The thesis explores the issue of fairness in the real-time (RT) control of battery energy storage systems (BESSs) hosted in active distribution networks (ADNs) in the presence of uncertainties by proposing and experimentally validating appropriate control frameworks for ADNs. In this respect, we identify two issues in the state-of-the-art of RT control of ADNs. The first one is the inadequacy of existing frameworks to handle fairness. We focus on the problem of controlling BESSs owned by different individuals to dispatch an ADN in RT. In this case, the distribution system operator (DSO) should, ideally, ensure that their states-of-charge (SoCs), relative to their energy capacities, are equal. In most relevant works in the literature, fairness can be controlled by tuning the controller's parameters, such as the weights of its objectives. Typically, the weights are manually tuned by evaluating the controller's performance a posteriori until a satisfactory result is obtained. These ``oracle-based'' approaches are impractical as they assume knowledge of the realization of the stochastic resources. To address the issues with these approaches, we propose a method to design cost functions for different RT objectives and compute their weights. The design is made such that dispatching is guaranteed with a given accuracy, and the control among BESSs is fair, i.e., their SoCs are as close as possible to each other, while a safe grid operation is guaranteed. The method translates the weights into parameters the user can intuitively choose without manual tuning. It is evaluated with simulations in various scenarios, showing its superiority over the oracle-based approaches in designing the controller's objective. The second issue not adequately addressed by the existing literature is quantifying the uncertainty of the stochastic energy resources, which is important to ensure the controller's reliable operation. Even though the problem of quantifying uncertainty has been extensively studied in the literature, the horizons targeted by most works are much longer than what is suitable for RT control. Instead, we target ultra-short-term horizons, i.e., up to a few seconds, and propose an algorithm to compute prediction intervals (PIs) for the prosumption of heterogeneous resources. The algorithm provides a model-free approach to estimating the statistical distribution of the uncertain power of prosumers without making any assumptions about the stochastic nature of the presumption. It has been evaluated on various prosumers with different power profiles and was shown to compute relatively narrow PIs with high confidence levels for time resolutions of up to 10 seconds. After the simulation validation, an extensive experimental campaign was conducted to evaluate the methods developed in the thesis in dispatching BESSs with a fair SoC regulation. We evaluated the performance of two control frameworks using the proposed methods for the PI computation and the design of the controller's objective. The experiments were carried out in a medium voltage utility-scale grid in Aigle, Switzerland, and the EPFL smart-grid platform and demonstrated how the methods developed in the thesis can successfully control BESSs fairly for dispatching while accounting for binding grid constraints.