A Novel Statistical Framework for Optimal Sizing of Grid-Connected Photovoltaic–Battery Systems for Peak Demand Reduction to Flatten Daily Load Profiles
Abstract
:1. Introduction
1.1. PV–Battery Optimal Sizing Approaches
1.1.1. Single-Objective Optimization
1.1.2. Multi-Objective Function Optimization
1.1.3. Stochastic Optimization
1.1.4. Robust Optimization
1.2. Contributions
- Development of a novel statistical model. We introduce a new statistical model specifically designed for optimizing PV–battery system sizes, with a primary focus on peak demand reduction. This model addresses a critical gap in the current literature by considering both energy consumption and peak demand costs, which are essential factors for utility companies.
- Incorporation of a modified Monte Carlo simulation. The study utilizes a modified Monte Carlo simulation approach to generate realistic and varied operational scenarios. This methodological innovation allows for a better understanding of PV–battery system performance under diverse conditions, enhancing the robustness of our optimization model.
- Operational and financial analysis for utilities. By providing a method to effectively flatten up to 95% of daily load demand profiles, the model offers a practical tool for utility companies. It enables them to make informed decisions regarding the optimal sizing of PV–battery systems, balancing technical feasibility with financial viability.
2. Materials and Methods
2.1. Data Collection and Assumptions
2.2. PV–Battery System Component Model
2.3. Required Daily Battery Size
2.4. Updated Peaks
2.5. Optimal PV–Battery Sizes
2.6. Statistical Analysis
- Selection of PV and battery sizes. We started by selecting a PV size of X1 and calculating the updated daily peak demand across various battery sizes ranging from Y1 to Ym.
- Histogram creation and PDF fitting. For each PV–battery size, we generated scaled histograms of daily peak demands following PV–battery installation. These histograms were then fitted with appropriate PDFs, specifically chosen for their relevance, characterized by PDF parameters.
- Determining the 0.95 threshold. For each PDF, we calculated the threshold value that corresponds to the 95th percentile. Mathematically, this is represented as:
- 4.
- Optimal sizing criteria. The objective was to find the PV–battery size combination that meets a predetermined threshold of T kW with a 95% probability. If the desired threshold, T, aligns with the thresholds found in Equation (9), the corresponding battery size is considered optimal for the PV size of X1. In cases where the desired threshold, T, did not align with the previously determined thresholds, we adjusted our approach and recalculated the parameters of a new PDF to match the desired threshold, T, with a 95% probability. This was achieved through the formula:
- 5.
- Optimal PV–battery system. By repeating all the aforementioned steps for a wide range of PV sizes, we eventually compiled an extensive set of optimal PV and battery combinations. Each of these combinations was capable of flattening 95% of the daily peaks up to a fixed threshold of T kW, which meets the technical requirement.
2.7. Economic Analysis
2.7.1. Initial Investment Cost
- PV installation [40]:
- Inverter cost [41]:
- Labor cost [42]:
- Equipment costs [43]:
- Overhead costs:
- Battery cost:
2.7.2. Operation, Maintenance, and Insurance Costs
2.7.3. Peak Demand and Energy Costs
2.7.4. Economic Benefit
2.8. Modified Monte Carlo Simulation
2.8.1. Time Series Clustering
- Data preprocessing. The first step involved comprehensive data preparation. This includes cleaning the data to remove any inconsistencies or errors, addressing outliers, and ensuring that all data are correctly formatted. Subsequently, we normalized the data values to fall between 0 and 1. This standardization is crucial for comparability analysis. Then, we grouped the data monthly, aggregating three years of data for further analysis.
- Similarity measures. The objective of time series clustering is to categorize time series datasets into clusters, where datasets within each cluster exhibit maximum similarity among themselves and minimal similarity with datasets in other clusters. A similarity measure is crucial in quantifying the degree of resemblance between two time series datasets. In this study, we employed dynamic time warping (DTW), a technique that has demonstrated significant efficacy in assessing similarity, particularly in the energy management sector [49]. DTW compares each point of one time series with multiple points of another, finding the best alignment by minimizing the cumulative distance between these matched points. By allowing such flexibility in the alignment, DTW effectively captures the inherent patterns and shapes within the time series data, even when these occur at different rates or phases.
- Clustering algorithms. The next step was to employ an appropriate time series clustering algorithm. Time series clustering is a complex process, and validation of the time series clustering results is challenging. For this purpose, we utilized two distinct clustering algorithms, including K-means and self-organizing maps (SOM), ensuring a robust and thorough examination of the time series data, enhancing the reliability of the results. The K-means clustering method is a partitioning clustering algorithm that has shown effective performance in various power system clustering applications [50]. It is adept at managing non-Euclidean similarity measures, demonstrates resilience against outliers, and has lower computational complexity relative to other partitioning clustering algorithms, making it a suitable choice for this study [51]. Despite its advantages in time series clustering, it cannot autonomously determine the optimal number of clusters. On the other hand, SOM is an unsupervised neural network that can inherently determine the optimal number of clusters as part of its training process. SOM visualizes high-dimensional data in a low-dimensional map and preserves the topological and temporal structure of the data. This capability of SOM facilitates the identification of patterns and trends within complex time series datasets [52]. However, SOM requires a careful selection of the appropriate map size and learning parameters. This combination of partitioning and neural network-based clustering methods helped us analyze load demand and solar irradiance clustering. K-means identifies distinct clusters based on similarity measures, while SOM captures complex patterns and relationships within the data through neural network layers. By leveraging the strengths of both approaches, we can gain a comprehensive understanding of the underlying structures in the dataset.
- Initialization—The process began by randomly selecting k data points as the initial centroids of the clusters.
- Assignment step—In this phase, each data point in the dataset was assigned to the nearest centroid. The closeness was determined based on the DTW distance.
- Update step—The centroids of the clusters were then recalculated as the mean of all points assigned to each cluster.
- Convergence—These steps were repeated until the positions of the centroids stabilized, indicating that the clusters had converged and were no longer significantly changing.
- Optimal number of clusters—Determining the optimal number of clusters is a critical aspect of the K-means algorithm. We employed the Elbow method to identify this number. To apply the Elbow method, we first executed the K-means algorithm over a range of K values, from 1 to a predefined maximum, then computed the Within-Cluster Sum of Squares (WCSS) for each K, and finally plotted these WCSS values against their cluster number. By observing the WCSS curve, we looked for a point where the rate of decrease in WCSS significantly slowed down, creating an elbow in the plot. The K value at this elbow point is considered the optimal number of clusters, as it indicates a trade-off between maximizing the number of clusters and minimizing WCSS [53].
- Initialization—We started by initializing the SOM neural network with weight vectors, through random selection.
- Competitive learning—For each data point in our dataset, SOM identified the Best Matching Unit (BMU) by finding the neuron with the closest weight vector to the data point.
- Weight adjustment—The weights of the BMU and its neighbors within the network were adjusted to become more similar to the input data point, with the adjustment magnitude decreasing over time and distance from the BMU.
- Iterative process—This cycle of competitive learning and weight adjustment was repeated across numerous iterations, allowing the SOM to evolve and form a map that reflects the intrinsic structure of the data.
- Cluster visualization—The final output was a map where similar data points were clustered together.
2.8.2. Modified Monte Carlo Simulation
- Assign probabilities to solar irradiance clusters. For each solar irradiance cluster (i = 1 to m), we calculated its probability as:
- Establish probability intervals. This was carried out by sequentially adding the probabilities of the clusters. For the first cluster , its probability interval, is:
- 3.
- Random cluster selection for solar irradiance. A random number R within the range [0, 1] was selected uniformly, selecting the solar irradiance cluster for which the random number R falls within its probability interval.
- 4.
- We determined the specific days that are included in the selected solar irradiance cluster.
- 5.
- Match the days with load clusters. For each identified day in the solar irradiance cluster , we found the corresponding days within the load demand clusters from to .
- 6.
- Calculate the conditional probability for load clusters. After selecting the solar irradiance cluster , the probability of each load demand cluster conditioned on the selection of was calculated. The conditional probability was calculated as:
- 7.
- We established probability intervals for each conditional probability, as in step 2.
- 8.
- We randomly selected a load cluster based on the conditional probability intervals.
- 9.
- Final scenario selection. From the selected solar irradiance cluster and the randomly chosen load cluster , a specific pair of solar irradiance and load demand profile was identified. If multiple profile pairs were available within the selected clusters, one pair was randomly selected. This random selection can be performed using a uniform distribution, ensuring each pair has an equal chance of being chosen.
3. Results
3.1. Data Analysis
3.2. Battery Operation—Required Daily Battery Sizes
3.3. Optimal PV–Battery System
3.4. Financial Analysis
3.5. Modified Monte Carlo Simulation
4. Conclusions
- Integrating electric vehicles into the grid—enhancing grid adaptability to manage the stochastic load and energy contributions from electric vehicles. This initiative aims to optimize financial benefits and energy efficiency through the development of dynamic charging strategies and vehicle-to-grid technologies.
- Examining the influence of various grid topology and related constraints. A deeper exploration of how grid configurations and limitations affect the deployment and performance of PV–battery systems will refine the accuracy of the proposed methodology. It will enable the model to account for physical and regulatory constraints, thereby improving the feasibility and reliability of the system.
- Extending the methodology to include other renewable energy sources. By incorporating technologies, such as wind turbines, into a hybrid system, the framework can provide a more comprehensive analysis of renewable energy potentials. This holistic approach will facilitate the development of optimized, multi-faceted energy solutions that better meet the needs of utilities and consumers, while also promoting a more sustainable energy mix.
- Refining the methodology to determine the desired demand threshold. Tailoring the model to align with specific utility company requirements and operational capacities will enhance its practical relevance and effectiveness. Customizing the methodology in this way ensures that the proposed solutions are not only theoretically sound but also practically implementable, leading to more efficient energy management strategies.
- Investigating the integration of emerging photovoltaic technologies. Incorporating advanced solar technologies, such as dye-sensitized and perovskite solar cells, could pave the way for leveraging cutting-edge innovations in solar energy, potentially transforming the economic landscape of solar power by reducing costs and increasing efficiency.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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PV Module (USD/W) 0.35 | Inverter (USD/W) 0.04 | Equipment (USD/W) 0.18 |
---|---|---|
Overhead (USD/W) | O&M (USD/kW) | Transformer (USD) |
0.1 | 15 | 150,000 |
Energy cost (USD/kWh) | Power cost (USD/kW) | Tax credit (%) |
0.025 | 22 | 30 |
Initial battery (USD/kWh) | Replacement battery (USD/kWh) | Project lifetime |
150 | 100 | 20 years |
Labor (USD/W) | Discount rate | Battery roundtrip efficiency |
0.1 | 0.08 | 0.9025 |
Inverter coefficient | Battery efficiency | Battery utilization |
1.2 | 0.95 | 0.7 |
Log-Normal | Gamma | Beta | ||||
---|---|---|---|---|---|---|
Battery | KS_Statistic | p-Value | KS_Statistic | p-Value | KS_Statistic | p-Value |
2000 | 0.041089 | 0.0480 | 0.031794 | 0.65124 | 0.08 | 0.005355 |
2500 | 0.03537 | 0.12579 | 0.029497 | 0.55618 | 0.11 | 0.027578 |
3000 | 0.040364 | 0.15470 | 0.026193 | 0.56416 | 0.09 | 0.005044 |
3500 | 0.026193 | 0.43233 | 0.053331 | 0.42345 | 0.12 | 0.04289 |
4000 | 0.062428 | 0.037 | 0.027589 | 0.65164 | 0.16 | 0.01455 |
4500 | 0.039343 | 0.06544 | 0.03461 | 0.32136 | 0.13 | 0.004353 |
5000 | 0.046593 | 0.01660 | 0.030778 | 0.32103 | 0.14 | 0.000539 |
5500 | 0.051666 | 0.00554 | 0.062428 | 0.27564 | 0.15 | 0.000127 |
6000 | 0.054413 | 0.00292 | 0.034076 | 0.24565 | 0.25 | 5.58 × 10−5 |
PV (kW) | 500 | 1000 | 1200 | 1500 | 2000 | 2500 | 3000 | 3500 | 4000 |
Battery (kWh) | NAN | NAN | 9200 | 4400 | 4000 | 3800 | 3600 | 3400 | 3300 |
Before PV–Battery | PV Only | After PV–Battery | |
---|---|---|---|
Equipment cost (USD) | 0 | 1,638,688 | 2,015,246 |
Energy cost (USD) | 3,788,907 | 3,036,927 | 3,023,569 |
Peak demand charge (USD) | 6,913,926 | 6,472,805 | 4,901,679 |
Benefit (USD) | 0 | −445,587 | 812,648 |
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Nematirad, R.; Pahwa, A.; Natarajan, B. A Novel Statistical Framework for Optimal Sizing of Grid-Connected Photovoltaic–Battery Systems for Peak Demand Reduction to Flatten Daily Load Profiles. Solar 2024, 4, 179-208. https://doi.org/10.3390/solar4010008
Nematirad R, Pahwa A, Natarajan B. A Novel Statistical Framework for Optimal Sizing of Grid-Connected Photovoltaic–Battery Systems for Peak Demand Reduction to Flatten Daily Load Profiles. Solar. 2024; 4(1):179-208. https://doi.org/10.3390/solar4010008
Chicago/Turabian StyleNematirad, Reza, Anil Pahwa, and Balasubramaniam Natarajan. 2024. "A Novel Statistical Framework for Optimal Sizing of Grid-Connected Photovoltaic–Battery Systems for Peak Demand Reduction to Flatten Daily Load Profiles" Solar 4, no. 1: 179-208. https://doi.org/10.3390/solar4010008
APA StyleNematirad, R., Pahwa, A., & Natarajan, B. (2024). A Novel Statistical Framework for Optimal Sizing of Grid-Connected Photovoltaic–Battery Systems for Peak Demand Reduction to Flatten Daily Load Profiles. Solar, 4(1), 179-208. https://doi.org/10.3390/solar4010008