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Demand response (DR) is a useful tool for end users, since it allows noticeable reductions in the electricity bill. However, some customers have stringent constraints in terms of hourly active power, which makes DR attractive only when performed with the contemporaneous use of battery energy storage systems (BESSs). When this option is used, it is desirable to optimally size the BESSs, since their high investment costs could make their use impracticable for DR purposes. The uncertainties related to many of the inputs required by the BESS-sizing procedure make it necessary to use a probabilistic framework for sizing. In this paper, a probabilistic approach is proposed for the optimal sizing of BESSs when time-of-use (ToU) pricing is applied. The sizing procedure takes into account the uncertainties that unavoidably affect the evaluation of the total cost incurred by the customer such as load demand, energy prices, and economic factors.

The optimal allocation of capacitors in unbalanced distribution systems can be formulated as a mixed integer, non-linear, constrained optimisation problem. Fuzzy-based approaches, simulated annealing, tabu search and genetic algorithms are some of the techniques used for solving the problem in deterministic scenarios. However, distribution systems are probabilistic in nature, leading to inaccurate deterministic solutions. As a result, a probabilistic optimization model is required to take into account the unavoidable uncertainties affecting the problem input data, primarily the load demands. Of the various techniques for the solution of the problem, one of the most frequently used is the genetic algorithm. However, the application of simple genetic algorithms to solve the probabilistic optimization model involves tremendous computational effort. To reduce the computational effort, this paper proposes a new single-objective probabilistic approach based on the use of a micro-genetic algorithm. Two different techniques, one based on the linearised form of the equality constraints of the probabilistic optimisation model and one based on the point estimate method, were tested and compared. The proposed approaches were tested on the IEEE 34-node unbalanced distribution system to demonstrate the effectiveness of the procedures in generating reduced computational efforts and increased accuracy of the results.