Application of Quantum Neural Network for Solar Irradiance Forecasting: A Case Study Using the Folsom Dataset, California

Abstract

Merging machine learning with the power of quantum computing holds great potential for data-driven decision making and the development of powerful models for complex datasets. This area offers the potential for improving the accuracy of the real-time prediction of renewable energy production, such as solar irradiance forecasting. However, the literature on this topic is sparse. Addressing this knowledge gap, this study aims to develop and evaluate a quantum neural network model for solar irradiance prediction up to 3 h in advance. The proposed model was compared with Support Vector Regression, Group Method of Data Handling, and Extreme Gradient Boost classical models. The proposed framework could provide competitive results compared to its competitors, considering forecasting intervals of 5 to 120 min ahead, where it was the fourth best-performing paradigm. For 3 h ahead predictions, the proposed model achieved the second-best results compared with the other approaches, reaching a root mean squared error of 77.55 W/m² and coefficient of determination of 80.92% for global horizontal irradiance forecasting. The results for longer forecasting horizons suggest that the quantum model may process spatiotemporal information from the input dataset in a manner not attainable by the current classical approaches, thus improving forecasting capacity in longer predictive windows.

1. Introduction

Recent developments in quantum computing have enabled the application of machine learning paradigms within a quantum framework [1,2]. This innovative field leverages quantum physics principles, like entanglement, superposition, and interference, to execute computational loads [3,4]. With these properties, it is theorized that quantum computers can leverage computational time in critical areas, such as financial transaction decisions, new materials and drug discovery, cryptography, and so on [5]. Unlike classical computers, quantum computers process information using quantum bits (qubits). The qubit’s state, represented by its orientation in a high-dimensional complex Hilbert ℂ^⊗n space (with n being the number of qubits), is managed by quantum gates. These unitary operators perform reversible operations on qubits, altering their orientation within the Hilbert space. These operations can implement superposition, entanglement, and interference properties to the quantum circuit, composed of qubits, allowing computers to generate unique and more complex patterns than those produced by classical computational systems. In this way, it is expected that quantum machine learning (QML) models will surpass the predictive performance of traditional ML approaches, reaching what is known as the quantum advantage, where a complex problem can be efficiently tackled with a quantum framework [1,6].

Despite being a newly developed field, it is possible to find many QML models, with their applications being broad and promising [7]. One of the possible applications lies within the renewable energy area, an important player for containing the adverse effects of climate change caused by the emission of greenhouse gases. The majority of greenhouse gases come from electricity production operations [8,9]. Their collateral effects have been linked to damage to ecological areas [10,11] and the occurrence of extreme weather events [12,13], among other negative consequences [14,15]. The adverse effects of pollutant gas emissions also negatively impact the global economy, causing the loss of billions of USD worldwide [16,17,18,19].

Within this context, renewable energies serve as a significant player in mitigating the harmful effects of climate change. Recently, there has been a global movement towards developing greener and renewable energy sources. One can highlight the importance of solar power, which has led to global investments in renewable energies [20,21].

While renewable energies offer significant benefits in cleaner energy production, their inherently stochastic and intermittent nature presents challenges. Therefore, developing precise and reliable tools for predicting energy output from these sources is crucial. In this field, there are three major approaches for solar irradiance forecasting: numerical weather prediction (NWP), statistical, and time-series models [22,23]. The NWP uses a physical-based approach involving weather attributes to determine future values of solar irradiance [24]. Statistical models rely on temporal correlation to forecast solar irradiance, being able to handle non-stationary data [25,26]. Finally, it is also possible to determine future values of solar irradiance using time-series data, where machine learning models can be implemented [22,23].

Machine learning (ML) has emerged as a valuable tool in this regard, demonstrating its ability to identify complex relationships within time-series data and enhancing the accuracy of renewable energy forecasting. Numerous works have employed ML to assess the behavior of key environmental indicators [27,28,29,30], including applications in the renewable energy area [31,32,33,34,35]. However, quantum machine learning (QML), remains in its early stages, with an absence of studies exploring its application in time-series scenarios and, consequently, in the modeling of renewable energy sources [36,37].

To address the knowledge gap in QML applications within renewable energy, this study proposes the development of a simulated quantum neural network (QNN) framework for short-term forecasting [22] of solar irradiance up to 3 h in advance. The framework utilizes the benchmarking dataset from Folsom, as introduced by Pedro et al. [38]. The main objectives of this work are as follows:

• To address the information gap in the scientific literature regarding solar irradiance estimation using QML predictive models.
• To assess the feasibility of a QML model designed for forecasting solar irradiance up to 3 h in advance.
• To evaluate different configurations for the QNN framework, investigating several parameters composing its structure and their impact on the model’s output.
• To investigate different optimization strategies and their impact on the model’s output.
• To evaluate the QNN performance compared to classical ML counterparts.

2. Materials and Methods

2.1. Solar Irradiance

Solar irradiance touching the Earth’s surface can be categorized into three distinct types. The first type, Direct Normal Irradiance (DNI), includes all solar irradiance reaching the Earth’s surface directly without dispersion. When dispersion occurs, it is referred to as diffuse irradiance (DI). The third type, Global Horizontal Irradiance (GHI), encompasses the total solar energy reaching the Earth’s surface from all directions, i.e., including DNI and DI [39].

Accurate forecasting of GHI and DNI is essential to the cost-efficient implementation of photovoltaic projects and the proper development of public policies or plans related to solar energy [40,41,42].

2.2. The Folsom Dataset

The Folsom dataset spans 2014 to 2016, containing meticulously checked minute-by-minute measurements of GHI and DNI from Folsom, California, USA. This region holds elevated interest in solar irradiance, given its importance in the US economy and its status as a technological HUB housing millions of people [32,38]. The Folsom dataset was conceived to be used as a benchmark dataset to promote the development of solar forecasting models [38], containing ground-based sky images, satellite data, and forecasts from numerical weather prediction models. The present project’s dataset was limited to GHI and DNI measurements and sky images.

As mentioned previously, solar irradiance is stochastic and intermittent. Additionally, its forecasting depends on many factors, such as the celestial body motion, weather, geographic location, cloud coverage, and albedo, to cite a few [31,43]. A way to mitigate these effects, along with daily and seasonal variances of solar irradiance [44], is using the normalization by the modeling of the clear-sky index, $k_{t,cs}$, a dimensionless number between 0 and 1, defined as below:

$$k_{t,cs}={\displaystyle \frac{I}{I_{cs}}}$$

(1)

In Equation (1), the term $I$ usually refers to GHI or DNI. The denominator $I_{cs}$ is a modeled value of clear-sky irradiance, representing the maximum incident irradiance reaching the Earth’s surface. For this dataset, the clear sky model developed by Ineichen and Perez [45] was implemented, taking into account the characteristics of the study location.

In the original work by Pedro and colleagues [38], from the calculated $k_{t,cs}$ clear-sky index value, three feature vectors were engineered, namely Backward average (B_i), Lagged average values (L_i), and clear-sky Variability (V_i). It is important to highlight that these features are calculated within the time window preceding the desired forecasting interval. Additionally, the sky images were collected from all sky-dome pixels, using 8-bit color images from a camera pointed toward the sky. The pixels were converted to a floating-point vector representing three different color channels: Red (R), Green (G), and Blue (B). Furthermore, the vectors relating the red-to-blue and the normalized red-to-blue ratios were also calculated. Finally, for each of the previous features, the mean value, standard deviation, and entropy were determined, resulting in fifteen features related to the sky images [38]. The total number of features is 18, including the engineered feature vectors.

Table 1. The attributes and their symbols as used in the present study.

Attribute Name	Attribute Symbol
Backward Average	Bi
Lagged Average	Li
Clear-sky Variability	Vi
Red Channel	R
Green Channel	G
Blue Channel	B
Red-to-Blue Ratio	RB
Normalized Red-to-Blue Ratio	NRB
Average	AVG
Standard Deviation	STD
Entropy	ENT

compiles the 18 attributes used by Pedro et al. [38].

At this point, it is essential to make clear that the solar irradiance forecasting in this project was made upon the prediction of the clear-sky index value for each irradiance, i.e., $k_{t,cs,GHI}$ and $k_{t,cs,DNI}$, which are later used to determine the values for GHI and DNI, respectively. Such an approach was used in the original work of Pedro et al. [38] and other solar irradiance forecasting works [8,31,32]. This approach is preferred over direct GHI and DNI forecasting, as the clear-sky index is a normalized and non-seasoned solar irradiance measurement, which may improve the determination of future values [46]. Statistical analysis and data visualization for the solar irradiance under forecast can be found in the Supplementary Materials file.

2.3. Classical Machine Learning Models

Traditional ML methodologies process data through their structure. To this end, different strategies have been developed in the previous decades, each with specific characteristics. For the present study, three classical methodologies were implemented.

The first one was the Support Vector Regression (SVR). This ML model is a variation of the Support Vector Machine (SVM) but designed specifically for regression scenarios. Similarly to the SVM, the SVR uses kernel functions to extrapolate the dataset to higher dimensions, allowing it to learn complex non-linear interactions within the dataset [47,48]. Furthermore, the SVR also uses convex optimization to improve its computational performance, mitigating the error during the training phase [49,50,51].

Another approach under consideration in this work was the Group Model of Data Handling (GMDH). Its configuration resembles a multilayer perceptron (MLP), and it has a self-organizing structure that allows it to select the best configuration for a given dataset [52,53,54]. The output from the GMDH is a polynomial approximation of the target variable, resulting from a combination of the dependent and independent attributes [30,55,56].

Another classical ML model implemented in this study was the Extreme Gradient Boost (XGBoost), a tree-based structure [57]. In essence, the XGBoost consists of sampling the outputs of smaller trees composing it, ultimately combining them into a larger and more complex structure. This unique feature categorizes it as an ensemble method, given that its structure aggregates multiple models. The ensemble strategy employed by the XGBoost model enhances its robustness and reduces its sensitivity to data variance. Additionally, it serves as a regularization technique, effectively mitigating the risk of overfitting [30,48].

2.4. Qubits and Quantum Gates

Quantum bits are the paramount basis of quantum information processing [58]. They serve much like classical bits, being able to store information depending on their state [59]. However, qubits have a unique property that sets them apart from their classical counterpart: they can assume different states simultaneously. Mathematically, qubits can be understood as vectors where $\left|0\right.⟩={\left[1 0\right]}^T$ and $\left|1\right.⟩={\left[0 1\right]}^T$ for the excited and non-excited states, respectively, considering only one qubit. For a general n-qubit system, the resulting quantum state can be described by Equation (2) as follows:

$$\left|\mathsf{\Psi }\right.⟩=\sum _{\mathrm{j}=0}^{\mathrm{n}-1}{\mathsf{\alpha }}_j{\left|j\right.⟩}_n$$

(2)

In Equation (2), the quantum state $\left|\mathsf{\Psi }\right.⟩$ results from a linear combination of ${\left|j\right.⟩}_n$ quantum states for a n-qubit system [60]. Additionally, the term ${\alpha }_j$, where ${\alpha }_j \in \mathbb{C}$, corresponds to the amplitude of a given quantum state and has a property such that $\left|{\alpha }_j\right|=1$. The absolute value amplitude can be understood as the probability for the qubit to assume a specific quantum state. Note that the quantum system allows superposition and entanglement of the qubit states.

In quantum computing, the information carried by the qubits is processed using quantum gates. Quantum gates are fundamental operators that apply unitary transformations in the qubit, describing its system’s evolution [3]. These operators can be mathematically represented by unitary matrices, defined as follows:

$${\mathbf{U}}^{-1}={\mathbf{U}}^{†}$$

(3)

Equation (3) describes that a unitary quantum gate operator has its inverse equal to its complex conjugate. This property is paramount to quantum information processing, as it reverses the operation and keeps the vector distance unchanged when applied to multiple qubits [3,60].

Applying one or more quantum gates in the qubits composing a quantum system is called a quantum circuit. The quantum circuits composing the QNN implemented in this work are discussed in the following subsection.

2.5. Quantum Machine Learning

The integration between classical ML and the newer quantum computer framework can be achieved in diverse ways, as initially proposed by Aimeur et al. [61], and this can be visualized in Figure 1 [3].

From Figure 1, we observe that the four combinations are based on the origin of the data (classical “C” or quantum “Q”) and its processing methodology (also classical or quantum). The top-left represents the classical machine learning approach, where a classical computer evaluates classical information. On the bottom-right of the chart, the quantum data is being processed by a quantum machine. In the context of this project, the “CQ” approach was adopted since the Folsom dataset, which was obtained through classical means, was processed using a quantum computer.

It is important to note that a simulated quantum computer was employed for the quantum processing component of this project. The use of simulated quantum circuits is a common practice given the current state of quantum computers, categorized as Noisy Intermediate-Scale Quantum (NISQ) machines. Although quantum computers have advanced to the point where a significantly large number of qubits is feasible, the real quantum hardware in NISQ machines still cannot provide sufficiently long coherence times (i.e., the qubit state is not maintained long enough) and is not entirely fault tolerant. Consequently, these devices are currently unable to deliver the theorized quantum advantage [62,63,64].

Additionally, the availability of real quantum hardware is minimal, restricted to a few enterprises that invest in such technology. These limitations hamper a deeper investigation of QML methodologies, as using the real NISQ devices requires monetary investments while offering limited computational power in its current state and may require an extended queue time in order to use them [3,62,63,65,66]. Therefore, to overcome these adversities, many researchers opt to simulate their quantum circuits, an approach adopted by the present study.

2.5.1. QNN Structure

As previously mentioned, quantum computers utilize qubits to process information, a fundamental distinction between quantum and classical ML frameworks. This results in an entirely new structure for the QML approach. In the context of QML, the QNN model consists of three basic components: a Feature Map, an Ansatz, and Measurement, which are constituents of quantum circuits where data are processed. A generic QNN architecture is depicted in Figure 2 [67].

In Figure 2, it is possible to visualize the structures composing the QNN model (encapsulated by the grey rectangle). The data coming from the n-qubits ($q_0$ to $q_{n-1}$) enter the QNN structure from left to right, passing through the feature map, the ansatz, and finally, the measurement stage, where the output is mapped to a classical bit configuration.

2.5.2. Feature Map

The feature map is a fundamental component of the QNN circuit. Since the Folsom dataset is of classical origin, it must be encoded in a way that allows it to be processed by a quantum framework. The feature map serves this purpose by mapping the classical inputs into a quantum space defined within a highly complex Hilbert space, enabling a quantum computer to process the data [68,69,70]. This step is arguably the most critical part of QML design, as it transforms the data structure deeply and non-linearly, achieving alterations that are not feasible through classical methods.

The current literature registers two methods for transforming classical data into a quantum format. The first one is called implicit, and it is related to directly applying a quantum circuit to implement the mapping strategy, defined as follows [69]:

$$\varphi :\mathbf{x}\to \left|\varphi \left(\mathbf{x}\right)\right.⟩=U_{\varphi }\left(\mathbf{x}\right)\left|000\dots 0\right.⟩$$

(4)

In Equation (4), a generic feature map $\varphi$ takes the classical data vector $\mathbf{x}$, where $\mathbf{x}\in {\mathbb{R}}^{\mathrm{m}\times \mathrm{p}}$, and encodes it into a quantum framework by using a parameterized quantum circuit $U_{\varphi }\left(\mathbf{x}\right)$ containing n-qubits with initial states $\left|0\right.⟩$. The second mapping strategy, called explicit, is related to using variational quantum circuits to determine the best mapping configuration. This study employs the implicit approach to facilitate the validation of proper encoding.

Considering the implicit feature mapping strategy, several distinct encoding strategies can be applied. In the context of this project, two feature mapping approaches were investigated. The first one is called angle encoding. This approach is straightforward, applying rotational quantum gates to all n-qubits in the quantum circuit. However, it still offers a powerful way to transform classical data into a quantum paradigm [3,59]. Each rotational gate is parameterized, taking as input the numerical input vector $\mathbf{x}$ from the classical dataset. Later, the input information is passed as the rotation angles for the quantum gates [59]. Therefore, the angle encoding uses the number of quantum qubits in its circuit with the same dimensions in the dataset, i.e., the total of predictors used to forecast solar irradiance [71]. The angle encoding mapping strategy can be represented mathematically by the following equation:

$$\left|\varphi \left(\mathbf{x}\right)\right.⟩={\displaystyle \underset{i=0}{\overset{n-1}{⨂}}}{R}_{\varphi }\left(x\right)\left|\varphi \right.⟩$$

(5)

Equation (5) is such that, for a n-qubit system, the rotational gate $R_{\varphi }$ is applied to each one of the input parameters of the original classical dataset, resulting in the quantum state $\left|\varphi \left(\mathbf{x}\right)\right.⟩$. Figure 3 represents a generic angle encoding mapping for n-qubits at an initial $\left|0\right.⟩$ state.

In this approach, $R_{\varphi }$ can assume any Pauli’s gate form, i.e., X, Y, Z, or I. Detailed information regarding these quantum gates can be found in dedicated references such as in Nielsen and Chuang [58].

The second feature mapping strategy investigated in this project was the ZZ feature map. It is the second-order Pauli expansion circuit [65]. This approach maps classical data to a quantum framework using a quantum circuit that transforms input data as follows [72]:

$$U_{\mathsf{\varphi }}\left(\mathrm{x}\right)=\exp \left(i\sum _{S\in I}{\mathsf{\varphi }}_S\left(\mathrm{x}\right)\prod _{i\in S}{P}_i\right)$$

(6)

In Equation (6), the mapping quantum circuit $U_{\varphi }\left(\mathrm{x}\right)$ results from a relationship between the data-mapping term ${\varphi }_S\left(\mathrm{x}\right)$ and the Pauli gates, $P_i=\{X, Y, Z, I\}$, having a diagonal form [73]. Furthermore, S represents a set of qubit indices in the feature map, and I is a set containing all these index sets. Commonly, the term ${\mathsf{\varphi }}_S\left(\mathrm{x}\right)$ is defined as

$${\mathsf{\varphi }}_S\left(\mathrm{x}\right)=\left\{\begin{array}{c}{\mathrm{x}}_i, if S= i\\ {\displaystyle \prod _{j\in S}}{P}_i\left(\mathsf{\pi }-{\mathrm{x}}_j\right), if |S|>i\end{array}\right.$$

(7)

The ZZ feature map is also a parameterized circuit, where the input values from its gates come from the classical dataset. The ZZ feature map circuit is presented in Figure 4 [59].

From Figure 4, the ZZ feature map initiates its circuit by applying Hadamard gates to the initial $\left|0\right.⟩$ state of the n-qubit system. It then proceeds to apply a rotation, where $P_i=Z$. After that, the circuit utilizes controlled-NOT gates pairwise, i.e., it takes {j, k} ⊆ {q₀, q₁, …, q_n−1} for j < k, to implement entangle states in the system [59].

2.5.3. Ansatz

The second component of the QNN structure is the ansatz. In quantum computing, ansatz refers to the initial guess for a quantum state, which will be used as a starting point for the computation. In the context of the QNN model, similarly to classical ML, it assumes the role of the training weights of the QML. To this end, the ansatz is implemented as a parameterized variational quantum circuit. The ansatz has parameters trained for each iteration through a classical optimization algorithm, reckoned by a classical computer [74]. A generic variational quantum circuit configuration is depicted as follows [3].

From Figure 5, showing the quantum circuit’s parameters, the ansatz takes as input the previously mapped classical data. Then, its parameters θ are updated in each iteration after consulting the cost function value. This process is usually performed by an iterative classical optimization algorithm, which aims to reduce its value in each iteration. Note that, at this point, the measurement shown in Figure 5 refers to the ansatz’s optimization process, not the final measurement performed at the end of the QNN structure (Figure 2).

For this project, three different ansatz were investigated. The first is a simple Ry ansatz, similar to the parameterized quantum circuit previously discussed for the feature map. It consists of applying parameterized Pauli Y quantum gates. The second strategy was the Two Local ansatz. This ansatz is composed of alternating rotations and entanglement layers using parameterized gates [75,76]. The rotational gates are applied individually for each qubit, while the entanglement is issued over two qubits according to the desired entanglement strategy. Two entanglement strategies were evaluated in this project: linear and full entanglement. The former consists of entangling an i-th qubit with its successor i + 1 for all qubits where $i\in \{\mathrm{0,1},\dots ,n-2\}$ for a n-qubit system. The latter entanglement strategy consists of entangling all qubits with each other. The qubit entanglement may be repeated. An example of the Two Local quantum circuit ansatz with linear entanglement and two repetitions is depicted as follows [77].

Figure 6 depicts a Two Local Ansatz with linear entanglement and two repetitions. It is possible to visualize that the rotational gates $R_{\varphi }$ accept parameterized values θ as inputs. These values represent the total change in qubit orientation, measured in radians. The total of parameters to be trained will depend on the total number of repetitions in the ansatz circuit. In the given example, since the quantity of qubits is the same as the number of dimensions in the classical dataset, defined as ${\mathbb{R}}^{\mathrm{m}\times \mathrm{p}}$, there is a total of p parameters in the first rotational layer (i.e., p = n − 1). Afterward, linear entanglement is implemented using controlled-NOT gates in a pairwise fashion. Later, the first repetition takes place, doubling the total count of parameters that can be trained. This is followed by another linear entanglement, which precedes the second repetition. In this example, the optimization algorithm needs to determine a total of 3p trainable weights.

The final ansatz used in this project was the Quantum Approximate Optimization Algorithm (QAOA). It is a hybrid ansatz approach combining classical and quantum frameworks [78]. The QAOA is a more specific form of variational quantum eigensolver (VQE) quantum circuits, which aim to find the lowest eigenvalue of the Hamiltonian of a system [59,79]. To this end, the parameterized ansatz circuit prepares the quantum state such that $\left|\mathsf{\Psi }\right(\theta )⟩=U(\mathsf{\theta })$, where $U\left(\theta \right)$ represents the parameterized ansatz. In this context, this relationship can be rewritten as follows [3]:

$$\mathrm{C}\left(\mathsf{\theta }\right)=⟨\mathsf{\Psi }\left(\mathsf{\theta }\right)|\mathrm{H}|\mathsf{\Psi }\left(\mathsf{\theta }\right)⟩$$

(8)

In Equation (8), the cost function, $\mathrm{C}\left(\theta \right)$, is calculated using the expected values of the system’s Hamiltonians. The cost function is minimized using classical optimizer algorithms [3,80]. Unlike the VQE, the QAOA approach uses a specially optimized ansatz, consisting of unique parameterizations of the problem Hamiltonian and rotational gates. The main configuration of QAOA is determined by a single integer parameter, which sets the depth of the ansatz circuit and directly influences the quality of the approximated result [78]. A generic quantum circuit representing the QAOA ansatz is presented as follows [3].

In Figure 7, C and B define the ansatz state starting with a uniform superposition, after implementing the Hadamard gates upon all the qubits. The term C is defined as $\mathrm{C}\left(\mathrm{z}\right)={\sum }_{\mathrm{k}=1}^{\mathrm{K}}{\mathrm{C}}_{\mathrm{k}}\left(\mathrm{z}\right)$, where $z={\{0, 1\}}^{\otimes \mathrm{n}}$represents a string satisfying a set of $C_k$, and the diagonal element of the Hamiltonian of a system, $H_{i,i}$, is defined as $H_{i,i}$ = C(z). The term B represents the sum of Pauli X operators in the quantum circuit $B={\sum }_{i=1}^n(\mathrm{P}\mathrm{a}\mathrm{u}\mathrm{l}\mathrm{i}{\mathrm{X})}_{\mathrm{i}}$. Finally, $U\left(B,\mathsf{\beta }\right)=e^{-i\mathsf{\beta }B}$ and $U\left(C,\mathsf{\gamma }\right)=e^{-i\mathsf{\gamma }B}$ [3].

Beyond the number of qubits and the quantum gates composing them, the number of circuit repetitions also significantly characterizes both feature maps and the ansatz. This repeats the circuit in sequence m-times to the same qubits [3]. This results in an embedded circuit in the form $U\left(\mathrm{x}\right)U\left(\mathrm{x}\right)\dots U\left(\mathrm{x}\right)=U{\left(\mathrm{x}\right)}^m$. Considering the ansatz structure, the repetition of its quantum circuit can be compared to classical layers in a classical neural network [81]. Works on this subject have shown that repeating circuits for feature maps can encode data to come up with universal approximation theorems of QML algorithms when $m \to \mathrm{\infty }$, improving their output [3]. In this project, repetitions of 1 and 2 were investigated when using the Two Local ansatz, as previously discussed.

2.5.4. Optimization Algorithms

One popular strategy adopted by researchers investigating the implementation of QML models is to not rely exclusively on a quantum paradigm but rather split it into quantum and classical architectures. The motivation behind this approach is to ease the computational efforts during the quantum circuit simulations. A possible implementation of this approach is to process the classical data using quantum circuits (feature map and ansatz), while the training phase is optimized using classical algorithms [3]. There are several strategies to optimize the cost function. In the context of this project, three optimization algorithms were investigated.

The first is the Constrained Optimization BY Linear Approximation (COBYLA) algorithm. It is a numerical optimization method used for problems with constraints, where the objective function’s derivative is unknown [82]. It minimizes a scalar function of one or more variables using linear approximation over a simplex, not requiring the cost function gradient to converge, thus mitigating errors related to gradient evaluation [83,84,85]. Its usage in QML models has been investigated in previous studies, showing that it requires a reduced number of iterations and provides good accuracy [85,86].

The second optimizer investigated was the Limited-memory Broyden-Fletcher-Goldfarb-Shanno Bound (L-BFGS-B). This is a quasi-Newton algorithm, meaning it does not require a second derivative of the objective function to minimize the value of a differentiable scalar function. The L-BFGS-B is an interactive model that solves unconstrained, non-linear optimization problems, updating the estimated value of the objective function with each iteration [87,88]. Application of this optimization strategy for quantum problems can be found in the literature, achieving good convergence speed and accuracy [89,90].

Finally, the last approach investigated in this project was the Quantum Natural Simultaneous Perturbation Stochastic Approximation (QN-SPSA) optimizer. Different from the previous optimizers, this algorithm is a gradient-based approach to minimize the cost function value. This is done by taking the derivative of an objective function, f(x), and changing the x-value to minimize it for each step. Considering the case for multiple inputs, f(x₁, …, x_n), the optimization must consider the partial derivatives of the function concerning each one of its inputs. In this case, one must calculate the gradient of the function ${\displaystyle \frac{\partial }{\partial x_i}}f\left(\mathrm{x}\right)$. Thus, for the latter case, the new value of x that minimizes f(x) for an arbitrary time step can be calculated as follows [47,49]:

$${\mathrm{x}}^{\mathrm{t}+1}={\mathrm{x}}^{\mathrm{t}}-ϵ{\nabla }_{\mathrm{x}}f{\left(\mathrm{x}\right)}^t$$

(9)

where, in Equation (9), the updated values of the attribute ${\mathrm{x}}^{\mathrm{t}+1}$ result from the variation of the previous ${\mathrm{x}}^{\mathrm{t}}$ by an amount $ϵ$ of the gradient of the function ${\nabla }_{\mathrm{x}}f{\left(\mathrm{x}\right)}^t$. Here, $ϵ$ is the learning rate, $\mathsf{ϵ}\in \left[0, 1\right]$, and it regulates the total of the change attribute to the new vector. The learning rate, a hyperparameter of the gradient descent optimizer, directly influences the convergence of the cost function. Slight values for the learning rate may require more steps, t, to achieve convergence. Conversely, larger learning rate amounts can cause the gradient change to become too unstable, even preventing the function from converging toward its minima [47].

The QN-SPSA optimizer is based on the SPSA one [91], but seeks to achieve convergence by sampling the natural gradient instead of the first-order one. This approach resembles the Monte Carlo method, where the higher the number of samplings, the more accurate the model. This may lead to an exact analytic result for the objective function [92]. The QN-SPSA is well suited for use in large-scale population models, adaptive modeling, and simulation optimization. This results in the QN-SPSA approach speeding up the natural gradient calculation by decreasing its accuracy [92].

The QN-SPSA implementation in Qiskit requires fidelity and perturbation values when a specific learning rate number is passed to its function. The fidelity refers to the measurement of the closeness between two quantum states, which in this case are the Hessian of the fidelity matrix defined by the ansatz and its real value. The perturbation refers to the magnitude of the fluctuations in the parameters of the quantum circuit for the finite difference approximation of the gradients. Perturbation and learning rate can be set to have the same values when implementing the QN-SPSA optimizer [91]. It achieves this by approximating the Hessian of the fidelity of the ansatz circuit.

2.5.5. Measurement

The final step of the QNN structure is measuring the achieved quantum state after previous data processing by the feature map and ansatz quantum. Differently from quantum gates, which apply reversible operations, the measurement of the qubit state is not reversible. The measurement of a qubit’s state makes it collapse into one of its possible states defined in Equation (2) [60]. As previously mentioned, the larger the amplitude value, the more probable the wave function is to collapse into a specific quantum state.

3. Results

The classical forecasting models were implemented using the Scikit-Learn library for Python [93], while the QNN approach was built and simulated using Qiskit [94]. The hyperparameters of the classical ML models were selected using the GridSearch strategy, but for GMDH, which is a self-organizing model. The dataset was split in the 80/20 Pareto fashion for the training and testing phases, without shuffling the dataset. The computer used to build and run the models had a 13th generation i7 processor, 32 Gb of RAM, and an RTX 4080 GPU.

The ML and QNN models’ performances were evaluated using the root mean square error (RMSE), the coefficient of determination (R²), and the forecast skill (FS). These are some of the commonly used metrics to analyze the performance of predictive models.

3.1. Parameter Selection

The best parameter subset from the eighteen total was selected for each forecasting horizon. To this end, the recursive feature elimination (RFE) was implemented [48]. It consists of selecting the best features, starting with the whole set of predictors and selecting the most relevant ones according to a previously selected ML model. The present study implemented the RFE using a random forest (RF) model to select the best subset of attributes. Subsequently, they were given importance after all the attributes were used. Finally, the least important predictor was removed, and the RFE restarted. This process was repeated until a desired number of features was achieved.

Different feature values were evaluated. It was observed that, for smaller numbers, RFE would return only attributes related to one type of data information. That is, the selected parameters would be temporal (B_i, V_i, and L_i) or spatial (R, B, G, and their derivates). Finally, the total of attributes was set to 5. The motivation behind this is twofold:

• To keep both information from statistical features, related to temporal information, and sky image features, related to spatial data.
• To keep the problem complexity low for quantum machine learning simulation.

From the RFE output, it was observed that for forecasting intervals of 5 and 10 min, most of the predictors were tied to the dataset’s temporal dimension, encompassing elements B, V, and L, along with two spatial predictors. However, as the forecasting window expanded, the balance shifted. This time, three out of five predictors were associated with the dataset’s spatial dimension, i.e., sky images, leaving only two tied to the temporal aspect. This suggests a dynamic interaction between spatial and temporal factors depending on the length of the forecasting horizon, where spatial information conveys more relevant information for solar irradiance forecasting.

As previously mentioned, the engineered temporal attributes were calculated considering the time interval prior to the desired prediction window. In this study’s methodology, we opted to select the time-series attributes considering the information for the desired forecasting horizon. Figure 8 illustrates the attribute selection process for a 5 min forecasting horizon, and the selected predictors are presented in

Table 2. Selected parameters using RFE.

Selected Features
Forecasting Horizon (min)	GHI	DNI
5	Bi, Vi, Li, ENT (R), ENT (RB)	Bi, Vi, Li, ENT (G), ENT (RB)
10	Bi, Vi, Li, ENT (G), ENT (RB)	Bi, Vi, Li, ENT (G), ENT (RB)
15	Bi, Vi, AVG (G), ENT (B), ENT (RB)	Bi, Vi, ENT (R), AVG (G), ENT (RB)
20	Bi, Vi, AVG (G), ENT (G), ENT (RB)	Bi, Vi, ENT (R), AVG (G), ENT (RB)
25	Bi, Vi, ENT (R), AVG (G), ENT (RB)	Bi, Vi, ENT (R), AVG (G), ENT (RB)
30	Bi, Vi, ENT (R), AVG (G), ENT (RB)	Bi, Vi, ENT (R), AVG (G), ENT (RB)
60	Bi, Vi, AVG (R), ENT (G), ENT (RB)	Bi, Vi, AVG (G), ENT (G), ENT (RB)
120	Bi, Vi, AVG (G), ENT (G), ENT (RB)	Vi, ENT (R), AVG (G), AVG (B), ENT (RB)
180	Bi, Li, AVG (G), ENT (B), ENT (RB)	Vi, Li, AVG (R), ENT (R), ENT (RB)

for each prediction interval.

3.2. Feature Map Quantum Circuit Configuration Selection

For the feature map quantum circuit, different rotations were assessed for the angle encoding strategy. The 5 min prediction window and global horizontal irradiance were used to benchmark the best feature map strategy. During this phase, we kept the ansatz constant, applying a simple rotational Pauli Y gate to each qubit in the circuit. We also used the COBYLA algorithm for optimization. It is worth noting that we did test different ansatz rotations and optimization algorithms during this phase. However, these alternatives did not yield better results than the chosen configuration. Due to its significant computational time requirements, the Two Local ansatz was not considered during this phase. The Pauli Y rotation yielded an RMSE of 56.91 W/m² and R2 of 95.99% for the selected feature map circuit configuration. At this point, Pauli X and Pauli Z gates were discarded as angle encoding strategies for the feature map quantum circuit. The graph showing the convergence of this approach is presented in the following figure.

The second feature map investigated was the ZZ feature map. Again, using the forecasting horizon of 5 min for GHI as a benchmarking dataset, with the Pauli Y ansatz and COBYLA optimizer, the ZZ data encoding output RMSE was 447.78 W/m², and the R² was −1.48. The metrics elucidate that the ZZ feature map yielded inferior values to the ones provided by the Ry angle encoding. The graph showing the convergence of this approach is presented as follows.

Figure 9 shows that the feature map for the given configuration presented downward fashion from iteration one until around iteration twenty. Afterward, the results showed convergence. Figure 9 and Figure 10 show that the objective function achieved convergence at around forty iterations in both cases. However, the model’s convergence was near zero when Pauli Y was implemented as a feature map quantum circuit. At the same time, in the ZZ approach, it occurred not long past the 0.65 value, indicating poorer performance than its predecessor. This can be observed by analyzing the metric values, where the first approach had a greater RMSE, by approximately 155%.

3.3. Ansatz Quantum Circuit Configuration Selection

For the ansatz quantum circuit, the configuration was first investigated using the previously selected Pauli Y as feature map angle encoding, with the Two Local ansatz with the Pauli Y gate for rotation, the controlled-x gate for entanglement, and COBYLA as the optimizer for the 5 min forecasting horizon dataset for GHI. Note that the Pauli Y rotation used as the ansatz had already been evaluated in the previous subsection (Figure 9).

Considering the Two Local approach, two different entanglements were investigated: linear and full. The former presented better results than the latter. This configuration returned an RMSE of 58.08 W/m² and R² of 95.82%. The convergence for this model can be visualized in the following figure.

Changing the rotation angle encoding to ZZ feature map and using the Two Local ansatz with the COBYLA optimizer, the QNN model returned subpar results, with an RMSE of 402.04 W/m² and R² of −1.00. Its convergence is presented in the following figure.

In Figure 11, the convergence occurs after around 40 iterations and is near to a zero value for the objective function, after the bumpy behavior of its objective function value. Figure 12 presents a smoother objective function value, with the convergence occurring only after 100 iterations and well before the zero value of the objective function, indicated in the image’s y-axis. The superior performance of the angle encoding approach for the feature map circuit is again observed from the given results. Considering the ZZ feature map results presented in Figure 10 and Figure 12, this mapping circuit was discarded as an encoding strategy for the QNN model.

Finally, the QAOA was evaluated using a similar configuration, considering only the Pauli Y as a feature map quantum circuit. For this ansatz, the circuit depth was set to two. The convergence of this ansatz circuit is presented in the following figure.

Figure 13 shows that QAOA convergence is similar to the one in Figure 11, which occurs around 30 iterations. Assessing the values present in the y-axis, the QAOA ansatz does not converge to a value near zero, indicating its suboptimal performance. The QAOA ansatz for the given configuration returned an RMSE of 498.86 W/m² and R² of −2.08, being the worst-performing approach so far.

3.4. Optimization Algorithm

The selection of the best-suited optimizer was benchmarked using the previously selected Pauli Y feature map approach for 5 min GHI forecasting. The Pauli Y, Two Local, and QAOA ansatz were also evaluated during this phase. The QN-SPSA optimizer used a learning rate and perturbation values of 0.5 for the Pauli Y ansatz, 0.1 for the Two Local ansatz, and 0.3 for the QAOA ansatz. Different values were tested for QN-SPSA, and when applied to each ansatz were not able to surpass the results from the selected learning rates.

Table 3. Results for different configurations of the QNN model using Pauli Y as feature map angle encoding and different ansatz and optimizers for 5 min GHI forecasting.

Pauli Y Feature Map
Ansatz, Optimizer	RMSE (W/m2)	R2
Ry, COBYLA	56.91	95.99%
Ry, L BFGS B	56.73	96.01%
Ry, QN-SPSA	56.55	96.04%
Two Local, COBYLA	58.08	95.82%
Two Local, L-BFGS-B	48.98	97.03%
Two Local, QN-SPSA	63.90	94.94%
QAOA, COBYLA	498.86	−2.08
QAOA, L-BFGS-B	498.85	−2.08
QAOA, QN-SPSA	501.21	−2.11

compiles the results for selecting the best optimization algorithm for each evaluated configuration.

Table 3 shows that the best configuration for the QNN model is composed by the Pauli Y angle encoding as a feature map, using the Two Local ansatz and the L-BFGS-B optimizer. The QN-SPSA optimizer achieved better results than those output by COBYLA and L-BFGS-B when used together with the Ry ansatz quantum circuit. However, when the Two Local approach was implemented, it was the worst-performing approach among the ones investigated. The QAOA ansatz also did not provide good outcomes for solar irradiance forecasting using the benchmarking configuration. It had a similar behavior as the one observed using the ZZ feature map, where the convergence occurred far from the zero value (Figure 10 and Figure 12). The following graph shows the convergence of the best-performing configuration present in the following figure.

Figure 14 showcases the convergence of the best-performing configuration for QNN. Compared with Figure 9, which uses COBYLA as an optimizer, it is noticeable that the convergence with the L-BFGS-B optimizer occurs faster, after around five iterations, having almost no oscillations along its path. The final configuration improved, by almost 15%, upon the output yielded by the second-best performing configuration, considering the RMSE metric (Table 3).

3.5. Number of Repetitions for the Ansatz

Additionally, the Two Local ansatz was implemented using two repetitions for its configuration, as shown in Figure 6, using controlled-NOT gates for entanglement. To this end, the best QNN configuration, previously defined and presented in Figure 14, was used. Using the Two Local ansatz with two repetitions yielded superior output metric values for 5 min GHI forecasting, achieving an RMSE of 39.74 W/m² and R² of 98.04%, meaning an enhancement of over 23% compared to the second-best configuration. However, its processing time proved prohibitive, as it took eight times longer (42 h) than the configuration presented in Table 3 (5 h). Therefore, the Two Local ansatz with two repetitions was not selected for the final QNN configuration. The selected QNN configuration is presented in the following Figure 15.

3.6. Solar Irradiance Forecasting Results

The best configurations for both classical and quantum neural networks were selected, and the models were implemented for solar irradiance forecasting up to 3 h ahead. Their outputs are compiled in

Table 4. Results for GHI forecasting by the classical and quantum ML models for all the forecasting horizons.

GHI
Forecasting Time (min)	Algorithm	RMSE (W/m2)	R2 (%)	Forecast Skill (%)
5	SVR	35.94	98.40	92.33
	XGBoost	28.74	98.98	93.87
	GMDH	34.82	98.50	92.58
	QNN	48.98	97.03	89.55
10	SVR	43.00	97.71	90.82
	XGBoost	32.20	98.72	93.13
	GMDH	45.56	97.43	90.28
	QNN	54.48	96.33	88.38
15	SVR	50.17	96.90	89.29
	XGBoost	40.68	97.96	91.31
	GMDH	51.62	96.72	88.98
	QNN	65.56	94.71	86.00
20	SVR	52.69	96.60	88.74
	XGBoost	42.69	97.77	90.88
	GMDH	54.15	96.41	88.42
	QNN	65.40	94.76	86.02
25	SVR	55.11	96.30	88.20
	XGBoost	44.23	97.62	90.53
	GMDH	57.57	95.97	87.68
	QNN	63.09	95.16	86.50
30	SVR	57.00	96.08	87.80
	XGBoost	45.70	97.48	90.20
	GMDH	58.30	95.90	87.50
	QNN	63.97	95.06	86.30
60	SVR	55.63	96.32	88.00
	XGBoost	34.76	98.56	92.50
	GMDH	61.08	95.56	86.82
	QNN	61.20	95.54	86.80
120	SVR	69.91	94.99	84.18
	XGBoost	54.67	96.93	87.63
	GMDH	74.91	94.26	83.05
	QNN	66.24	95.50	85.01
180	SVR	81.94	94.00	79.84
	XGBoost	58.93	96.90	85.50
	GMDH	81.90	94.01	79.85
	QNN	77.55	94.63	80.92

and

Table 5. Results for DNI forecasting by the classical and quantum ML models for all the forecasting horizons.

DNI
Forecasting Time (min)	Algorithm	RMSE (W/m2)	R2 (%)	Forecast Skill (%)
5	SVR	80.31	93.33	87.16
	XGBoost	61.96	96.03	90.09
	GMDH	72.08	94.63	88.47
	QNN	103.03	89.02	83.52
10	SVR	95.17	90.70	84.75
	XGBoost	77.08	93.90	87.65
	GMDH	92.49	91.22	85.18
	QNN	117.21	85.89	81.22
15	SVR	105.25	88.77	83.08
	XGBoost	87.76	92.19	85.90
	GMDH	105.86	88.64	82.99
	QNN	127.15	83.61	79.57
20	SVR	112.31	87.43	81.88
	XGBoost	93.69	91.25	84.89
	GMDH	114.72	86.88	81.49
	QNN	130.76	82.96	78.91
25	SVR	117.58	86.49	80.95
	XGBoost	98.19	90.58	84.09
	GMDH	121.78	85.51	80.27
	QNN	133.70	82.53	78.34
30	SVR	122.84	85.54	80.00
	XGBoost	102.33	89.97	83.34
	GMDH	127.79	84.36	79.20
	QNN	136.34	82.19	77.81
60	SVR	123.77	85.69	79.53
	XGBoost	84.33	93.35	86.05
	GMDH	134.96	82.98	77.68
	QNN	157.23	76.91	73.99
120	SVR	172.05	77.31	69.11
	XGBoost	107.24	91.19	80.74
	GMDH	168.16	78.33	69.80
	QNN	243.06	56.35	54.72
180	SVR	192.58	74.81	61.46
	XGBoost	145.54	85.61	70.87
	GMDH	177.88	78.51	64.40
	QNN	168.81	80.64	66.22

for GHI and DNI predictions, respectively. They are discussed in the follow-up section. The unit for the RMSE metric is W/m².

4. Discussion

4.1. QML Models Found in the Literature

As previously stated, this study successfully implemented a simulated QNN model utilizing the Qiskit library in Python. Existing studies have applied simulated quantum machine learning to renewable energy, whether using PennyLane [95], Qiskit, or other available libraries.

In the work by Correa-Jullian et al. [96], a QML model using SVM was implemented for wind turbine fault detection. Their work proved that the quantum approach could improve the results over traditional ML methodologies, being able to outperform some of them. Another quantum application in the renewable energy area can be found in the work by Sagingalieva et al. [97]. There, the authors forecasted the photovoltaic energy output using a hybrid model, combining both classical and quantum paradigms. Their methodology achieved excellent results, proving the feasibility of this architecture in solar power prediction. The usage of QML strategies for microgrid control was investigated in the work of Jing and colleagues [98]. In their work, the authors used a hybrid QML model based on the Harrow Hassidim Lloyd (HHL) algorithm for microgrid management, which proved to yield remarkable results for such a task.

Considering that QML is still in its nascent phase, few works addressing solar irradiance forecasting can be found. In work by Senekane and Taele [99], the authors implemented a quantum SVM model for solar irradiation forecasting. Their work proved the efficacy of using QML for solar irradiance forecasting. In a more recent work, the authors Sushmit and Mahbubul [81] explored the implementation of a hybrid QML methodology for solar irradiance forecasting. Their methodology was tested across various locations and demonstrated its reliability as a forecasting tool. The structure employed in the current study mirrors the approach taken by Sushmit and Mahbubul. They utilized a parameterized quantum circuit with angle encoding as a feature map and an ansatz closely resembling the Two Local circuit used in this study. Their findings indicated that the QNN alone could not surpass the hybrid configurations examined, yielding RMSE and R² values comparable to those in Table 4. Another study published by Yu et al. [70] investigated the combination of a quantum circuit embedded in a recurrent neural network for 1 h GHI forecasting. Their methodology also holds similarities to the one presented in this study, like the use of angle encoding. On the other hand, their ansatz circuit was somewhat distinct, presenting entanglement and rotation operations over the qubits. Their results for the annual prediction performance returned average results for RMSE of 61.76 W/m² and 94.6% for R², showing a significant improvement over the classical ML evaluated.

The present study yields similar outcomes to those of the published studies in QML for solar irradiance forecasting. Notably, references [70,81] utilize hybrid QML configurations to enhance their solar irradiance predictions [100]. Nevertheless, based on the evaluation metrics, our study matches these predictions effectively.

4.2. GHI and DNI Forecasting Results

From Table 4 and Table 5, the superior performance of the XGBoost approach is noticeable when compared to that of its competitors. It managed to overcome each of the other models for every assessed time horizon, especially for GHI, where it reached the best values of 28.74 and 98.98% for RMSE and R², respectively. This is not a surprise, given that the XGBoost is a well-established ML model, and its superiority for solar energy forecasting has been reported in several studies [31,101,102,103]. The remarkable XGBoost performance can be explained as twofold:

• The parameter selection was performed by a random forest model, which is a tree-based model and therefore may have added some bias toward the improved performance of the XGBoost.
• The inherent capacity of tree-based models to outperform other ML and deep learning approaches on tabular data has been noted, as investigated in [104,105].

Considering the forecasting interval between 5 and 120 min, the models SVR and GMDH could also provide similar results to those of XGBoost, though differing slightly. For this same time window, the QNN model was the fourth-best-performing approach, also able to provide competitive results for GHI and DNI forecasting. The superiority of the classical models over the QNN approach is due to their much more mature implementation in a classical device, whilst QML software remains in its early stages [106], employing a very limited number of qubits. Additionally, classical ML approaches explicitly for a target task often overcome non-hybrid QML structures. Nevertheless, the quantum paradigm still can provide competitive results, as reported in [96,106].

Additional investigation of Table 4 and Table 5 reveals that for the longer forecasting horizon of 3 h, the QNN approach managed to reach the second-best performance among the assessed models. Commonly, in time-series forecasting of spatiotemporal environmental indicators, the predictive performance of the models is expected to decrease as the forecasting horizon expands. This behavior can be explained by the lack of additional information from the input parameters, causing the model’s performance to deteriorate for longer horizons [27,107]. This phenomenon occurs for the assessed models, as it is possible to observe for the forecasting comprising the interval of 60 to 180 min. However, the QNN approach presents a much more robust performance than the classical ones. For instance, in the 60 to 120 min interval for GHI forecasting, the QNN model exhibits less performance deterioration, as indicated by the performance metrics. This pattern does not hold for DNI forecasting, where the QNN approach displays a significantly more volatile behavior relative to other models. Remarkably, when forecasting both types of irradiances for a 3 h period, the QNN model secured the second-best performance among the evaluated paradigms. This fact can be an indication that the quantum model may identify and retrieve relevant spatiotemporal information from the input dataset in such a manner not attainable by the current classical approaches. This capability significantly enhances its performance over extended prediction windows.

Finally, from the results in Table 4 and Table 5, it is also possible to observe that the overall performance for all the evaluated models decreased considerably for DNI predictions. Such a behavior is common among predictive models. The GHI represents the total radiation that arrives at the Earth’s surface from all directions. On the other hand, the DNI is a highly directional measurement, where even minor obstacles can prevent sun rays from reaching the Earth’s surface, resulting in inaccuracies in its modeling [31,108].

Although the outcomes are suboptimal for shorter forecasting windows, i.e., from 5 to 120 min in advance, when compared to conventional ML frameworks, the quantum paradigm provides a compelling perspective on its utility in forecasting solar irradiance, more specifically for longer prediction intervals. This approach could potentially contribute significantly to advancing investigations in the domain of renewable energies. As the current quantum computing technology evolves, the application of quantum circuit models using real quantum hardware is expected to become more tangible [7]. In this context, the implementation of QML models on real quantum devices will lead to faster processing times with superior accuracy to that currently possible. This would ultimately allow the achievement of quantum supremacy [5].

5. Conclusions

This study investigated the potential of applying quantum machine learning to develop a forecasting model for the GHI and DNI to optimize solar energy production. The widely recognized Folsom dataset served as the benchmark for this investigation. In this analysis, the most effective subset of predictors, encompassing both spatial and temporal attributes, were chosen for each forecasting horizon ranging from 5 to 180 min. The attribute selection was made using the RFE approach. Different configurations for the quantum model were examined, and the best-performing one was selected. The QML results were compared to classical ML algorithms, namely SVR, XGBoost, and GMDH.

Results for the parameter selection showed that the influence of spatial-related predictors increased as the time window escalated. For the QML, the best configuration for the QNN model was achieved using Pauli Y as an angle encoding for the feature map, the Two Local ansatz with Pauli Y, and controlled-NOT gates, containing one repetition, with the L-BFGS-B optimizer. The overall best results for irradiance forecasting were reached using the XGBoost model for all investigated time horizons. The QNN model, however, was able to provide competitive results for horizons spanning from 5 to 120 min, especially for forecasting GHI. However, the expansion of the forecasting horizon to 180 min revealed that the QNN model managed to overcome the classical approaches of SVR and GMDH, indicating a superior capacity for identifying and extracting spatiotemporal information from the dataset compared to its competitors. The overall performance of the models for the DNI forecasting was inferior when compared to the GHI results.

For future works, the proposed QNN methodology could be evaluated for other renewable energy applications, such as wind speed forecasting for wind energy production. In addition, a more complex ansatz configuration containing more than one repetition could be investigated. Finally, it is expected that further investigation of the QNN model applied to real quantum hardware may allow the usage of deeper quantum circuits. This would permit the implementation of more complex quantum circuit configurations, such as larger repetitions for the ansatz, promoting a more expressive quantum machine learning model, while having improved processing time and more accurate outcomes.