Calculation of Probability of Storage Battery Energy Obtaining Based on the Use of Martingale Theory

In the presented article, a novel technique for calculation of probability of storage battery energy obtaining based on the use of martingale theory to ensure its effective operation is proposed. A simplified block diagram of a distributed generation system with a storage battery is given. The process of the energy level fluctuating at the output of a renewable source relative to its average value and, accordingly, the sequence of the deviation of the amount of energy falling on the charge or discharge of the storage battery from its average value is proposed to consider as a martingale. Formulas for calculating the probability of the storage battery excessive discharge and recharge for symmetric (when probabilities of energy charge and discharge are equal) and asymmetrical (when probabilities of energy charge and discharge are not equal) cases are given. The graphs of the dependence of the probability of an excessive discharge of a storage battery on the amount of an insufficient level of discharge energy are plotted, as well as the probability of a battery recharge from the excess charge energy level for both cases. It is shown that with the growth of both the magnitude of the excessive level of charge energy and the size of the insufficient level of discharge energy, the probability of both recharge and excessive discharge of the storage battery are increased. In the asymmetric case the function characterizing the storage battery energy can not be considered as a martingale. Therefore, it is necessary to use a different function for calculations that takes into account the unequal values of the probabilities of charge and discharge, is a martingale, and to which a theorem of a stop can be applied. It is noted that with an increase in the ratio between the probability of storage battery charge and battery level, the likelihood of excessive discharge and recharge of the battery are increased. At the same time, the tendency to increase the likelihood of excessive discharge and recharge of the battery from the values of insufficient level of discharge energy and excessive level of charge energy while increasing its level of storage remains. Thus, the application of the theory of martingales allows us to estimate, first, the amount of energy that may be excessive in charge (which may lead to overcharge); and secondly, the amount of energy that may be insufficient in discharging (which can lead to excessive discharge) and determine the probabilities of its overcharge and excessive discharge, provided the maximum energy selection from renewable energy source. Ref. 4, fig. 2.