Prediction of Solar Irradiance One Hour Ahead Based on Quantum Long Short-Term Memory Network

The short-term forecasting of photovoltaic (PV) power generation ensures the scheduling and dispatching of electrical power, helps design a PV-integrated energy management system, and enhances the security of grid operation. However, due to the randomness of solar energy, the output of the PV system will fluctuate, which will affect the safe operation of the grid. To solve this problem, a high-precision hybrid prediction model based on variational quantum circuit (VQC) and long short-term memory (LSTM) network is developed to predict solar irradiance 1 hour in advance. VQC is embedded in LSTM to iteratively optimize the weight parameters of four gates (forgetting gate, input gate, cell state, and output gate) to improve prediction accuracy. To evaluate the prediction performance of this model, five solar radiation observatories located in China are selected, together with widely used models including seasonal autoregressive integrated moving average, convolution neural network, recurrent neural network (RNN), gate recurrent unit, (GRU), and LSTM; comparisons are made under different seasons and months. The experimental results show that the annual average root mean square error of the quantum long short-term memory model is 61.756 <inline-formula><tex-math notation="LaTeX">$\text{W/m}^{2}$</tex-math></inline-formula>, which is reduced by 10.7%, 13.9%, 8.1%, 3.8%, and 3.4%, respectively, compared with other models; the annual average mean absolute error is 24.257 <inline-formula><tex-math notation="LaTeX">$\text{W/m}^{2}$</tex-math></inline-formula>, which is reduced by 28.1%, 28.9%, 24.1%, 12.2%, and 12.8%, respectively, compared with other models; the annual average R-Square (<inline-formula><tex-math notation="LaTeX">$R^{2}$</tex-math></inline-formula>) is 0.946, which is improved by 1.5%, 1.9%, 1.2%, 0.4%, and 0.4%, respectively, compared with other models.


I. INTRODUCTION
Solar energy engineering is one of the energy alternatives for global sustainable development goals [1]. Photovoltaic (PV) panels are the biggest driving force in the rapid growth of solar power generation [2]. However, PV systems are susceptible to environmental factors such as sunlight, season, and geographic location, resulting in the PV characteristics of being stochastic, intermittent, and variable [3], [4]. These characteristics are required to reduce the efficiency of PV output [5]. The reliability and stability of the power supply connected to the main grid will also be affected [6], [7]. Reliable PV prediction can provide important data support for power system operation, which is the basis and key to realize the large-scale utilization of solar energy resources [8]. And it is very important for the safe and stable operation of the PV system.
Generally, PV prediction methods mainly focus on PV output and solar irradiance. In order to enable the prediction model to mine the inherent characteristics between the data, the deep analysis of a large number of historical PV or solar irradiance data is required [9]. A PV system is affected by environmental conditions and geographic location, which lead to a strong correlation between meteorological characteristics and PV output. Meteorological data can be obtained from local meteorological stations. In addition, global horizontal irradiance (GHI) in the next period of time can be obtained through the developed prediction model.

Engineering uantum
Transactions on IEEE Yu et  In the existing literature, the methods for solar irradiance prediction are mainly divided into three categories: physical model, statistical model, and machine learning (ML) model. The physical model utilizes mathematical equations to describe the physical state of meteorological conditions and the dynamic movement of the atmosphere, thereby, deriving the mathematical relationship between the physical state and solar irradiance [10]. In general, these models compute solar irradiance by the formulation of mathematical analysis of physical and dynamic atmospheric variables. The performance of these physical models highly depends on the climate and the dynamic solar irradiance of the particular target area, which makes it almost impossible to predict accurately. Moreover, these numerical models are more complicated and time consuming.
Compared with physical models, statistical models are designed to observe the changing trend of historical timeseries data to find the mathematical relationship between historical solar irradiance and meteorological parameters. The statistical methods commonly used in solar irradiance prediction include autoregressive (AR) [11], autoregressive moving average, and autoregressive integrated moving average (ARIMA) [12]. Alsharif et al. [13] developed a seasonal autoregressive integrated moving average (SARIMA) model to predict the daily and monthly solar radiation in Seoul, South Korea, based on the hourly solar radiation data obtained from the Korean Meteorological Administration over 37 years (1981-2017). According to the findings, the expected average monthly solar radiation ranges from 176 to 377 Wh/m 2 , which is compatible with the general trends of the highest and lowest monthly values and daily fluctuations. Basmadjian et al. [14] proposed a set of ARIMA-based models for each month of the year, and introduced exogenous variables for analysis. They designed different models for each month, which is an excellent method, and finally achieved an error of no more than 10%. Atique et al. [15] used the ARIMA model to forecast the total daily solar energy generated by a solar panel located in a research facility. The model was developed by using techniques like differencing, ACF, PACF, and ADF test, and was validated by using the Akaike information criterion and error sum of squares. This work can be used as a good building block for further research into forecasting of renewable energy generation. Basmadjian et al. [16] proposed a methodology to generate accurate predictions, which consists of two main phases. In the first phase, the most relevant variables having impact on the generation of the renewables were identified using correlation analysis. The second phase consists of 1) estimating model parameters, 2) optimizing and reducing the number of generated models, and 3) selecting the best model for the method under study. The average absolute error based on ARIMA, SARIMA, and SARIMA with exogenous factor (SARIMAX) models is between 6.76% and 11.57%. Vagropoulos et al. [17] compared four practical methods for electricity generation forecasting of grid-connected PV plants, namely, SARIMA, SARIMAX, modified SARIMA, and artificial neural network (ANN)-based model. According to the results, the ANN models, the SARIMAX model, and the modified SARIMA model are superior in terms of the day-ahead forecasting performance compared to the persistence model (PM) and the SARIMA model. Eldali et al. [18] employed an ARIMA model and real data from ERCOT for demonstrating an improved wind power forecasts (WPFs) effectiveness and usability. The results show WPF improvements as vector norms of the relative error and how the error is reduced in all the cases. Generally, statistical models perform well in predicting future values of stationary solar irradiance time series. However, time-series data of solar irradiance show nonstationary behavior due to cloud and seasonal effects. As a result, these models cannot accurately capture fluctuations in data.
To overcome the drawbacks of statistical models, the datadriven-based ML algorithm was extensively used to predict solar irradiance and PV power. ML-based schemes can process the nonlinear relationship of meteorological data. For instance, Mellit et al. [19] proposed an ANN-multilayer perceptron architecture by using the Levenbeg-Marquardt backpropagation algorithm to predict solar irradiance based on 24-h ahead for the GCPV plant in Trieste, Italy. The input parameters applied were air temperature, daily solar irradiance, and day of the months. The simulation results demonstrated that the metric performances of correlation coefficient are 98% and 94% for sunny and cloudy days, respectively. Shi et al. [20] developed the PV power output prediction for 1-day ahead based on support vector machine and the features of weather categorization at a PV station in China. The historical model data were categorized into four classes of historical weather conditions, i.e., cloudy, sunny, rainy, and foggy. As a result, the sunny model was emerging as the best among the others in PV power forecasting, and the average forecasting errors of root mean square error (RMSE) and MRE are 2.10 MW and 8.64%, respectively. However, the conventional ML methods have limited intensive feature extraction of large amounts of diverse data, which is challenging to predict accurately in nonidentical distribution parameters.
In recent years, deep learning models have gained extensive attention as they perform well over conventional ML models due to the following reasons: they can extract features automatically with no or little knowledge of background details, their strong generalization power, and the ability to handle Big Data. A recurrent neural network (RNN) is a class of ANNs specially designed to model sequential or timeseries data. Cao et al. [21] used the RNN model to predict the solar irradiance one day in advance. They used the data training model with a time interval of 1 h from 1995 to 2000, and finally obtains the RMSE value of 44.326 W/m 2 . Aslam et al. [22] used RNN, gate recurrent unit (GRU), and long short-term memory (LSTM) to predict the solar irradiance one hour and one day in advance. The data used were the data of 10 years from 2007 to 2017, with a time interval of 1 h. In the prediction experiment one hour in advance, RNN, GRU, and LSTM obtained RMSE values of 105.277, 99.722, and 108.888 W/m 2 , respectively, and GRU shows the best prediction performance. In the prediction experiment one day in advance, RNN, GRU, and LSTM obtained RMSE values of 63.125, 55.821, and 55.277 W/m 2 , respectively, and LSTM shows the best prediction performance. Both LSTM and GRU are variants of RNN model, and the difference between them is not big. The choice of which model depends on the specific task. In addition, Yu et al. [23] proposed a prediction model based on LSTM to predict solar radiation under complex weather conditions. They reported that the proposed LSTM model shows an RMSE in a range of 45.84 and 41.37 W/m 2 for the considered locations. Qing et al. [24] utilized the LSTM for GHI forecasting using weather forecast and meteorological data. The developed model shows a good accuracy over other considered models, with an RMSE value of 76.245 W/m 2 . The independent LSTM model has a singularity and cannot perform all work effectively. The combined hybrid network model not only reduces the limitations of the independent model, but can also combine multiple models to achieve a more comprehensive execution of the prediction task [25]. Therefore, some researchers combined LSTM and other models to extract features from the data to improve the accuracy of time-series forecasting. Zhang et al. [26] introduced a hybrid forecasting model with a combination of autoencoder LSTM (AE-LSTM) and PM. The robustness of the deep learning model is boosted by AE-LSTM, which enhances the uncertainty estimation. Wang et al. [27] proposed a hybrid deep learning model LSTM-convolutional network and applied it to a PV power prediction. In the proposed hybrid model, the temporal features of the data are extracted first by the LSTM, and then, the spatial features of the data are extracted by the convolution neural network (CNN).
In addition, as quantum computers have made breakthroughs in computing scale and stability, ML based on quantum algorithms has attracted attention. Quantum machine learning (QML) has become a rapidly developing research direction. A number of studies used quantum technology to improve ML algorithms, which provide references for future QML research. Due to the emergence of noisy intermediate-scale quantum (NISQ) [28] machines, the VQC algorithm [29], [30], [31], [32] has become the main candidate for finding a shallow quantum algorithm that surpasses the classical computer. By taking advantage of quantum computing with much fewer parameters [33], the VQC on NISQ has successfully implemented classical classification and clustering algorithms on benchmark datasets [34]. Chen et al. [35] explored VQC for deep reinforcement learning, which uses low-depth shallow VQC to reshape each module of the model, such as experience playback and target networks. Giani et al. [36] pointed out that there is a disconnect between quantum computing and practical applications, so it is necessary to find out relevant application fields to prove the potential of quantum computing to solve problems. They believe that the expansion of renewable energy will bring many new challenges to the power grid, so quantum computing will play an important role in the field of renewable energy. They also recommend potentially fruitful areas of crossover research to advance applications of quantum computing and renewable energy. Estimates from the New England grid operator (ISO-NE) suggest potential savings in the tens of millions of dollars for reducing the error of solar forecasting [37], [38]. Solar irradiance forecasting, windblades interactions, and hydrodynamics follow the viscous fluid physical law and obey the Navier-Stokes nonlinear partial differential equations. This is a new application area for quantum computing, and we have already seen initial studies using both the gate model [39], [40], [41] and annealing [42].
Inspired by the VQC, this work proposed a novel hybrid quantum long short term memory (QLSTM) model architecture, which embeds the VQC into the LSTM. Then input the meteorological data sequence into the QLSTM model to test its ability to predict short-term solar irradiance. The main contributions of this study can be summarized as follows. 1) We extend the classical LSTM to the quantum field, and replace the classical neural network in the LSTM cell with VQCs, which would play the role of feature extraction and data compression. We then applied the QLSTM model to the field of solar energy prediction, which provides a reference for the application of quantum computing in the field of renewable energy. 2) We compare the QLSTM model with the commonly used SARIMA, CNN, RNN, GRU, and LSTM models under different experimental conditions. The results show that the overall performance of the QLSTM model is better than these baseline models. This shows that quantum computing has a very broad application prospect in the field of renewable energy.
The rest of this article is organized as follows. Section II introduces the structure of QLSTM. Section III presents data preprocessing. The experimental results are presented and analyzed in Section IV. Finally, Section V concludes this article.

II. QLSTM MODEL A. VARIATIONAL QUANTUM CIRCUIT
A variational quantum circuit (VQC) is a quantum circuit with adjustable parameters optimized iteratively. The main function of the VQC circuit is feature extraction and data compression. Its generic structure is shown in Fig. 1. The VQC consists of three parts: data encoding layer (with R x ), variational layer, and quantum measurement layer. The encoding module first encodes the data information into the quantum state of qubit, and then, the gate operation of the entanglement module contains trainable weights. The evolved quantum state is measured to derive the classical output.

1) DATA ENCODING LAYER
A quantum embedding represents classical data as quantum states in a Hilbert space via a quantum feature map. It takes a classical datapoint x and translates it into a set of gate VOLUME 4, 2023 Engineering uantum  parameters in a quantum circuit, creating a quantum state |ψ x .
In the amplitude-embedding technique, data are encoded into the amplitudes of a quantum state. A normalized classical N-dimensional datapoint x is represented by the amplitudes of an n-qubit quantum state |ψ x as where N = 2 n , x is the ith element of x, and |i is the ith computational basis state.

2) VARIATIONAL LAYER
After the classical state data encoded into a quantum state, it is passed to the variable layer. The variant layer is represented by a dashed box in Fig. 1. After the data enter the variant layer, a number of entanglement operations are performed. These entanglement operations are composed of several controlled not (cnot) gates and single-qubit rotation gates. cnot gates are used for each pair of qubits with fixed adjacency of 1 and 2 (in a cyclic manner) to produce multiqubit entanglement. The single-qubit rotation gates with three parameters is shown as follows: where parameters α j i , β j i , and γ j i are the parameters for the iterative optimization of the VQC.

3) QUANTUM MEASUREMENT LAYER
Behind the variable layer is the quantum measurement layer. On the basis of calculation, the expected value of each qubit is obtained through Pauli matrix σ Z as shown in (3). The expectation values of each qubit can be computed using classical computers through the utilization of PennyLane, which is a quantum simulation software. After a series of quantum entanglement operations, the measured VQC output is a fixed-length classical vector. In the proposed QLSTM, the measured values from each of the VQC will be processed within a QLSTM cell

B. QLSTM NETWORK
There are four VQCs in a QLSTM unit, as shown in Fig. 2.
From VQC 1 to VQC 4 , the input is the concatenation v t of the x t from the current time step, and h t−1 from the previous time step, the output is a vector obtained from the quantum measurement layer. The measured values, which are Pauli Z expectation values of each qubit by design, then go through nonlinear activation functions (sigmoid and tanh). The four VQCs are the same, each of them are defined by a basic VQC from Fig. 1. The VQC includes a trainable fully connected layer that squeezes the dimensional space of the information down to the circuit size of the VQC. After each VQC, a quantum measurement of the expectation for each wire is fed to the trainable fully connected expansion layer. The expansion layer increases the size from the quantum circuit back to the  required dimensional space of the classic network. This is then processed using classical LSTM approaches. The QLSTM architecture can be defined as follows:  Fig. 4 by the form of box diagrams, and the information on the studied cities is shown in Table 2. The purpose is to observe and filter anomalous data, as shown by the small orange dots in Fig. 4. For the abnormal data, we use the mean value correct method to process it, that is, use the average value of consecutive observations to correct the outlier. It can be seen intuitively that the median distribution of GHI in Urumqi, Hohhot, and Harbin gradually rises from January to June and descends from July to December. From the median distribution diagram of Wuhan, it can be observed that it increases gradually from January to July, and then decreases gradually

TABLE 2. Information of the Studied Cities
from August to December. The median distribution of GHI is similar convex smooth curves for Urumqi, Hohhot, Harbin, and Wuhan, but the curve of Haikou is not smooth.
From the geographical point of view, the five locations can be described as follows. The Heihe-Tengchong line [43] is the dividing line of China's population density, which describes the macro pattern of population distribution in China with more in the East and less in the West. Urumqi and Hohhot above the line belong to low population density cities, and Harbin, Wuhan, and Haikou belong to high population density cities, which are below the line. Urumqi is located in Northwest China, which is deep inland and far away from the sea. It is surrounded by mountains on three sides and the northern plain is open. The ocean air flow is not easy to reach, forming an obvious temperate continental climate. Throughout the year, Urumqi has been mainly affected by the northwest air flow from the inland of Asia, with sufficient sunshine time and less precipitation (annual precipitation is 310.8 mm). Hohhot is located in the northern part of China, in the midtemperate inland area. The Daqingshan Mountain in the north is a mountainous terrain, and the Tumochuan plain in the south. The terrain gradually slopes from north to south. Hohhot has a temperate continental climate. In spring and summer, it is dry and windy, with drastic changes in cold and warm. In autumn and winter, it is cold and dry, with plenty of sunshine. In addition, the annual precipitation is low (335.2-534.6 mm), and the precipitation is mainly distributed between July and August. Harbin is located in northeastern China, with the Zhangguangcailing branch hills in the southeast, the Xiaoxing'an Mountains in the north, and the Songhua River in the middle. The climate of Harbin is a midtemperate continental monsoon climate. The winter is 3100815 VOLUME 4, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. mainly affected by the northeast monsoon, and it is cold and long. In summer, it is mainly affected by the Pacific Ocean's westward stretch from the north to the southwest, and the formed convective clouds will produce precipitation. The annual average precipitation is 569.1 mm, and the precipitation is mainly concentrated in June-September. Wuhan is located in central China, which is located in the east of the Jianghan Plain and the middle reaches of the Yangtze River. Wuhan has a north subtropical monsoon climate, with the characteristics of abundant rainfall, sufficient heat, cold winter, and hot summer, and four distinct seasons. In Wuhan, the convective cloud is caused by the intersection of the warm and humid air from the southwest and the cold air from the north. This has resulted in sufficient rainfall throughout the year in Wuhan, with annual precipitation ranging from 1150 to 1450 mm. Haikou is located in the south of China, with most of the mountains and hills in the south. Haikou is located in southern China and has a tropical monsoon climate. When the northeast monsoon and typhoon blow inland, topographic clouds are formed in Haikou as the topography rises. It has resulted in abundant rainfall in the region, with an average annual rainfall of 2067 mm, which is mainly distributed in summer and autumn. Mountains (orographic clouds), lakes, and seas (convective clouds) are the main topographic factors for the formation of regional clouds in the target city.

B. DATA DIVISION AND STANDARDIZATION
To utilize the deep learning technique for solar forecasting, the collected data are divided into a training set (2016-2018) and testing set (2019). As the scale of the retrieved data is inconsistent, this may influence the accuracy of the prediction.
Thus, the input time series is normalized according to the Min-Max method and converted to [0,1]. In addition, the denormalization of data is also achieved through the Min-Max method. The calculation method is as follows: (10) where X norm , X, X min , and X max is the normalized, original, minimum, and maximum input values. The solar irradiance prediction framework based on QL-STM is shown in Fig. 5. As can be seen from Fig. 5, the collected data are preprocessed first and then input to the input layer. After passing through the input layer, the data are input into QLSTM, and then passed through the output layer, and finally, the predicted value is obtained. Hence, this is a quantum-classical hybrid approach. For each QLSTM block, its details are shown in Fig. 2. In each QLSTM block, the first sigmoid activation is known as the forget gate that is used to decide whether to update context c t to include the new input. After the sigmoid, the result is multiplied onto c t−1 . The second sigmoid and the tanh activations are known as the input gates that are used to write the new input into the context. The result of the sigmoid and tanh activations is multiplied to either be added to c t−1 , or to ensure that the input is not added to c t−1 . The last sigmoid activation is known as the output gate where the actual prediction is performed. This output is also used to update the hidden layer h t .

C. PARAMETERS DESIGN OF DEVELOPED MODEL
In this study, the developed model is based on Python 3.7, PennyLane software package, TensorFlow 2.0, and Keras 2.3.1 framework. In addition to the fully connected layer, the rectified linear unit [44] is selected as the activation function in all layers. In order to reduce the possibility of overfitting, the regularization method of dropout [45] is applied in each training batch, which ignores a few neurons with a certain probability. The hyperparameters of the model include the dropout value, batch size, and epoch. Their values are optimized by the grid search algorithm and are ultimately set to 0.1, 256, and 150, respectively. Mean absolute error (MAE) is selected as the loss function to measure the approximation between the predicted value P f and actual value P a . Adam (Adaptive moment estimation) [46] is selected as the optimizer, which is founded on the training data iteratively to update the neural network weights. The output obtained from the fully connected layer of the prediction model will be denormalized to obtain the true prediction GHI values after the model completed the prediction.

D. MODEL PERFORMANCE METRICS
As all the models are dealing with a regression problem, the quality of predictive accuracy of each model is evaluated through three basic error metrics, such as RMSE, MAE, and R-Square (R 2 ), which are widely used for predictive analysis.
RMSE: It is used to measure the standard deviation between the predicted GHI and the actual GHI as follows: MAE: It can be explained as the average of absolute error value between the predicted GHI and the actual GHI. It is defined as follows: R 2 : It determines the degree of correlation between predicted GHI and the actual GHI as shown below: (13) where N represents the total number of datasets. P f and P a is the predicted GHI and actual GHI, respectively.

E. PV SYSTEM
A PV system is shown in Fig. 6 . According to the Code for Design of Photovoltaic Power Plants, the PV power generation can be calculated according to the solar energy resources of the site where H A represents the total solar radiation on the horizontal plane (kW · h/m 2 , peak hours), E P indicates the power generation (kW · h), E S is the irradiance under standard conditions (constant=1 kW · h/m 2 ), P A Z represents the installed capacity of the module (kWp), and K represents the comprehensive efficiency coefficient (value 75%-85%).

IV. RESULTS AND DISCUSSION
This section is mainly about the experiments of the QLSTM model under different conditions and the analysis of the experimental results. The data are divided into a training set (2016-2018) and testing set (2019) with an interval of 1 h. To achieve solar irradiance prediction, the input given in all models consists of the meteorological information in the previous 24 h, and the output was the solar irradiance for the next hour. As all the models are dealing with a regression problem, the quality of predictive accuracy of each model is evaluated through three basic error metrics, such as RMSE, MAE, and R 2 , which are widely used for predictive analysis. First, the annual prediction results of all models in five locations were analyzed. Second, in order to demonstrate the influence of seasonal changes on model predictions, the prediction performance of developmental model is calculated for four seasons (spring, summer, autumn, and winter). Finally, in order to explore the influence of different months on the prediction performance of the model, we evaluated the model performance under different months.

A. MODEL ANNUAL FORECAST PERFORMANCE EVALUATION
The experimental results of the solar irradiance prediction of each model are presented in Table 3, and the best evaluation values are indicated in bold. The experimental results show that the annual average RMSE of the QLSTM model is 61.756 W/m 2 , compared with other models, it decreased by 10.7%, 13.9%, 8.1%, 3.8%, and 3.4%, respectively, the annual average MAE is 24.257W/m 2 , which is reduced by  28.1%, 28.9%, 24.1%, 12.2%, and 12.8%, respectively, and the annual average R 2 is 0.946, which is improved by 1.5%, 1.9%, 1.2%, 0.4%, and 0.4%, respectively. The observed values of solar irradiance were compared with the forecasted values in Urumqi, as shown in Fig. 7. It can be seen from Fig. 7 that the predicted value of QLSTM is almost consistent with the observed value curve.
As described in Section III-A, significant changes in model performance may be attributed to geographic and climatic differences in all studied cities. The bad performance at Haikou can be interpreted by the diversity climatic changes and topography of the region. Haikou is the southernmost capital city of China with the lowest dimension and the highest temperature. Haikou is close to the sea and the air humidity is relatively high. Therefore, it is easy to form clouds at high altitude, which affects the accuracy of the prediction model. In Wuhan, the southeast monsoon from the Indian Ocean and the South China Sea was blocked by the mountains at the junction of Hubei and Sichuan provinces, which lead to the rapid formation of convective and topographical clouds. This makes it particularly difficult for models to forecast. As for Harbin, which shows relatively bad performance, it is the northernmost metropolis with the highest latitude and lowest temperature in China. Harbin is affected by the Pacific southwest air flow in summer and controlled by the Mongolian northwest air flow in winter. This is the main reason for precipitation and snowfall, which is relatively difficult for models to predict. Due to its distance from the ocean, Hohhot and Urumqi are mainly affected by the northwest air flow rather than the marine monsoon. This causes convective clouds less in these two regions, and the prediction difficulty of the models is relatively small compared with the other three regions. As Urumqi is surrounded by mountains on three sides, it blocks part of the northwest airflow. The sky conditions of Urumqi are relatively stable than Hohhot, which lead to more accurate prediction results. In general, the QLSTM model has higher annual prediction performances than other comparable models, which proves that adding VQCs to LSTM has a positive effect of extracting the temporal correlation of data.

B. MODEL SEASONAL PREDICTION PERFORMANCE EVALUATION
Based on the annual performance, in this section, we study the influence of seasonal changes on model performance. The prediction results of the studied models for four seasons (spring, summer, autumn, and winter) in five cities are shown in Table 4. For each season, the best evaluation values obtained in all models are indicated in bold. According to the empirical results shown in the table, the proposed QLSTM achieved excellent results in terms of RMSE, MAE, and R 2 .
In general, our model shows a stable performance of all models in different seasons, among 60 sets (i.e., five sites, four seasons, and three performance metrics) of results achieved the best performance in 58 groups. Furthermore, the solar prediction ability of QLSTM is slightly better than LSTM because QLSTM embeds VQC in LSTM, which enables LSTM to obtain more abundant features when processing memory information.
Besides, in order to better visualize the performance of the proposed QLSTM model and other comparable models, taking Wuhan as an example, we plotted the prediction curves of representative dates in four seasons, as shown in Fig. 8. It can be seen from Fig. 8 that the QLSTM proposed by us can fit the observed values well. Wuhan has a north subtropical monsoon climate with abundant rainfall, sufficient heat, rain, and heat in the same season, light and heat in the same season, cold in winter and hot in summer, and four distinct seasons. The prediction curve is consistent with the climatic characteristics of Wuhan.

C. MODEL PREDICTION PERFORMANCE EVALUATION UNDER DIFFERENT MONTHS
Moreover, the month prediction error of the GHI model in Table 5 was calculated to explore the impact of month alternation on the prediction results. The evaluation values of developed models for 12 months from January to December in all cities was given in Table 5, where the evaluation metrics marked in bold are equivalent to the best prediction performance. The results in Table 5 demonstrate the proposed hybrid QLSTM model shows the highest prediction accuracy in 50 of 60 cases (five cities, 12 months, and the metric of R 2 ). The prediction performance of QLSTM model outperforms other developed models in each month in Hohhot. For Urumqi, only in January and November, the QLSTM model did not get the highest R 2 , and in other months, it got the highest R 2 . For Harbin, QLSTM achieved the highest R 2 in 10 months, and LSTM achieved the highest R 2 in two months, which shows that the prediction performance of QLSTM is better than that of LSTM. In Wuhan, the QLSTM model achieved the highest R 2 in eight months, LSTM achieved in one, GRU achieved in two, and RNN achieved in three. Compared with SARIMA and CNN models, RNN series and   its variants have better processing capacity for time series. In RNN series models, the performance of the QLSTM model is better than that of other models. In Hainan, the QLSTM obtained 10 highest R 2 , LSTM obtained four highest R 2 . This is due to the tropical monsoon climate in Hainan, which has an obvious rainy season and less rainy season, leading to large fluctuations in solar radiation and reduced prediction accuracy of the model. Figs. 9 and 10 show the monthly values of RMSE and MAE. It can be seen from Fig. 9 that the RMSE of QLSTM has achieved the lowest value of 50 out of 60 groups of results. At the same time, it can be seen from Fig. 10 that the MAE value of QLSTM is the lowest in 59 groups of 60 results. All in all, the best prediction performance still belongs to the proposed hybrid QLSTM model irrespective of months and cities.

D. COMPARISON OF LSTM AND QLSTM
This article is a follow-up study based on the paper titled "An LSTM short-term solar irradiance forecasting under complicated weather conditions." Table 6 shows the comparison of the two papers. This article proposes a high-precision hybrid prediction model for solar irradiance based on VQC and LSTM networks. The proposed model embedded VQC in LSTM to iteratively optimize the weight parameters of four gates, namely, forgetting gate, input gate, cell state, and output gate, to improve the forecast accuracy. The performance of the proposed model is evaluated and compared with widely used models, including SARIMA, CNN, RNN, GRU, and LSTM, under different seasons and months using data from five solar radiation observatories in China. Results show that the proposed model significantly outperforms other models, with reductions in RMSE and MAE ranging from 3.4% to 28.9% and 10.7% to 28.1%, respectively, and improvements in R 2 ranging from 0.4% to 1.9%. The proposed model can enhance the security of grid operation by ensuring the scheduling and dispatching of electrical power and designing a PV-integrated energy management system. Overall, this article provides a promising approach for improving the accuracy of solar irradiance predictions.
However, that paper focuses on predicting solar irradiance under complicated weather conditions. The proposed approach improves the accuracy of predictions on cloudy days. This article introduces the use of a clearness index to classify the type of weather during data processing, which helps to improve the accuracy of predictions. The performance of different models, including BPNN and SVR, is compared, and the results show that the LSTM-based approach is more accurate. The proposed approach is demonstrated to be generalizable across different regions through a cross-regional study. Overall, this article makes a valuable contribution to the field of PV forecasting methods and provides insights for future research in this area.

V. CONCLUSION
This article proposed a hybrid quantum LSTM model for one-hour-ahead solar irradiance prediction, which combines VQC and LSTM to obtain richer time-dependent information in meteorological and solar radiation data. Different meteorological data consisting of solar radiation, temperature, pressure, relative humidity, wind speed, cloud type, dew point, and solar zenith angle related to solar irradiance are considered to improve the prediction accuracy of the QL-STM model. The performance of the QLSTM model is evaluated based on the RMSE, MAE, and R 2 results and compared with other existing models such as SARIMA, CNN, RNN, GRU, and LSTM. The experimental results show that the annual average RMSE of the QLSTM model is 61.756 W/m 2 , which is reduced by 10.7%, 13.9%, 8.1%, 3.8%, and 3.4%, respectively, compared with other models; the annual average MAE is 24.257 W/m 2 , which is reduced by 28.1%, 28.9%, 24.1%, 12.2%, and 12.8%, respectively, compared with other models; the annual average R 2 is 0.946, which is improved by 1.5%, 1.9%, 1.2%, 0.4%, and 0.4%, respectively, compared with other models. Our work provides a reference for the application of quantum computing in the field of renewable energy. In the future, we will try to implement this predictive system in a NISQ computer. Generally, the implementation of quantum algorithms on NISQ computers can provide valuable insights into the behavior of quantum systems and help identify practical applications for these devices. However, the performance obtained may be limited by the noise and errors of the system, and achieving useful results may require significant optimization and error correction techniques.