surface of the solar panel:
$$ E_e = \frac{O}{G_o}[I_b + I_d + I_r] $$
Here, $O$ is a soiling factor, and $G_o$ is the reference irradiance. The beam irradiance $I_b$ is critically dependent on the angle of incidence $\theta$ of sunlight on the solar panel:
$$ I_b = DNI \cdot \max(\cos\theta, 0) $$
The cosine of the incidence angle is calculated from the solar panel’s orientation and the sun’s position:
$$ \cos\theta = \cos\beta_s \cos Z + \sin\beta_s \sin Z \cos(\gamma – \gamma_s) $$
where $Z$ is the solar zenith angle and $\gamma$ is the solar azimuth angle. The condition $\cos\theta > 0$ (i.e., $\theta < 90^\circ$) is essential for the solar panel to receive direct beam radiation. When $\cos\theta \le 0$, $I_b = 0$. By decomposing the trigonometric expressions for $I_b$, $I_d$, and $I_r$, the power output model can be reformulated into a linear expression with respect to the solar panel’s orientation, valid for periods when $\cos\theta > 0$:
$$ P_{mp} = C_1 \cos\beta_s + C_2 \sin\beta_s \cos\gamma_s + C_3 \sin\beta_s \sin\gamma_s + C_4 $$
In this formulation, the coefficients $C_1$, $C_2$, $C_3$, and $C_4$ are random variables that aggregate all factors independent of the solar panel’s installation angles:
$$ \mathbf{C} = [C_1, C_2, C_3, C_4]^T = \frac{C_o V_{mpo} I_{mpo} O}{G_o} [A_1, A_2, A_3, A_4]^T $$
The vector $[A_1, A_2, A_3, A_4]^T$ is determined solely by the solar geometry (DNI, DHI, GHI, $Z$, $\gamma$) and the ground albedo ($\rho$). Therefore, for a cluster of solar panels experiencing the same ambient weather conditions at a given time, the vector $\mathbf{C}$ should be identical for all healthy solar panels, regardless of their individual tilt and azimuth angles. This vector $\mathbf{C}$ is defined as the characteristic value for fault detection.
The practical calculation of $\mathbf{C}$ from operational data requires solving a linear system. For a set of four solar panels with known, non-coplanar orientations ($\beta_s^i, \gamma_s^i$), their power outputs at a timestamp where $\cos\theta^i > 0$ for all four satisfy:
$$
\begin{bmatrix}
P_{mp}^1 \\ P_{mp}^2 \\ P_{mp}^3 \\ P_{mp}^4
\end{bmatrix}
=
\begin{bmatrix}
\cos\beta_s^1 & \sin\beta_s^1\cos\gamma_s^1 & \sin\beta_s^1\sin\gamma_s^1 & 1 \\
\cos\beta_s^2 & \sin\beta_s^2\cos\gamma_s^2 & \sin\beta_s^2\sin\gamma_s^2 & 1 \\
\cos\beta_s^3 & \sin\beta_s^3\cos\gamma_s^3 & \sin\beta_s^3\sin\gamma_s^3 & 1 \\
\cos\beta_s^4 & \sin\beta_s^4\cos\gamma_s^4 & \sin\beta_s^4\sin\gamma_s^4 & 1
\end{bmatrix}
\begin{bmatrix}
C_1 \\ C_2 \\ C_3 \\ C_4
\end{bmatrix}
$$
Thus, $\mathbf{C}$ can be obtained by inverting the orientation matrix $\mathbf{M}$:
$$ \mathbf{C} = \mathbf{M}^{-1} \mathbf{P}_{mp} $$
It is crucial to filter the historical power data, using only timestamps where $\cos\theta > 0$ for the specific solar panel in question. Using data from periods when the panel is not receiving direct irradiance (e.g., early morning or late afternoon for certain orientations) violates the linear model assumption and introduces significant error into the calculated $\mathbf{C}$.
Due to the inherent randomness in weather, the power output $P_{mp}$ and consequently the characteristic vector $\mathbf{C}$ are random variables. To compare the operational state of solar panels, we compare the probability distributions of their $\mathbf{C}$ vectors. The Gaussian Mixture Model (GMM) is employed to characterize these multivariate distributions because of its flexibility in modeling complex, non-Gaussian shapes and its useful property of linear invariance. The parameters of the GMM for a solar panel’s $\mathbf{C}$ can be estimated from historical data using the Expectation-Maximization (EM) algorithm.
The proposed fault detection method leverages the collective behavior of presumably healthy solar panels to identify faulty ones. The procedure is as follows:
- Selection of Reference Panels: Choose $N$ (e.g., $N=6$) solar panels believed to be healthy, ensuring their orientation angles provide a diverse and invertible set in the matrix $\mathbf{M}$.
- Construction of Reference Distributions: From the $N$ reference panels, form all possible combinations of 4-panel groups ($C_N^4$ groups). For each group $j$ and for a target panel $i$ under test, filter timestamps where $\cos\theta > 0$ for all five involved solar panels. Fit a GMM to the joint power output of the reference group, and using the linear invariance property and the known matrix $\mathbf{M}_j$, derive the reference distribution for $\mathbf{C}$, denoted as $f_{\mathbf{C}}^{ref, j}$. Then, transform this back using panel $i$’s orientation to get a reference output power distribution for panel $i$, $f_{P,i}^{ref, j}$.
- Comparison and Fault Judgment: Calculate the Jensen-Shannon (JS) divergence between the actual output power distribution of panel $i$ ($f_{P,i}^{actual}$) and each of its $C_N^4$ reference distributions $f_{P,i}^{ref, j}$. The JS divergence measures the similarity between two probability distributions. If the majority of the calculated JS divergences for panel $i$ exceed a predefined threshold, the panel is flagged as faulty.
The effectiveness of this method was validated through detailed simulation studies. A plant with 15 solar panels having varied orientations was simulated. Three panels were artificially set to a faulty state (soiling factor $O=0.85$). Using $N=6$ reference panels, the method successfully identified the three faulty units. The results are summarized in the heatmap below, where the color intensity represents the JS divergence value. Faulty panels (1, 9, 13) show consistently high JS divergence across most reference groups, unlike healthy panels.
| Panel Under Test | Ref Group 1 | Ref Group 2 | Ref Group 3 | … | Ref Group 15 | Fault Status |
|---|---|---|---|---|---|---|
| 1 | 0.52 | 0.48 | 0.55 | … | 0.50 | Faulty |
| 2 | 0.02 | 0.01 | 0.03 | … | 0.02 | Healthy |
| 3 | 0.01 | 0.02 | 0.01 | … | 0.03 | Healthy |
| … | … | … | … | … | … | … |
| 9 | 0.45 | 0.47 | 0.49 | … | 0.44 | Faulty |
| 13 | 0.51 | 0.53 | 0.48 | … | 0.52 | Faulty |
The accuracy of the method depends on several factors. The selection of the JS divergence threshold balances precision and recall. Furthermore, the linear model simplification introduces some error. Analysis shows that the discrepancy between the ideal linear model and the real non-linear behavior of solar panels is minimized when the reference panels and the panel under test have similar azimuth angles. Therefore, for practical implementation, it is advisable to group solar panels by azimuth quadrant and perform the reference selection and comparison within these groups to enhance detection accuracy.
Key advantages of this method include its independence from on-site irradiance or temperature sensors, reducing hardware costs. It is inherently robust to the random variability of weather by relying on statistical comparisons. The method is specifically designed for the realistic scenario of complexly installed solar panels, filling a gap in existing fault detection literature. By analyzing the distribution of the characteristic vector $\mathbf{C}$, the method not only detects the presence of a fault but can also indicate its severity, aiding in maintenance prioritization for large farms of solar panels.
In conclusion, this article details a novel, comparative state analysis framework for fault detection in photovoltaic plants with non-uniform installation conditions. By deriving an installation-angle-invariant characteristic value from a linearized power model and employing statistical divergence measures on its distribution, the method effectively identifies abnormal solar panels without relying on extensive environmental sensing. The methodology provides a practical, scalable, and cost-effective solution for the operational maintenance of next-generation, topographically complex PV power stations.