Optimal battery capacity in electrical load scheduling

Highlights

• Proposed a novel framework for electrical load forecasting that accounts for the use of battery system.

• Incorporated different costs for positive and negative prediction errors.

• Proposed a novel asymmetric support vector regression capturing the dynamic battery status.

• Derived optimal battery capacity by minimizing the average long term economic costs.

Abstract

With a rapid decline in cost of battery energy storage, a battery system plays an increasingly important role in managing imbalance between ordering and consumption in the electricity wholesale market. We develop an innovative electricity demand forecasting framework for calculating the optimal battery capacity that maximizes the profit of an electricity retailer. The framework allows different costs associated with over- and under-prediction errors, and two insensitive parameters to capture the battery residual capacity and remaining storage space. An application to Australia National Electricity Market in New South Wales shows that the use of a battery system with the optimal capacity can save up to AUD $65 millions annually under reasonable battery unit cost assumptions.

Introduction

Electrical load scheduling is essential, and accurate prediction of the electrical load can reduce economic costs from the imbalanced ordering [1]. While one should aim for prediction as accurate as possible, under- and over-prediction is the norm due to many unpredictable factors and perturbations in nature. In cases of under-ordering, the electricity retailers have to buy extra loads from Australia National Electricity Market (NEM) at spot price. However, if more loads are ordered than needed, retailers would waste the remaining loads if they were not stored. Electricity spot prices are extremely volatile and heavily skewed to the right [2]. Under-ordering can be especially costly to retailers as they might need to pay extremely high spot prices to meet the demand.

The loss from the imbalanced ordering can be buffered by greater battery storage that can offset errors due to under- and over-predictions and hence mitigate the loss. However, large capacity batteries are costly, and setting the optimal battery size requires accounting of other relevant cost factors. This cost management issue becomes even more challenging given that it is hard to predict electrical loads with high accuracy [3], [4], [5].

With the rapid decease of battery costs nowadays, battery storage systems have become popular for managing the imbalance and reducing the economic loss from prediction errors. The key issue for such a storage system to be effective lies in carefully balancing battery costs (hence battery capacity because the relationship between battery costs and capacity is monotonically increasing) and economic gains from reducing the economic loss from prediction errors.

Apparently, if the prediction is 100% accurate, there is no need to invest in battery storage. However, the optimal capacity generally depends on the goodness of prediction, i.e., the distribution of the errors associated with the predictions. We aim to offer a novel model framework to derive the optimal battery capacity in this aspect.

Under the assumption of symmetric loss from under- and over-prediction errors, two classes of machine learning approaches have been widely applied in electrical load scheduling, namely, artificial neural networks (ANN) and support vector regression (SVR) models [4]. Examples of ANN applications to load scheduling problem include wavelet neural networks [6], RBF neural networks [7], boosted neural networks [8], and extreme learning machines [9], and recurrent neural networks [10]. The SVR approach is more flexible for designing the objective of load scheduling with economic cost incorporated. Examples of SVR applications to load scheduling problem include the basic form of SVR [11], weighted SVR [12], on-line SVR [13], and vector field-based SVR [14].

The assumption of symmetric loss in the aforementioned forecasting models is inconsistent with the asymmetric losses in reality and hence is generally inappropriate for minimizing economic costs in load scheduling [15], [16]. Models with asymmetric loss have been proposed in the literature [16], [17], [18]. In these models, over-prediction and under-prediction are distinguished by different economic costs with under-prediction typically being associated with higher costs. In particular, [19] develop a cost-oriented asymmetrical SVR framework for load forecasting that incorporated an insensitive parameter in linear–linear cost (LLC) [20] which they show are much better for load scheduling. The insensitive parameter in their model framework is a pre-given small value to from an error tolerating band. For small training errors fall into this tolerating band will be ignored. Statistically, an insensitive parameter allows the $\epsilon$-SVR to tolerate small training errors in its model training process and hence reduces potential over-fitting. However, $\epsilon$ has to be given in machine learning algorithms.

We will establish a physical meaning for this $\epsilon$ in the context of electrical load scheduling with a battery storage system — this insensitive parameter has a neat physical meaning — the capacity of the battery system. For small prediction errors, as long as they are within the storage capacity of a battery system, they can be absorbed without any additional costs because the system can be used to store over-ordered load and supply under-ordered load. Hence a battery system effectively creates an error tolerance band for load prediction, which serves the same purpose as the insensitive parameter in an $\epsilon$-SVR. Using the battery capacity as the input of the insensitive parameter required in the $\epsilon$-SVR, we can then obtain the predictions from the resultant SVR. To be more specific, we propose a novel asymmetric SVR framework that uses two insensitive parameter as the loss function. This accommodates the need for an insensitivity interval which is asymmetric for negative and positive parts. The negative part represents the regular load, and the positive part is the available capacity. Note that the rental costs for the battery system are also incorporated in the objective function of our asymmetric SVR framework. The battery capacity, regular load, and model parameters are obtained by optimizing the overall objective function including the battery cost itself.

This contributions/highlights of the study are threefold. First, we develop a novel asymmetric cost function for a battery storage system where two insensitive parameters capture the capacities of the system for discharging and recharging. Second, we develop a new asymmetric SVR framework for load scheduling which yields the solution for the optimal capacity of the battery storage system. Third, we explore the cost-effectiveness of use a battery storage system with the optimal battery capacity by our model in load scheduling. An application to Australia National Electricity Market in New South Wales shows that the use of a battery system with the optimal capacity can save up to AUD $65 millions annually under reasonable battery unit cost assumptions.

The organization of this paper is as follows. Section 2 reviews the LLC literature and proposes the asymmetric insensitive LLC from the perspective of electrical load scheduling. The proposed asymmetric insensitive LLC is illustrated in the framework of the SVR, and the model procedure is also presented. Then, in Section 3, the performance of our proposed framework is evaluated in different scenarios with the state of New South Wales electrical load data. Finally, Section 4 concludes the paper.