Hourly Photovoltaic Power Forecasting Using Exponential Smoothing: A Comparative Study Based on Operational Data

Abstract

The accurate forecasting of solar power generation is becoming increasingly important in the context of renewable energy integration and intelligent energy management. The variability of solar radiation, caused by changing meteorological conditions and diurnal cycles, complicates the planning and control of photovoltaic systems and may lead to imbalances in supply and demand. This study aims to identify the most effective exponential smoothing approach for real-world PV power forecasting using actual hourly generation data from a 9 MW solar power plant in the Kyiv region, Ukraine. Four exponential smoothing techniques are analysed: Classic, a Modified classic adapted to daily generation patterns, Holt’s linear trend method, and the Holt–Winters seasonal method. The models were implemented in Microsoft Excel (Microsoft 365, version 2408) using real measurement data collected over six months. Forecasts were generated one hour ahead, and optimal smoothing constants were identified via RMSE minimisation using the Solver Add-in. Substantial differences in forecasting accuracy were observed. The Classic simple exponential smoothing model performed worst, with an RMSE of 1413.58 kW and nMAE of 9.22%. Holt’s method improved trend responsiveness (RMSE = 1052.79 kW, nMAE = 5.96%), but still lacked seasonality modelling. Holt–Winters, which incorporates both trend and seasonality, achieved a strong balance (RMSE = 1031.00 kW, nMAE = 3.7%). The best performance was observed with the modified simple exponential smoothing method, which captured the daily cycle more effectively (RMSE = 166.45 kW, nMAE = 0.84%). These results pertain to a one-step-ahead evaluation on a single plant and an extended validation window; accuracy is dependent on meteorological conditions, with larger errors during rapid cloud transi. The study identifies forecasting models that combine high accuracy with structural simplicity, intuitive implementation, and minimal parameter tuning—features that make them well-suited for integration into lightweight real-time energy control systems, despite not being evaluated in terms of runtime or memory usage. The modified simple exponential smoothing model, in particular, offers a high degree of precision and interpretability, supporting its integration into operational PV forecasting tools.

1. Introduction

Photovoltaic (PV) generation has become a cornerstone of global efforts to decarbonise the energy sector, and many countries have significantly expanded installed PV capacity to reduce greenhouse gas emissions and enhance energy security. As the share of solar energy in the electricity mix increases, the inherent variability and intermittency of PV power pose operational challenges for power systems. In this context, accurate short-term forecasting of PV output is essential for maintaining grid stability, demand–supply balancing, scheduling reserves, and optimising energy storage systems. Moreover, modern deployment configurations—such as bifacial PV system and agrivoltaics—introduce additional albedo- and geometry-driven effects (e.g., back-side irradiance, ground-cover management, row spacing, and tracker geometry), that amplify short-term variability and further strengthen the need for reliable forecasting [1]. In parallel, recent studies of national energy-consumption forecasting under wartime conditions has been studied for Ukraine highlight the importance for robust and adaptable forecasting models in crisis scenarios [2].

Prior works [3,4] highlights the growing role of micropower systems based on renewables in strengthening power system resilience through distributed generation, storage, and intelligent control. At the community scale, an energy balance study for Smila (Ukraine) reported that raising the renewable share from 55% to 69% while reducing electricity imports by 30% produced a 3.5-fold decrease in CO₂ emissions [5], underscoring the value of forecasting models that align variable renewable output with daily load fluctuations in decentralised systems.

Hybrid PV–wind–H₂ architectures further illustrate settings where accurate PV power forecasting directly informs energy management system (EMS) decisions [6,7]—electrolyser dispatch, storage operation, curtailment control—making even computationally simple models (such as modified exponential smoothing approach considered in this study) relevant for real-world deployment.

While solar irradiance is a primary driver of PV power [8]. Many forecasting approaches model irradiance explicitly as an intermediate step [9,10], this study focuses on forecasting the PV power. In contrast, we frame the task as direct PV power forecasting in a time series settings, which integrates naturally into operational EMSs without a separate irradiance model and supports efficient, scalable, and grid-responsive control.

Methods for short-term PV power forecasting span numerical weather prediction, hybrid machine learning techniques, and statistical time series models [11,12,13,14]. In practice—especially under constraints of limited data, modest hardware, or rapid deployment—statistical models remain a strong alternative.

As grid-connected PV capacity expands, particularly in regions where solar penetration exceeds 20%, the economic and operational costs of forecast error grow. Balancing costs attributable to solar variability can reach up to approximately 15% of dispatch expenditures, motivating more precise and responsive forecasting methodologies [15,16,17,18,19].

Ultra-short-term horizons are particularly challenging due to transient clouds and atmospheric disturbances that can induce hundreds of kilowatts of power swing within minutes, directly affecting grid stability and reserve allocation strategies [20,21].

Integrating all-sky imagery and advanced textural convolution techniques can reduce forecasting errors during cloudy conditions by up to approximately 30%, evidencing the value of visual data [20,21,22].

Recent advances also include hybrid CNN–LSTM architectures, where convolutional layers extract spatial features (e.g., from sky images) and LSTM layers model temporal dynamics; the canonical multi-head CNN–LSTM structure is described in detail in [23]. Related hybrid pipelines used in photovoltaic forecasting, such as wavelet-LSTM and other time-frequency/deep learning combinations, have also shown improved accuracy [24,25,26].

In particular, time–frequency domain approaches that incorporate wavelet decomposition alongside LSTM structures have achieved normalised RMSE reductions exceeding 25% compared to standalone exponential smoothing models [25,27]. Additionally, transfer learning frameworks based on foundation models have demonstrated promising results in zero-shot forecasting for newly commissioned PV installations, with accuracy improvements reaching 70% in early-stage deployments [28].

From an operational perspective, models must balance accuracy with computational efficiency and interpretability. Edge-computing platforms indicate that hybrid statistical and machine learning ensembles can meet sub-minute update requirements on modest hardware, supporting both centralised grid operations and local EMSs [16,17,18,19].

The use of auxiliary data sources—such as satellite-derived irradiance and all-sky imaging—further strengthens model robustness under variable meteorological conditions. These multimodal approaches reflect a broader shift toward domain-aware forecasting frameworks that are well-suited to the evolving demands of solar energy integration [20,21,22,23,29].

While advanced machine learning techniques continue to push the boundaries of forecasting accuracy, their complexity and resource demands often limit their applicability in embedded or real-time control environments. In many operational contexts—particularly those constrained by hardware limitations, deployment speed, or the need for model interpretability—statistical methods remain indispensable. Among these, exponential smoothing techniques have retained their relevance due to their simplicity, transparency, and adaptability to high-frequency data streams [30].

A comprehensive review [31] emphasised the continued relevance of exponential smoothing methods in forecasting applications where computational efficiency and transparency are paramount. The study identified simple exponential smoothing (SES), Holt’s linear method, and Holt–Winters’ seasonal method as competitive options under stable operational conditions, especially for embedded or resource-constrained systems. Similarly, ref. [32] categorised exponential smoothing as a core family of statistical methods suitable for rapid deployment in real-time grid environments. These models remain widely used in forecasting both load and generation, particularly where high-frequency data and fast forecast updates are required.

Exponential smoothing methods generate forecasts as weighted averages of past observations, with more recent values given higher weight. Depending on the structure of the underlying data, different model variants are applicable: SES for level prediction, Holt’s method for linear trends, and Holt–Winters for seasonal time series. These models have been successfully applied across a wide range of energy forecasting tasks, including electricity consumption, wind energy output, and PV power.

In [33], authors validated the practical value of exponential smoothing techniques, particularly the Holt–Winters method, in short-term load forecasting under variable socio-economic conditions. Their study compared the performance of Holt–Winters and Prophet algorithms for forecasting hourly electricity demand in Houston, Texas, during the COVID-19 pandemic. Despite the simplicity of the Holt–Winters approach, the authors demonstrated its robustness and reliability across multiple intra-day forecasting intervals. While Prophet provided marginally better generalisation, Holt–Winters was praised for its simplicity and ease of implementation, especially in embedded control systems. Furthermore, study [34] extended the application of exponential smoothing to probabilistic forecasting in the Australian National Electricity Market, using it to construct prediction intervals for solar and wind generation. The study demonstrated that exponential smoothing could match or exceed the performance of more complex approaches, such as quantile regression and ARCH/GARCH models, in generating reliable confidence bounds for renewable generation.

Although exponential smoothing methods may lack the sophistication of deep learning algorithms, they remain competitive in many practical contexts, particularly when forecasting performance must be balanced against implementation constraints. Their inherent interpretability and ability to respond to dynamic trends make them an attractive choice for real-time energy forecasting in operational environments.

The goal of the paper is to conduct a comparative analysis of four exponential smoothing techniques for PV power forecasting—the classic SES model, a modified SES approach adapted to capture diurnal patterns, Holt’s linear trend method, and the Holt–Winters seasonal method—and to evaluate their forecasting accuracy using actual hourly generation data, ultimately identifying the most suitable method for practical PV power forecasting applications where model transparency, ease of implementation, and computational efficiency are prioritised. This work makes the following contributions:

• We develop a modification of Simple Exponential Smoothing tailored to hourly PV power with an explicit treatment of the diurnal pattern, improving short-term forecasts while preserving interpretability.
• We benchmark the modified SES against classic SES, Holt’s linear method, and multiplicative Holt–Winters on real operational hourly data from a 9-MW PV plant (Kyiv region, Ukraine).
• We conduct a detailed seasonal performance analysis (winter, spring, summer), demonstrating the adaptability of each model under varying irradiance conditions and confirming the robustness of the modified SES in both stable and transitional periods.
• We provide a transparent, Excel-based implementation of all models with smoothing parameters optimised via validation RMSE, making the methodology reproducible and suitable for low-resource or embedded environments.
• We discuss the applicability and potential limitations of the proposed and benchmarked models in other geographic and operational contexts—including systems with different installed capacities (e.g., residential-scale PV), diverse latitudes, and climate regimes—highlighting the generalisation potential of the modified SES under various irradiance conditions.

2. Data Description

This study utilised real-world data obtained from a ground-mounted photovoltaic power plant (PVPP) located in Velyka Dymerka, Kyiv region, Ukraine. The facility has an installed capacity of approximately 9 MW and operates in a grid-connected configuration. The dataset comprises active power output measurements recorded at 10 min intervals over a continuous seven-month period, from 1 January to 1 July 2021.

Given the high temporal resolution and large volume of the raw data, a normalisation step was applied by aggregating the 10 min measurements into hourly averages. This transformation reduced the dimensionality of the dataset while preserving essential temporal characteristics, facilitating analysis and aligning with the operational requirements of EMSs. For each hour h, the average active power output was computed as:

$${\overline{\mathrm{P}}}_{\mathrm{h}}={\displaystyle \frac{1}{6}}{\displaystyle {\sum }_{\mathrm{i}=1}^6{\mathrm{P}}_{\mathrm{i}}},$$

(1)

where ${\overline{\mathrm{P}}}_{\mathrm{h}}$ denotes the average PVPP power outputs for hour h, ${\mathrm{P}}_{\mathrm{i}}$ represents the PVPP power output measured at the i-th 10 min interval within hour h, i is the index of the six 10 min intervals in each hour.

The resulting time series captures both intra-day (diurnal) and inter-day (seasonal) variability typical of PV systems in mid-latitude regions. Due to the solar cycle and geographic location, the PVPP exhibited zero or near-zero generation during nighttime hours, typically spanning approximately six hours per day, resulting in nearly half of the dataset containing zero values. These periods of inactivity alternate with clear and sharply defined daytime generation profiles, as illustrated in Figure 1 and Figure 2.

Figure 1 illustrates the hourly PVPP power output for three representative months—January (a), April (b), and July (c)—based on the cleaned and aggregated dataset. This comparison highlights the pronounced temporal variability and seasonal characteristics of PV power. In January, the PVPP generates low and irregular energy outputs, with many days displaying near-zero generation. Short daylight hours and frequent winter cloud cover contribute to the limited generation. Peak values rarely exceed 5500 kW, and daily profiles are sparse and inconsistent, highlighting the challenge of forecasting under winter conditions. By April, the generation profile becomes more defined and stable. The days are longer, and daily peak outputs frequently range between 6000 kW and 8000 kW. The curves exhibit greater consistency, reflecting improved insolation and fewer weather-related disruptions. The typical bell-shaped diurnal patterns begin to emerge, signifying a seasonal shift toward more predictable generation. In July, the PV power reaches its highest levels and greatest regularity. The plant operates close to its nominal capacity, with many days showing smooth, symmetric diurnal curves. The hourly peaks are consistently above 8000 kW, with minimal variability from day to day. This stability reflects optimal solar irradiance and clear-sky conditions characteristic of midsummer.

Figure 2 offers a comparative view of the same months as Figure 1, but with data overlaid on a single plot. The overlapping curves highlight the monthly differences in both magnitude and duration of daily generation. This representation reinforces the visual understanding of seasonal amplitude shifts and underlines the importance of capturing both intra-day structure and inter-month transitions in forecasting models.

Together, these graphs demonstrate a clear evolution from low, variable winter output to high, stable summer generation. The seasonal pattern emphasises the importance of capturing trend and seasonality in forecasting models. Given the observed regularity and structure in the summer months and the noisier, intermittent behaviour in winter, the time series is well-suited for exponential smoothing methods that can adapt to both persistent trends and recurring diurnal cycles.

Figure 3 presents the average daily PV power profiles by hour of day for each month from January to July. Each curve represents the mean hourly active power output aggregated over all days within the corresponding month. This figure highlights the distinct diurnal nature of solar generation and its seasonal modulation in both intensity and duration.

Across all months, the generation profile follows a consistent unimodal shape, with zero production during the night, followed by a rapid rise in the early morning, a peak around solar noon, and a gradual decline towards evening. However, the shape and magnitude of these profiles vary significantly by season. In the winter months (January and February), generation is constrained to a narrow time window between approximately 8:00 and 16:00, with modest peak values not exceeding 2000 kW. This limited output reflects the combination of short daylight periods, low solar elevation angles, and potentially frequent cloud cover. March and April show notable increases in both the peak power and the width of the daily curve, indicating longer days and improved irradiance conditions. The generation window extends from roughly 7:00 to 18:00, with peak values ranging between 4000 and 6000 kW. By late spring and early summer (May to July), the generation profile becomes broader and more symmetric, with the earliest onset and latest cutoff of production. In these months, peak output frequently exceeds 7000 kW, and the system remains at high generation levels for several consecutive hours around midday. These months correspond to the most favourable solar conditions, characterised by high irradiance and minimal atmospheric interference.

The evolution of the curves in Figure 3 demonstrates both intra-day regularity and inter-month variability of PV power. The pronounced seasonal shift in the height and width of the profiles underscores the necessity of incorporating both trend and seasonality into forecasting models.

Figure 4 visualises the evolution of daily average PV power output throughout the observation period, highlighting both the seasonal trajectory and the variability in daily generation within each month. Unlike aggregated hourly profiles (Figure 3), this representation captures the net energy yield of each day, revealing how short-term fluctuations and broader climatic trends shape PVPP performance.

In the early months (January and February), daily generation values are generally low, and the curve is characterised by high irregularity. Sharp transitions between near-zero and moderate-output days suggest a strong influence of atmospheric variability, such as passing clouds or snowfall, which can significantly diminish irradiance under already constrained winter conditions. Notably, some days achieve brief peaks around 1000 kW, but these are interspersed with extended periods of negligible production. As advancing into spring (March and April), the Fig. reveals a dual effect: an upward shift in the baseline generation and a reduction in volatility. This period marks the transition from erratic to more sustained power output, reflecting longer daylight hours and a gradual stabilisation of weather patterns. However, periodic declines remain visible, underscoring the persistence of intermittent meteorological disruptions.

The most stable and elevated performance is evident from May onward. Both June and July display tightly clustered data points in the upper range of the vertical axis, with daily averages frequently exceeding 3000 kW. This concentration suggests favourable and consistent solar conditions over prolonged intervals, allowing the PVPP to operate near its design capacity. Yet, even in these peak months, a few outliers indicate the influence of localised weather events or transient shading.

What makes this visualisation particularly valuable is its capacity to reflect inter-day reliability and generation continuity—metrics that are crucial for energy planning, especially in systems with high PV penetration. The gradual flattening of fluctuations from winter to summer demonstrates the increasing predictability of solar output, which has direct implications for forecast model training, resource allocation, and grid integration strategies.

In summary, Figure 4 provides evidence of a strong seasonal signal coupled with diminishing short-term variability over time. It supports the conclusion that forecasting models must be capable not only of modelling trends and seasonality but also of responding to non-systematic deviations in daily energy yield caused by short-lived weather events.

3. Data Preprocessing

Before applying forecasting methods, the raw dataset was subjected to a comprehensive preprocessing procedure to ensure data quality, temporal consistency, and suitability for time series analysis. A visual overview of the preprocessing workflow is presented in Figure 5.

The diagram illustrates the sequential steps applied to clean and prepare the PV time series data before forecasting, including covering aggregation, anomaly removal, timestamp validation, and sorting.

The original time series, recorded at 10 min intervals, was first aggregated into hourly averages. This step reduced the dimensionality and retained the essential temporal dynamics relevant to PV power.

The preprocessing workflow included several key steps. Missing or incomplete records were identified and removed to eliminate discontinuities. Spurious spikes in active power output—typically caused by the sensor noise, inverter fluctuations, or communication errors—were detected through visual inspection and filtered using empirically defined upper bounds based on system performance. Additionally, the dataset was examined for duplicate entries and inconsistent time stamps, which were corrected to maintain uniform hourly intervals and strict chronological order.

Unlike in machine learning tasks involving gradient-based optimisation or neural networks, normalisation was not applied in this study. The exponential smoothing methods used in this analysis—such as simple, double, and triple smoothing—are scale-independent and operate directly on the raw magnitudes of the time series. Normalisation could reduce interpretability in the context of physical units (e.g., kW) and was therefore intentionally avoided to maintain the clarity and traceability of forecasted values in real operational terms.

To reconstruct short, isolated missing values that remained after initial filtering, linear interpolation was applied at the final stage of preprocessing. This method estimates a missing value based on its nearest valid neighbours in time. In this context, the horizontal axis t represents time (in hours), and the vertical axis P denotes the active power output. The interpolated value P(t) was estimated using Equation (2):

$$\mathrm{P}\left(\mathrm{t}\right)={\mathrm{P}}_1+{\displaystyle \frac{\left(\mathrm{t}-{\mathrm{t}}_1\right)\left({\mathrm{P}}_2-{\mathrm{P}}_1\right)}{{\mathrm{t}}_2-{\mathrm{t}}_1}},$$

(2)

where P(t) is the interpolated power at time t, and (t₁, P₁), (t₂, P₂) are the nearest valid observations before and after the missing time point.

Since only five values were missing across the entire dataset, the overall share of missing data was statistically negligible. This further justified the use of a simple and transparent interpolation method over more complex approaches. Linear interpolation was therefore selected for its robustness, interpretability, and adequacy in restoring isolated gaps without distorting the overall signal dynamics. Given the hourly resolution and the small proportion of missing values, this method was considered both efficient and sufficiently accurate.

As a result, a clean and consistent dataset was obtained, accurately reflecting both intra-day and seasonal dynamics in PV power.

4. Forecasting Models

4.1. Simple Exponential Smoothing Method

SES is a widely used forecasting method suitable for time series data that fluctuates around a relatively constant mean, without trend or seasonal effects. It is particularly effective in short-term forecasting tasks where the underlying process is stable or changes gradually.

SES generates forecasts by applying exponentially decreasing weights to past observations, with the most recent data having the strongest influence on the forecast. The fundamental idea is that older data points carry less information about the future, and their impact should decay over time.

The basic SES formula is given by Equation (3) [30]:

$${\widehat{\mathrm{P}}}_{\mathrm{t}+1}=\mathsf{\alpha }{\mathrm{P}}_{\mathrm{t}}+\left(1-\mathsf{\alpha }\right){\widehat{\mathrm{P}}}_{\mathrm{t}-1},$$

(3)

where ${\widehat{\mathrm{P}}}_{\mathrm{t}+1}$ is the forecasted power at time t + 1, ${\mathrm{P}}_{\mathrm{t}}$ is the actual observed value at time t, ${\widehat{\mathrm{P}}}_{\mathrm{t}-1}$ is the forecast made for time t − 1, $\mathsf{\alpha }\in \left[0,1\right]$ is the smoothing constant controlling the weighting.

This decay mechanism enables the method to respond to recent changes while maintaining stability in the presence of noise. A low α value results in strong smoothing, making the model slow to respond to changes, while a high α value increases sensitivity to recent fluctuations.

In the context of PV power, SES is best applied under conditions of stable irradiance, such as clear-sky days with minimal meteorological variability. However, it cannot capture long-term upward or downward trends or seasonality, which are typical in PV power due to diurnal and annual cycles.

From a comparative evaluation standpoint, ref. [35] demonstrated that SES performed well in terms of execution time and basic accuracy metrics (RMSE, MAE), especially under low-complexity scenarios. However, its predictive power diminished in the presence of structural trends, as evidenced by negligible or undefined R² values and zero correlation in datasets with large variability. Similarly, ref. [36] emphasised that the performance of SES strongly depends on the proper selection of the smoothing constant α. This parameter is typically optimised by minimising error metrics. For instance, their study showed that α values in the range of 0.26–0.29 yielded excellent accuracy (MAPE < 10%) when forecasting electricity production data.

Its comparative simplicity and interpretability make it a useful benchmark, especially in systems where computational constraints are present or model transparency is required.

4.2. Modified Simple Exponential Smoothing Method

While classic SES provides a baseline approach for time series forecasting without trend or seasonality, it may not adequately reflect the repetitive daily patterns observed in PV power. To address this, a modified version of SES is proposed by the authors, designed specifically for PV power prediction by incorporating historical data from previous days for the same hour.

This modification extends the SES framework by averaging actual or forecasted values from the same hour of the previous day, thus embedding diurnal structure directly into the forecast. The modification is particularly suitable for time series exhibiting strong daily regularity but limited long-term trend, as in the case of hourly PV power under relatively stable weather conditions.

The method forecasts the active power output P at a given hour h and day d by incorporating the actual or previously forecasted value from the same hour on the preceding day. The updated smoothing equation is defined as:

$${\widehat{\mathrm{P}}}_{\mathrm{d}+1,\mathrm{h}}=\mathsf{\alpha }{\mathrm{P}}_{\mathrm{d},\mathrm{h}}+\left(1-\mathsf{\alpha }\right){\widehat{\mathrm{P}}}_{\mathrm{d}-1,\mathrm{h}},$$

(4)

where ${\widehat{\mathrm{P}}}_{\mathrm{d}+1,\mathrm{h}}$ is the forecasted power for day d + 1 and hour h, ${\mathrm{P}}_{\mathrm{d},\mathrm{h}}$ is the actual observed power at the same time, ${\widehat{\mathrm{P}}}_{\mathrm{d}-1,\mathrm{h}}$ is the forecasted power for the previous day at the same hour, $\mathsf{\alpha }\in \left[0,1\right]$ is the smoothing constant controlling the influence of recent values.

This structure preserves the simplicity of the original SES model while enhancing its temporal consistency by embedding knowledge of recurrent daily cycles. It improves adaptability to short-term fluctuations in solar irradiance due to transient weather conditions, while remaining computationally efficient and interpretable.

The proposed modification is especially beneficial when actual measurements are delayed or unavailable, as it allows forecasts to rely on previous estimations without disrupting continuity. The method maintains a balance between accuracy and efficiency, making it well-suited for deployment in local EMSs or real-time control environments.

4.3. Holt’s Exponential Smoothing Method

Holt’s exponential smoothing method extends the classic SES approach by explicitly incorporating a trend component. This allows the model to forecast time series that exhibit a systematic increase or decrease over time, behaviour frequently observed in PV power due to gradually changing irradiance during the day or seasonal transitions.

The method maintains two recursive components:

• Level (L_t): captures the smoothed value of the series at the current time.
• Trend (T_t): estimates the rate of change from one time step to the next.

The power output forecast ${\widehat{\mathrm{P}}}_{\mathrm{t}}$ for k time steps ahead is computed as [37]:

$${\mathrm{L}}_{\mathrm{t}}=\mathsf{\alpha }{\mathrm{P}}_{\mathrm{t}}+\left(1-\mathsf{\alpha }\right)\left({\mathrm{L}}_{\mathrm{t}-1}+{\mathrm{T}}_{\mathrm{t}-1}\right),$$

(5)

$${\mathrm{T}}_{\mathrm{t}}=\mathsf{\beta }\left({\mathrm{L}}_{\mathrm{t}}-{\mathrm{L}}_{\mathrm{t}-1}\right)+\left(1-\mathsf{\beta }\right){\mathrm{T}}_{\mathrm{t}-1},$$

(6)

$${\widehat{\mathrm{P}}}_{\mathrm{t}+\mathrm{k}}={\mathrm{L}}_{\mathrm{t}}+\mathrm{k}{\mathrm{T}}_{\mathrm{t}},$$

(7)

where L_t is the smoothed level component at time t, T_t is the trend estimate at time t, ${\widehat{\mathrm{P}}}_{\mathrm{t}+\mathrm{k}}$ is the forecast for k hours ahead, $\mathsf{\alpha },\mathsf{\beta }\in \left[0,1\right]$ are smoothing constants for the level and trend components, respectively.

In the context of PV forecasting, Holt’s method is particularly effective in capturing intra-day ramp-up in the morning and ramp-down in the evening, especially on clear-sky days. It is also useful during transitional months such as early spring or late summer, when solar altitude changes produce gradual variations in hourly power output. Unlike SES, which assumes a stationary mean, Holt’s method dynamically adapts to persistent changes in power generation. This makes it more suitable for periods with a systematic trend in solar irradiance. However, its inability to model repetitive intra-day patterns limits its performance during months characterised by strong daily seasonality.

From a practical standpoint, Holt’s method offers favourable computational efficiency. This is crucial for real-time applications or deployment on embedded systems. Still, the quality of predictions heavily depends on the appropriate tuning of α and β. These parameters control the model’s responsiveness to recent changes—higher values enable faster adaptation, while lower values yield smoother, more stable trend estimations.

A recent study [38] successfully applied Holt’s method to model mid-term trends in Ukraine’s renewable energy deployment. Their results demonstrated that the method effectively captured upward capacity growth trajectories for wind, solar, and biomass energy. This evidence supports Holt’s model as a viable tool not only for time series forecasting but also for energy sector planning and scenario development.

In summary, Holt’s exponential smoothing method offers a balanced approach between adaptability and simplicity. Its ability to handle non-stationarity without introducing excessive complexity makes it a strong candidate for forecasting PV power under conditions of evolving irradiance but lacking regular cyclic patterns.

4.4. Holt–Winters Exponential Smoothing Method

The Holt–Winters exponential smoothing method extends Holt’s trend-corrected model by adding a third component—seasonality. This enhancement makes the method particularly suitable for forecasting time series that exhibit both trend and recurring seasonal patterns, such as those observed in PV power.

There are two common variants of Holt–Winters: additive and multiplicative.

The model decomposes the series into three components:

• Level (L_t) is the smoothed estimate of the central tendency.
• Trend (T_t) is the smoothed estimate of the slope.
• Seasonality (S_t): the cyclic pattern repeating every s periods (e.g., daily).

The additive Holt–Winters model is defined by the following recursive Equations (8)–(11) [39]:

$${\mathrm{L}}_{\mathrm{t}}=\mathsf{\alpha }\left({\mathrm{P}}_{\mathrm{t}}-{\mathrm{S}}_{\mathrm{t}-\mathrm{s}}\right)+\left(1-\mathsf{\alpha }\right)\left({\mathrm{L}}_{\mathrm{t}-1}+{\mathrm{T}}_{\mathrm{t}-1}\right),$$

(8)

$${\mathrm{T}}_{\mathrm{t}}=\mathsf{\beta }\left({\mathrm{L}}_{\mathrm{t}}-{\mathrm{L}}_{\mathrm{t}-1}\right)+\left(1-\mathsf{\beta }\right){\mathrm{T}}_{\mathrm{t}-1},$$

(9)

$${\mathrm{S}}_{\mathrm{t}}=\mathsf{\gamma }\left({\mathrm{P}}_{\mathrm{t}}-{\mathrm{L}}_{\mathrm{t}}\right)+\left(1-\mathsf{\gamma }\right){\mathrm{S}}_{\mathrm{t}-\mathrm{s}},$$

(10)

$${\widehat{\mathrm{P}}}_{\mathrm{t}+\mathrm{k}}={\mathrm{L}}_{\mathrm{t}}+\mathrm{k}{\mathrm{T}}_{\mathrm{t}}+{\mathrm{S}}_{\left(\mathrm{t}-\mathrm{s}\right)+\mathrm{k}},$$

(11)

where s is the seasonality length (e.g., 24 for hourly data with daily cycles), k is the forecast horizon, $\mathsf{\alpha },\mathsf{\beta },\mathsf{\gamma }\in \left[0,1\right]$ are smoothing constants for level, trend, and seasonality, respectively.

The multiplicative Holt–Winters model is defined by the following recursive Equations (12)–(15) [40]:

$${\mathrm{L}}_{\mathrm{t}}=\mathsf{\alpha }{\displaystyle \frac{{\mathrm{P}}_{\mathrm{t}}}{{\mathrm{S}}_{\mathrm{t}-\mathrm{s}}}}+\left(1-\mathsf{\alpha }\right)\left({\mathrm{L}}_{\mathrm{t}-1}+{\mathrm{T}}_{\mathrm{t}-1}\right),$$

(12)

$${\mathrm{T}}_{\mathrm{t}}=\mathsf{\beta }\left({\mathrm{L}}_{\mathrm{t}}-{\mathrm{L}}_{\mathrm{t}-1}\right)+\left(1-\mathsf{\beta }\right){\mathrm{T}}_{\mathrm{t}-1},$$

(13)

$${\mathrm{S}}_{\mathrm{t}}=\mathsf{\gamma }{\displaystyle \frac{{\mathrm{P}}_{\mathrm{t}}}{{\mathrm{L}}_{\mathrm{t}}}}+\left(1-\mathsf{\gamma }\right){\mathrm{S}}_{\mathrm{t}-\mathrm{s}},$$

(14)

$${\widehat{\mathrm{P}}}_{\mathrm{t}+\mathrm{k}}=\left({\mathrm{L}}_{\mathrm{t}}+\mathrm{k}{\mathrm{T}}_{\mathrm{t}}\right){\mathrm{S}}_{\left(\mathrm{t}-\mathrm{s}\right)+\mathrm{k}}.$$

(15)

These formulations allow the model to adapt to fluctuations caused by meteorological variability, changes in solar angles, and the daily cycle of irradiance. The seasonal component S_t plays a crucial role in capturing the regular, diurnal rhythm of PV power, where solar power generation rises and falls predictably with the sun’s position.

The method has seen widespread application in both academia and industry. For instance, ref. [37] demonstrated its effectiveness in modelling electricity demand patterns exhibiting strong weekly and monthly cycles. Similarly, in [33] was employed the Holt–Winters method for intra-day load forecasting during the COVID-19 pandemic, showing that it maintained stable performance under volatile conditions. Their findings indicated that the additive model provided more consistent results than ARIMA and LSTM under normal weather conditions, while offering the benefits of lower computational demand and easier interpretability.

Although Holt–Winters is more computationally demanding than SES or Holt’s linear trend model, it still retains a structurally simple and interpretable form, especially when compared to machine learning techniques. At the same time, it offers full transparency regarding the contribution of each component to the forecast.

4.5. Metric for Model Performance Evaluation

To evaluate the forecasting performance of the exponential smoothing methods, the following statistical metrics are employed: the coefficient of determination (R²) (16), rootmean square error (RMSE) (17), normalised RMSE (nRMSE) (18), mean absolute error (MAE) (19), and normalised MAE (nMAE) (20):

$${\mathrm{R}}^2={\displaystyle \frac{{\displaystyle {\sum }_{\mathrm{t}=1}^{\mathrm{n}}{\left(\left({\mathrm{P}}_{\mathrm{t}}-\overline{\mathrm{P}}\right)\left({\widehat{\mathrm{P}}}_{\mathrm{t}}-\overline{\widehat{\mathrm{P}}}\right)\right)}^2}}{\sqrt{{\displaystyle {\sum }_{\mathrm{t}=1}^{\mathrm{n}}{\left({\mathrm{P}}_{\mathrm{t}}-\overline{\mathrm{P}}\right)}^2}}\sqrt{{\displaystyle {\sum }_{\mathrm{t}=1}^{\mathrm{n}}{\left({\widehat{\mathrm{P}}}_{\mathrm{t}}-\overline{\widehat{\mathrm{P}}}\right)}^2}}}},$$

(16)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle {\sum }_{\mathrm{t}=1}^{\mathrm{n}}{\left({\mathrm{P}}_{\mathrm{t}}-{\widehat{\mathrm{P}}}_{\mathrm{t}}\right)}^2}},$$

(17)

$$\mathrm{n}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}{{\mathrm{P}}_{\max }-{\mathrm{P}}_{\min }}}\cdot 100\%,$$

(18)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle {\sum }_{\mathrm{t}=1}^{\mathrm{n}}\left|{\mathrm{P}}_{\mathrm{t}}-{\widehat{\mathrm{P}}}_{\mathrm{t}}\right|},$$

(19)

$$\mathrm{n}\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{\mathrm{M}\mathrm{A}\mathrm{E}}{{\mathrm{P}}_{\max }-{\mathrm{P}}_{\min }}}\cdot 100\%,$$

(20)

where ${\mathrm{P}}_{\mathrm{t}}$ are the actual power values, ${\widehat{\mathrm{P}}}_{\mathrm{t}}$ are the corresponding estimated values, $\overline{\mathrm{P}}$ is the mean of measured power data points, and n is the number of measurements.

In contrast to RMSE, which is influenced by the absolute values of measured power, the normalised RMSE (nRMSE) serves as a scale-independent indicator of forecasting accuracy for power output. This metric evaluates the consistency between predicted and actual power values, with an nRMSE of 100% representing perfect predictive alignment. Lower nRMSE values reflect diminished forecasting precision.

Similarly, the normalised MAE (nMAE) complements nRMSE by expressing the average magnitude of forecasting errors relative to the range of the measured data. While RMSE penalises larger deviations more severely, nMAE provides a more interpretable, absolute measure of average forecast deviation. This metric is particularly useful in comparing model accuracy across systems with different scales of generation. Lower nMAE values indicate improved alignment between forecasted and observed values, especially during periods with moderate or steady production.

Together, reduced RMSE and MAE values, combined with R² values approaching 1, indicate stronger agreement between the predicted and actual power output and thus improved model accuracy in PV power forecasting.

5. Results and Discussion

This section compares the predictive performance of each method based on performance metrics, discusses the influence of model structure and parameter selection, and highlights the strengths and limitations of each approach. Special attention is given to the proposed modification of the SES model, which integrates diurnal characteristics of solar generation and is evaluated against the baseline and more complex smoothing methods.

5.1. Forecasting Setup and Parameter Optimisation

For each model, forecasting was conducted on a cleaned and preprocessed hourly time series representing active power generation from a 9 MW PVPP. To assess forecasting performance and avoid overfitting, the dataset was chronologically divided into training and validation subsets: the training subset included all observations up to 31 March 2021, and the validation subset began on 1 April 2021 and continued until the end of the dataset. Each model was evaluated using a rolling one-step-ahead procedure, whereby forecasts at time t only used information available up to t − 1. This forward-only setup emulates operational forecasting and strictly prevents look-ahead bias. Given the low parameterization of exponential smoothing models relative to their sample size, the risk of overfitting is small, especially compared to high-performance machine learning models that support robust generalisation without significant hyperparameter tuning or regularisation.

Classic SES and modified SES use a single smoothing constant α was tuned within the range [0.0, 1.0]. Holt’s method estimates level and trend with α, β ∈ [0.0, 1.0]. Holt–Winters additionally models seasonality with γ ∈ [0.0, 1.0] and a diurnal seasonal period m = 24 h, consistent with the daily PV cycle. The forecasting horizon was one hour ahead. No data normalisation was applied to preserve the interpretability of the power values in kilowatts.

Parameter tuning was performed exclusively on the training subset. For each model, Microsoft Excel Solver Add-in with the generalised reduced gradient (GRG) nonlinear method minimised the training-window RMSE (MAE monitored as a secondary check) by changing the smoothing constants under bound constraints:

• Classic SES and modified SES: minimise RMSE by changing α ∈ [0,1];
• Holt: minimise RMSE by changing α, β ∈ [0,1];
• Holt–Winters: minimise RMSE by changing α, β, γ ∈ [0,1].

Solver settings: central derivatives; convergence tolerance 10e-4; function “Make unconstrained variables non-negative” enabled; bound constraints enforced on all decision variables. The tuned parameters were then fixed and used to generate rolling one-step-ahead forecasts on the held-out validation window without re-estimation. This forward-only protocol prevents look-ahead bias and keeps the evaluation strictly out-of-sample.

As emphasised in [41], selecting an appropriate smoothing constant α is critical for achieving accurate forecasts using the SES methods. Their study demonstrated that optimising α through mean squared error minimisation notably improves prediction performance. While their application focused on supply chain forecasting, the principles are equally valid for energy systems with repetitive temporal structures, such as PV power. This supports the current study’s choice to calibrate α (and β, γ where applicable) using on the training window of Excel Solver Add-in for each model to minimise RMSE across the validation subset.

5.2. Comparative Analysis of Forecasting Methods

5.2.1. Simple Exponential Smoothing Method

The Classic Simple Exponential Smoothing method was evaluated across a wide range of smoothing constant, α ∈ [0.0, 1.0], in increments of 0.1, to analyse the model’s sensitivity and performance at different responsiveness levels. Forecasts were generated for each α value using the cleaned and aggregated hourly PV power dataset.

Each SES variant used only the previous actual value and the last forecast to compute the next prediction. The model’s simplicity allows it to react to recent changes in data, with the degree of responsiveness governed by the smoothing parameter α.

The performance of each α configuration was visually assessed using plotted forecast profiles and numerically evaluated by calculating RMSE on the validation subset. The optimal value of α was selected based on the configuration that minimised the RMSE, balancing short-term reactivity with noise suppression.

Despite its interpretability and computational simplicity, SES was limited in accurately forecasting the PV time series due to its inability to capture recurring patterns such as daily seasonality or long-term trends. These structural limitations led to notable errors, particularly during transitions between low and high irradiance periods.

5.2.2. Modified Simple Exponential Smoothing Method

The methodological foundation of the modified SES is to enhance the responsiveness of SES to the recurring diurnal patterns typical of PV power. Unlike the classic SES, which bases its forecast exclusively on the previous value and the smoothing level, the modified version also integrates the same-hour prior-day observation, directly capturing the daily cycle of irradiance-driven power.

This approach demonstrated a remarkable capacity to balances responsiveness and stability, particularly around sunrise/sunset, while preserving computational simplicity.

A similar conceptual framework was explored in [42], which proposed two SES-based models for solar irradiance and load forecasting in hybrid energy systems. Their second model, which also used same-hour values from the previous day, resembles our modification. However, there are key methodological differences. In [42] it implemented models on synthetic irradiance generated in the HOMER simulation platform, without evaluation on measured PV power. Authors did not optimise α using an RMSE-based criterion, nor a systematic search over multiple values. Their analysis, though innovative for its time, lacked empirical validation against actual PV power, and the model’s performance was not benchmarked using standard error metrics across multiple scenarios.

In contrast, the present study’s modified SES model was rigorously tested on real operational data and systematically optimised for forecasting accuracy. While the conceptual foundation, in [42], support integrating prior-day data, our methodological rigour, empirical validation, and out-of-sample benchmarking make the proposed variant practically applicable.

At the model level, the first variant in [42] (“previous-hour SES”) is classic SES with no explicit diurnal mechanism, which cannot represent hour-of-day structure and tends to lag during daytime ramps. The second (“same-hour, prior-day SES”) smooths within each hour-of-day subseries, preserving a 24 h pattern but decoupling hours, and maintaining 24 parallel smoothers. Our model, instead, keeps a single SES level on the full stream and applies a fixed diurnal envelope inside the forecast. The envelope is estimated once on the training window and then held fixed (no separate seasonal state). Thus, modified SES captures the hour-of-day structure without fragmenting the series, borrows strength across hours, and preserves SES sensitivity to ramps while the envelope supplies the daily shape.

5.2.3. Holt’s Exponential Smoothing Method

Holt’s method extends the classic SES by introducing a second equation to explicitly model the trend component of the time series. This makes it particularly suitable for capturing gradual increases or decreases in solar generation over time, which often arise due to changing cloud cover, sunrise/sunset transitions, or seasonal shifts in solar angles.

For each pair (α, β), forecasts were generated iteratively across a validation subset, and the corresponding RMSE was calculated.

Visual inspection of forecast profiles confirmed that Holt’s method successfully captured medium-term linear trends in the PV power data, such as ramp-up and ramp-down periods during morning and evening hours. However, the method exhibited limitations in modelling recurring diurnal cycles and abrupt changes in irradiance, resulting in reduced accuracy compared to models with seasonal components.

While Holt’s method provided a significant improvement over classic SES by accommodating trend dynamics, its inability to model seasonality directly limited its performance. Nevertheless, the model remains valuable for forecasting in contexts where linear trends dominate or computational simplicity is a priority.

5.2.4. Holt–Winters Exponential Smoothing Method

Next, the Holt–Winters method was investigated as a natural extension, offering both trend and seasonal components. To ensure optimal performance, both the additive and multiplicative forms were compared, aiming to identify which better aligns with the seasonal dynamics of hourly PV power. As previously shown in Figure 1, the active power time series exhibits strong daily seasonality, with a notable variation in seasonal amplitude. The winter months are characterised by lower daily peaks, while the summer months show higher and broader daily fluctuations. This behaviour is consistent with the physical nature of solar irradiance, which changes seasonally due to astronomical and meteorological factors. To verify the type of seasonality, seasonal decomposition of the time series was performed using both additive and multiplicative models. The decomposition results are presented in Figure 6 and Figure 7, respectively.

The additive Holt–Winters model assumes a constant seasonal amplitude, which does not align with the physical characteristics of PV power. This assumption results in non-uniform residuals (Figure 6) and fails to reflect the observed fluctuations in solar energy output, especially across different seasons. In contrast, the multiplicative Holt–Winters model scales the seasonal component in proportion to the level of the time series, which more accurately captures the underlying dynamics of solar power generation. This model not only reflects seasonal variations more realistically but also adapts to gradual changes in energy output over time. Moreover, it produces residuals with more stable behaviour and lower dispersion (Figure 6), indicating a better fit to the data. Based on this analysis, the multiplicative Holt–Winters model was selected for forecasting, as it provides a more accurate and robust representation of the seasonal and trend behaviour of PV power.

5.2.5. Visual Forecast Comparison

To complement the numerical evaluation, a visual inspection of the forecasted results was conducted. Figure 8 displays a representative three-day period (1–3 April 2021), comparing actual PV output with the predictions generated by each of the four models.

The three-day comparison in Figure 8 reveals distinct performance patterns among the evaluated models, especially in terms of their ability to replicate the diurnal structure of solar generation. Visual inspection of the forecast trajectories highlights how each model responds to sharp irradiance transitions, peak timing, and overnight stability.

On 1 April, the classic SES model exhibits a delayed response to the morning ramp-up and significantly overestimates evening values. Its forecast trajectory lacks the necessary reactivity to track the rapid power increases and decreases typical of clear-sky days. Conversely, the modified SES and Holt–Winters models follow the actual power curve closely, accurately reflecting both the magnitude and shape of the generation peak. This suggests superior adaptability to daily solar patterns.

2 April introduces a day with partially cloudy or variable conditions, as indicated by irregular midday dips in the actual power curve. The Holt’s model responds with excessive sensitivity, producing a fluctuating forecast with visible overshooting. The classic SES again trails behind, lagging during the afternoon peak. In contrast, the modified SES maintains a smoother trajectory, aligning well with the overall trend while dampening short-term noise. Holt–Winters also performs reliably but shows minor deviations during peak instability periods.

On 3 April, the differences become even more pronounced during the midday plateau. While most models capture the steep morning rise with reasonable accuracy, the classic SES continues to underestimate peak generation. Holt’s method overshoots during the noon hours, while the modified SES and Holt–Winters remain consistently close to the actual values. Their ability to track both the amplitude and timing of generation transitions underlines their robustness.

In summary, this visual comparison confirms the conclusions drawn from the statistical evaluation. The modified SES and Holt–Winters models exhibit the highest accuracy and adaptability across a range of irradiance conditions. The classic SES, while computationally simple, fails to capture the dynamics of PV power, particularly under rapidly changing conditions. Holt’s method, although better suited for trend tracking, occasionally produces unstable predictions when faced with nonlinear variability.

Table 1. Forecasting performance metrics for all models.

Model	RMSE, kW	nRMSE, %	MAE, kW	nMAE, %	R2
Classic SES	1413.58	15.35	848.80	9.22	0.41
Holt’s	1052.79	11.43	548.59	5.96	0.94
Holt–Winters	1031.00	11.20	340.99	3.70	0.96
Modified SES	166.45	1.81	77.46	0.84	0.99

presents the comparative results across various accuracy metrics, highlighting significant differences in the predictive performance among the models.

5.3. Seasonal Analysis of Model Accuracy

To evaluate the adaptability of the forecasting models under varying seasonal conditions, the test dataset was divided into three representative periods that reflect typical changes in solar irradiance and PV power dynamics throughout the year:

• Winter period (January–February): this segment corresponds to the months with the lowest levels of solar irradiance, due to reduced daylight duration, low solar altitude angle, and frequent cloud cover. These conditions result in shorter generation windows, high intra-day variability, and increased PV power unpredictability. Forecasting during this period is particularly challenging due to frequent fluctuations, fog, snow coverage, and rapid irradiance changes, which can cause models without robust adaptability to produce significant errors;
• Spring period (March–May): this transitional season is characterised by gradually increasing irradiance levels, longer daylight hours, and greater variability in weather patterns. As atmospheric conditions become more dynamic—due to temperature swings, passing clouds, and intermittent storms—PV power experiences a mix of high-power days and partially clouds events. This period provides a testbed for assessing a model’s ability to handle non-stationary generation behaviour and transition smoothly between winter-type and summer-type profiles;
• Summer period (June–July): these months offer the most favourable and stable solar irradiance conditions, with long days, high sun angles, and generally consistent clear skies. PV power during this time is marked by strong diurnal regularity and predictable peak generation periods, making it ideal for models that leverage daily repetition (e.g., those incorporating seasonal or prior-day logic). However, heatwaves and occasional overproduction curtailments may still introduce nonlinear effects. This period tests the model’s capacity to accurately reproduce consistent patterns without overfitting to specific anomalies.

Seasonal performance analysis is illustrated in Figure 9a (RMSE) and Figure 9b (nMAE), which reveal significant differences in model accuracy across seasonal conditions.

In the winter period (January–February), the PVPP faces extreme forecasting challenges due to low solar altitude, limited daylight hours, frequent overcast skies, and snow accumulation on PV panels. These conditions lead to short and irregular generation windows, often marked by abrupt irradiance drops and erratic ramps during sunrise and sunset transitions. The classic SES model, being structurally limited to past-value smoothing, fails to adapt to these non-repetitive variations. Its RMSE reached 567.28 kW, with nMAE of 2.73%, reflecting systematic lag and overshooting during rapid changes in PV power. In contrast, the modified SES model—incorporating previous-day, same-hour data—performed notably better, achieving 79.93 kW RMSE and only 0.29% nMAE. Its ability to reference typical daily cycles enable it to anticipate irradiance behaviour more effectively, even under variable cloud cover. Holt’s method, by capturing linear trends, reduced RMSE to 385.98 kW, but its trend assumption often led to overreactions to short-term fluctuations, particularly when morning generation rose sharply and then plateaued due to fog or snowfall. The Holt–Winters model, combining trend and seasonality, balanced these effects best in winter, with RMSE of 309.30 kW and nMAE of 1.36%. It successfully adapted to days with clear midday spikes and days with flat, suppressed output, offering smoother forecasts across irregular sequences. During the spring period (March–May), PV power becomes more volatile due to transitional weather patterns: alternating clear and overcast days, passing cumulus clouds, and unstable thermal layers causing solar irradiance flickering. The classic SES model broke down under this instability, producing the highest RMSE of 1518.43 kW and nMAE of 10.47%, often overestimating during cloudy mornings and underreacting during sudden clearings. The modified SES, however, maintained remarkable robustness, with RMSE of 202.21 kW and nMAE of 1.16%, demonstrating its structural advantage in handling semi-regular but fluctuating daily profiles. Its dual-reference mechanism (forecast and prior-day observation) smoothed out noise without erasing pattern predictability. Holt’s method (RMSE 1200.71 kW, nMAE 7.28%) showed difficulty adapting to nonlinear ramps and midday dips, particularly on days with intermittent cloud passages, due to its fixed linear trend assumption. Meanwhile, Holt–Winters offered a balanced forecast with RMSE of 412.40 kW and nMAE 3.30%, tracking seasonal shifts reasonably well. Still, it occasionally lagged during rapid irradiance transitions, especially when early mornings resembled winter profiles, but midday resembled summer conditions—highlighting the limits of fixed-period seasonality in dynamically changing months.

In summer period (June–July), conditions become more stable: high solar angles, clear skies, and predictable diurnal curves dominate, forming the ideal environment for daily-pattern-based models. The modified SES delivered its strongest performance here, with RMSE of 167.82 kW and nMAE of just 0.92%, closely following the generation curve’s amplitude and timing. It successfully captured full solar irradiance ramps, as well as midday plateaus, maintaining accuracy even during partial cloud-induced dips or saturation plateaus near system limits. Holt–Winters also thrived in this environment, reaching RMSE 370.00 kW and nMAE 3.00%, as its multiplicative seasonal component effectively adjusted to amplitude-proportional peaks. However, it occasionally overfit midday peaks or underfit low-generation mornings due to fixed seasonal weights. Holt’s method (RMSE 1227.31 kW, nMAE 8.21%) again struggled with the flat midday outputs, often producing linear over-extensions beyond the actual plateau. The classic SES, although somewhat improved due to lower noise, still underperformed with RMSE 1771.37 kW and nMAE of 13.86%, frequently misaligning the timing and magnitude of generation peaks. It could not adapt to sharp sunrise ramps or midday flattering—common in hot, high-irradiance conditions.

In summary, this seasonal analysis underscores the critical importance of model selection based on seasonal context:

• The modified SES model consistently delivered the most accurate forecasts across all three periods, with exceptionally low error metrics in both spring and summer.
• The Holt–Winters model, especially in its multiplicative form, proved most versatile under variable and low-light conditions, especially in winter, where it outperformed all others except the modified SES.
• Holt’s method was limited by its linear assumptions and was most effective only under moderately trending profiles.
• Classic SES lagged behind in all cases, affirming that over-simplified models lack the flexibility required for real-world PV power forecasting.

These findings support the need to align forecasting model architecture with expected seasonal variability and highlight that easy improvements, such as modified SES, can significantly improve performance without adding complexity, especially in embedded or SCADA-limited applications.

5.4. Comparative Evaluation of Forecasting Performance Across All Models

Additionally, for enhanced clarity, graphical comparisons of forecasting accuracy metrics are provided. Figure 10 presents a bar-line combination chart showing RMSE and MAE (in kW) alongside R² for each model. Figure 11 displays a grouped bar chart comparing nRMSE and nMAE (in %).

The classic SES method demonstrated the least accurate forecasting performance, evident from its highest RMSE (1413.58 kW) and lowest R² (0.41). Such results indicate that classic SES struggled significantly in capturing the dynamics of the PV power series, primarily because of its inherent assumption of a stationary time series without seasonal or trend variations. Given the clear seasonal and diurnal cycles observed in the PV data, it is unsurprising that this model produced substantial forecasting errors.

While SES remains popular for its simplicity and fast implementation, its practical limitations are increasingly recognised in the context of renewable energy forecasting. Several studies have attempted to evaluate its suitability for solar forecasting. For instance, ref. [43] applied the SES method for forecasting PV power from a 5 MW solar power plant in India. Their findings indicated that although SES was easy to implement, its performance degraded significantly in the presence of seasonal or trend-related patterns.

In contrast, Holt’s method offered a noticeable improvement in accuracy, with RMSE and MAE reduced to 1052.79 kW and 548.59 kW, respectively, and R² improving significantly to 0.94. This improvement can be attributed to Holt’s incorporation of a trend component, enabling the model to adapt better to gradual fluctuations over time. However, despite these improvements, Holt’s method did not account explicitly for seasonal patterns, limiting its accuracy, especially during periods of pronounced daily or seasonal variability.

Studies such as [44] demonstrated that Holt’s method can yield satisfactory results for short-term forecasting of PV power, particularly when the primary concern is modelling linear trends over brief intervals. Their implementation in a practical context of solar forecasting highlights the method’s ease of use and relatively stable performance under mild seasonal variation. Similarly, in [45] it was explored a broader class of exponential smoothing, including triple exponential smoothing, but acknowledged the relevance of simpler models like Holt’s under constrained computational setups or minimal seasonal dynamics. When comparing these insights with our results, it becomes evident that while Holt’s model performs adequately in trend-dominant scenarios, it is less suitable when pronounced daily and seasonal cycles dominate, as is the case with our PV dataset.

The Holt–Winters model, particularly chosen in its multiplicative form due to the clear multiplicative seasonality observed in initial data analysis, achieved notable forecasting accuracy. With an RMSE of 1031.00 kW, an MAE of 340.99 kW, and a robust R² value of 0.96, Holt–Winters excelled in capturing both trend and seasonal fluctuations inherent in PV power data. The multiplicative variant was specifically selected after preliminary model comparisons indicated that the PV power exhibited proportional seasonal variations that aligned closely with multiplicative model assumptions. The resulting model thus effectively mirrored the seasonal peaks and troughs, as demonstrated by its relatively low nMAE (3.7%) and nRMSE (11.2%).

While the multiplicative Holt–Winters model was selected in this study due to its ability to accurately capture the proportional seasonal variations inherent in PV power generation, alternative approaches in the literature have favoured the additive form. For instance, some studies have revisited the additive Holt–Winters method, proposing enhancements to its initial value calculations to improve forecasting accuracy in certain contexts [46]. These studies suggest that the additive model, which assumes constant seasonal fluctuations, can be effective in scenarios where the seasonal effect does not vary significantly with the level of the time series. However, in the context of PV power generation, where seasonal effects are multiplicative—i.e., the amplitude of seasonal variations increases with the level of solar irradiance—the multiplicative model provides a more realistic representation. This is particularly evident during summer months when higher irradiance levels lead to proportionally larger power outputs. Therefore, despite the existence of literature supporting the additive model in specific applications, the multiplicative Holt–Winters model is more appropriate for modelling PV power generation due to its capacity to accommodate the proportional nature of seasonal variations observed in the data. This is further supported in study [45], which demonstrated the capability of triple exponential smoothing to respond effectively to solar irradiance variability, underscoring the benefits of dynamic seasonal adjustment.

Most remarkably, however, the modified SES model, explicitly tailored in this study by incorporating previous-day, same-hour values into its forecasting structure, significantly outperformed all other models. With the lowest recorded RMSE (166.45 kW) and MAE (77.46 kW), along with exceptionally high R² (0.99), this modification demonstrated superior predictive accuracy and consistency. The substantial improvement in forecasting precision can be explained by its direct alignment with the repetitive daily cycle inherent in PV power. This modification not only leveraged recent historical data but also effectively integrated the critical diurnal pattern, dramatically enhancing forecast reliability.

The efficiency and interpretability of the modified SES model are also noteworthy. Compared to the more computationally intensive Holt–Winters model, the modified SES maintained a simple computational framework, crucial for real-time operational contexts such as grid management or local energy storage optimisation. This model offered the best trade-off between computational simplicity and forecasting precision, thus emerging as a highly practical solution for hourly PV power forecasting.

Figure 10 and Figure 11 support this evaluation. Figure 10 shows RMSE and MAE values for all models, overlaid with their respective R² scores. The modified SES model distinctly leads across all three dimensions. Figure 11 complements this by comparing normalised errors, clearly indicating the modified SES model’s dominant accuracy even after scale adjustment.

While SES remains popular for its simplicity and fast implementation, a review of the recent literature (2020–2024) revealed a lack of modern studies that focus on adapting or modifying SES specifically for PV power forecasting without hybridisation. Most works either apply SES in its classical form or integrate it into hybrid frameworks. To the best of our knowledge, there are no studies since 2020 that have proposed a comparable modification of SES as implemented in this research, namely, the integration of same-hour previous-day values into the smoothing process.

Summarising the comparative analysis, it is clear that simple stationary-assumption models (e.g., classic SES) are significantly limited when forecasting data exhibiting pronounced daily or seasonal variations. Methods incorporating trend (Holt’s) or seasonality (Holt–Winters) markedly improve predictive accuracy but at an increased computational cost. By contrast, the proposed modified SES is highly effective, offering very high one-step-ahead accuracy, minimal computational burden, and strong interpretability. This combination of strengths strongly supports its adoption for practical forecasting applications in solar energy management. Nevertheless, performance naturally varies with meteorological conditions, with larger errors during rapidly changing cloud cover and ramp events. Section 6 discusses portability and limitations across climates, system sizes, and volatility regimes.

In conclusion, while traditional exponential smoothing methods provide baseline forecasting capabilities, the integration of domain-specific modifications significantly enhances their predictive performance. The findings underscore the necessity of adapting classical forecasting approaches to specific operational characteristics—in this case, the daily and seasonal patterns inherent to PV power.

6. Generalisation Potential and Limitations

Although this study focuses on the forecasting of hourly PV power at a 9 MW ground-mounted PV plant in the Kyiv region of Ukraine, the evaluated exponential smoothing methods, including modified SES and Holt–Winters models, are designed with general applicability in mind. Their transparent, model-independent structure, low computational requirements, and reliance solely on historical time series data make them suitable for deployment across a wide range of PV installations.

In terms of installed capacity, these methods are fundamentally independent of the size of installation. They can be easily applied to both utility-scale systems and distributed small-scale PV arrays. Since the algorithms operate on the temporal dynamics of generation rather than on the physical configuration, their logic remains valid regardless of the capacity. However, in the case of small rooftop arrays, higher relative variability is often introduced by factors such as localised shading, inverter clipping, and orientation heterogeneity. These conditions may necessitate additional pre-processing (e.g., smoothing filters or outlier detection) to improve input data quality before forecasting.

Regarding geographical portability, in low-latitude regions with high sun elevation angles and more stable diurnal irradiance profiles, the modified SES model may perform even better, as it can exploit stronger and more predictable diurnal patterns. In contrast, regions with tropical or oceanic climates, where PV power is often disrupted by convective clouds, short-lived rains, or monsoon patterns, pose a greater challenge for purely statistical approaches. In such environments, typically involve nonlinear and abrupt changes in irradiance that exceed the models’ implicit memory, the performance of exponential smoothing methods may deterio-rate if they are not supplemented with auxiliary input data (e.g., numerical weather fore-cast, sky imagery, or satellite-derived irradiance data).

From a climate-adaptability standpoint, the Holt–Winters method effectively captures long-term seasonality and gradual variability, while the modified SES model is robust in transition months with semi-regular behaviour; however, both are limited in handling rapid, short-term fluctuations driven by fast-moving weather fronts, local cloud formations, or microclimatic effects, so in highly variable conditions—especially without external inputs, and hybrid models integrating physical or visual predictors are likely more appropriate.

Thus, while the proposed models demonstrate broad cross-context applicability, their stand-alone use may be best suited for regions with relatively predictable weather conditions or systems with low tolerance for computational costs. For environments with high short-term volatility or for advanced applications such as forecasting PV power constraints, grid support, or dynamic demand response, hybridising these exponential smoothing methods with real-time auxiliary inputs (e.g., irradiance sensors, sky cameras, or weather forecasts) can significantly improve forecast robustness.

In summary, the generalisation potential of the studied models is strong, especially in lightweight or embedded scenarios. Their ability to provide interpretable and timely forecasts with minimal computational overhead makes them attractive for a variety of PV deployments. Nevertheless, future research and implementation strategies should consider application-specific improvements, especially in terms of noise handling and adaptation to climate complexity.

7. Conclusions

This study conducted a comparative evaluation of four exponential smoothing methods—Classic Simple Exponential Smoothing, Modified Simple Exponential Smoothing, Holt’s linear, and the Holt–Winters seasonal—for short-term forecasting of PV power using real hourly operational data from a ground-mounted PV plant in Ukraine. Unlike many studies that focus on solar irradiance, this research directly targeted the active power, aligning the objective with the operational needs.

Classic Simple Exponential Smoothing, though simple and computationally efficient, provided inadequate for PV power forecasting due to the absence of a diurnal mechanism. Its inability to adapt to recurrent daily patterns led to the poorest performance across all metrics (R² = 0.41, high RMSE and nMAE values), underscoring its limitations for non-stationary, seasonally affected data. Adding a trend component, Holt’s linear method improved accuracy (R² = 0.94) but still struggled during periods of rapid generation changes typical in PV power. Its lack of a seasonal component made it prone to inaccuracies, particularly during the steep morning and evening transitions. Holt–Winters method, applied in its multiplicative form following analysis of the seasonal characteristics of the data, demonstrated strong forecasting accuracy (R² = 0.96, nMAE = 3.7%), suiting series with proportional seasonal fluctuations.

The most significant advancement was achieved with the Modified Simple Exponential Smoothing proposed in this study. By integrating previous-day, same-hour information into the SES framework, it captured diurnal regularities while maintaining a simple, transparent structure. It achieved the highest accuracy (R² = 0.99, lowest RMSE and nMAE = 0.84%), outperforming both classical and more complex smoothing approaches. At the same time, we emphasise that performance naturally varies with meteorological conditions: errors increase during rapidly changing cloud cover and ramp events, and models perform best when diurnal structure is strong.

Overall, the results underscore the importance of tailoring forecasting models to the specific temporal structure of PV power. Classic Simple Exponential Smoothing, benefits from adaptation to daily cycles of solar output. The Modified Simple Exponential Smoothing shows that even simple model, when aligned with domain-specific patterns, can offer an attractive accuracy–efficiency–interpretability balance for real-time use in grid-connected and decentralised settings.

The analysis covers a single plant and a seven-month window; thus, part of the seasonal cycle (autumn/winter) is not included. Therefore, it is interpreted reported metrics as period-specific rather than universal year-round values. To illuminate seasonal effects, it provides a seasonal breakdown (January–February, March–May, June–July) showing higher errors in winter/low-light and gains in summer. Modified Simple Exponential Smoothing is particularly strong in transitional months.

Future work could explore expanding the Modified Simple Exponential Smoothing approach to multi-step forecasting horizons, combining it with probabilistic modelling to address forecast uncertainty, or integrating exogenous weather information to enhance robustness under highly variable conditions.