A hierarchical clustering decomposition algorithm for optimizing renewable power systems with storage
Introduction
In 2019, renewable energies comprised 11% of the global primary energy consumption [1]. For renewable energies to increase their penetration into the power grid and avoid curtailment & over-generation issues, such as those currently observed in California [2], [3], greater investment in energy storage technologies is necessary. As a result, global energy storage capacity is expected to triple by 2030 [4]. However, supply and demand uncertainties pose significant investment risks [5] for renewable power systems with energy storage, which are large capital intensive projects. Likewise, without considering the future operations of a renewable power system simultaneously with its design during the investment planning phase, undersizing or oversizing power and storage capacities become costly miscalculations [6], [7]. Due to supply-demand discrepancies (Fig. 1) and oversizing, capacity factors of renewable power plants are much lower than their fossil fuel counterparts [8].
To address these issues, decision-making strategies that systematically model the design and operation of renewable power systems with energy storage, while minimizing capital and operational costs, are required [9]. Integrated design and scheduling models in the form of mixed-integer optimization problems [10], [11] are useful for this purpose. Such methods are also called capacity expansion [12] and unit commitment [13] models, where investments into new power and storage capacities (expansion) are concurrently decided with how the units are operated to meet the demand (commitment).
The results from these design and scheduling models give the decision-maker important information about the optimal sizing of solar photovoltaic (PV) modules, wind turbines, and energy storage options such as batteries. In addition, key operational choices like optimal battery charging and discharging are elucidated. The disadvantage is that the models grow very large from the hourly time discretization and long time horizons that are needed to describe weather, load, and price dynamics. This makes them very computationally demanding to solve. As the discretized granularity and time horizon increases, the models become even more complicated.
Time aggregation [14] or temporal clustering [15] have been applied to reduce the complexity of these models. The main concept is to decrease the number of unique hours that are modeled through finding patterns or clusters of similar weather, load, and price behavior in the time series data. As an example, Fig. 2 shows the similarity between solar and wind availabilities for two independent days in College Station, TX. Instead of optimizing over the entire time horizon, it is assumed that optimizing over an aggregated number of these clusters offers a good enough approximate solution to the problem [16]. Several authors have investigated clustering in mixed-integer optimization models for renewable energy systems.
Gabrielli et al. [17] used k-means clustering to optimize multi-energy systems with seasonal storage through developing designs off of clustered time periods and operating them for the full time horizon. Zhang et al. [18] do not specify the clustering method, but utilized representative seasons to optimize a production network of fuels and power from renewables. Lara et al. [19] utilized k-means clustering to group together generators in a multi-scale electric power infrastructure planning model. Heuberger et al. [20], [21] employed k-means clustering in optimizing capacity expansion and analyzing the system value of energy storage. Peng et al. [22] optimized a concentrated solar power (CSP) design across scenarios identified through k-means clustering. Wang et al. [23] utilized k-means clustering to reduce the number of scenarios used as input to a deep belief neural network for short-term wind power forecasting. Pineda and Morales [24] compared different representative hour, day, and week profiles from agglomerative hierarchical clustering (AHC) for a capacity expansion planning problem. Teichgraeber and Brandt [25] used k-shape clustering to optimally schedule a battery storage operation. Tejada-Arango et al. [26] compared different short- and long-term battery storage models through k-medoids clustering. Domínguez et al. [27] selected typical days using k-medoids clustering to optimize the configuration and operation of a residential heat & power system.
The above is not an exhaustive list, but a representative set of various studies where different clustering methods have been applied to reduce the optimization model complexity. In most studies, an underlying assumption is that when the clusters, which are calculated through minimizing the within cluster variance during data processing, are incorporated into an optimization problem, the resulting optimal objective value will also have the least error difference from the true optimal solved using the full time horizon. This is not true; the optimal time aggregation based on clustering error does not necessarily give the best approximation to the true optimal solution. Teichgraeber and Brandt [25] and Bahl et al. [28] have also noticed this error in optimal objective values between the aggregated and full time horizons.
Likewise, while k-means clustering and its variants are common techniques for time aggregation, they do not preserve the time chronology of the input data. Information from non-sequential time points are grouped together into the same cluster. Because time chronology is not kept, clever techniques are then applied to reconstruct the right timeline [16], [17], [29], increasing the model complexity. Instead, an AHC approach as utilized by Pineda and Morales [24] maintains time chronology in its cluster assignments through a connectivity matrix, enforcing the consecutive ordering of clusters. Having a correctly sequenced time aggregation is important for energy storage studies because inventory levels carry over between adjacent time periods.
This work attempts to address the aforementioned issues. A decomposition algorithm that selects the proper number of representative periods to aggregate a time horizon based on objective value error is developed from elements of previous works [17], [28]. The main aim of this algorithm is to quantify how adequate of an approximation is a specific aggregated time horizon, with respect to the objective value. The algorithmic concept is irrespective of clustering method, but AHC is chosen over k-means clustering because it is a more appropriate choice for energy storage [24]. The decomposition algorithm is then applied to a renewable power system with battery storage to demonstrate how sensitive the levelized cost of electricity (LCOE) is to the number of clusters in time aggregation.
The rest of the paper is organized as follows. In Section 2, an overview of AHC is provided, and the decomposition algorithm is described. In general, a similar approach is applicable to any optimization problem with time discretization. In Section 3, a mixed-integer linear programming (MILP) design and scheduling model for a renewable power system with battery storage in New York City (NYC) is discussed. Two case studies with different load demand profiles on this small example are presented in Section 4 to highlight the decomposition algorithm results. Finally, some concluding remarks are made and future directions are suggested.
Section snippets
Agglomerative Hierarchical Clustering (AHC)
Let X be a matrix of various time series data, where each column is a different temporal feature (i.e. solar irradiance, wind speed, load demand, electricity price) and each row represents a particular time period (i.e. hour, day, week). A data point, , is then all feature values that occur during the ith time period. The AHC method is a bottom-up approach, where each of the data points start off in their own cluster and are successively merged together to create new clusters as one
Problem definition
The problem definition for a renewable power system with battery storage in NYC is described below.
Given:
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time series data of local solar & wind availabilities, demand loads, and electricity prices
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land, capital, and operating costs for solar PV, wind turbine, battery, and DC-AC inverter
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process performance coefficients for the above units
Solve for:
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minimal levelized cost to meet electricity demand (LCOE)
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optimal capacity sizing for all units
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optimal operating schedule for all units
Process description & model
A resource-task
Case studies
While the 15 representative days selected in Section 3.3 minimized WCSS%, they might not necessarily comprise the best time aggregation to approximate the true optimal LCOE from an objective error perspective. Moreover, how sensitive the optimal solution is to the number of clusters is unknown, and what is the least number of representative days needed to adequately approximate the true optimal LCOE is unanswered.
To address these issues, the 15 representative days are used as a baseline to
Conclusion
Time aggregation through clustering is one method to solve capacity expansion and unit commitment optimization models for renewable power systems with storage. In this work, a decomposition algorithm through agglomerative hierarchical clustering (AHC) is developed and applied to investigating a solar PV and wind turbine system with battery storage in New York City (NYC). The renewable power system is described using a previously developed mixed-integer linear programming (MILP) design and
CRediT authorship contribution statement
William W. Tso: Conceptualization, Methodology, Software, Formal analysis, Data curation, Writing - original draft, Writing - review & editing, Visualization. C. Doga Demirhan: Conceptualization, Methodology, Software, Writing - review & editing, Visualization. Clara F. Heuberger: Conceptualization, Supervision. Joseph B. Powell: Project administration, Funding acquisition, Supervision. Efstratios N. Pistikopoulos: Project administration, Writing - review & editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors declare no competing interests. WWT and CDD gratefully acknowledge financial support from Royal Dutch Shell and the Texas A&M Energy Institute.
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