The long-term forecasting of interval-valued time series (ITS) is vital for applications such as financial risk warning and energy planning. Modeling ITS requires capturing not only long-term temporal dependencies but also the coevolution between interval bounds as well as between centers and radii, which together characterize the intrinsic structure of interval data. Recently, trend fuzzy information granulation methods have been introduced to ITS forecasting, enhancing the uncertainty modeling capabilities. However, these methods impose predefined assumptions on the trend-patterns described by Trend Fuzzy Information Granules (TFIGs), limiting their adaptability to depict nonlinear dynamic trends and interval structural interactions within an ITS. To address these issues, we propose a new kind of TFIGs, that is, the Joint Interval Trend-Pattern Unlimited Fuzzy Information Granules (JITPUFIGs), as well as their construction method, which leverages the AutoEncoder neural network based on Gated Recurrent Unit (GRU-AE) to jointly learn the coevolution patterns of interval centers and radii in a data-driven manner, avoiding predefined trend-pattern assumptions and incorporating an interval structure constraint to ensure logical consistency in JITPUFIGs. Building on this, the JITPUFIG-based Long Short-Term Memory (LSTM) neural networks are developed for long-term ITS forecasting. Experimental results on multiple benchmark ITS datasets show that our proposed model achieves superior forecasting accuracy while ensuring interval logical consistency—effectively preventing counterintuitive outputs such as predicted interval lower bounds exceeding upper bounds. This establishes a theoretically rigorous and effective paradigm for long-term ITS forecasting within the fuzzy system framework.

Impedance-based stability assessment (IBSA) is an effective method to study sub-synchronous control interaction (SSCI) with inverter-based resources (IBRs) in modern power systems. An IBR's required frequency-dependent impedance model can be obtained via the positive sequence scanning method. In this paper, single-sinusoidal signal and multi-sinusoidal signal-based methods for frequency scanning are investigated and evaluated. To benefit from the advantages of both schemes, hybrid scanning is proposed including perturbation signal, amplitude, and frequency segmentation settings. Additionally, a modified quadratic phase shift scheme is proposed for reducing the high magnitudes of multi-sinusoidal signals. All methods are validated and compared using the average value model of a doubly fed induction generator. The guidelines and recommendations are provided for the proper usage of the proposed hybrid scanning in the sub-synchronous frequency range.

Fuzzy cognitive maps (FCMs) are directed graphs with multiple nodes, making them well-suited for addressing multivariate time series (MTS) forecasting problems. When forecasting MTS, it is crucial to treat each vector of the MTS as a whole, considering both the causalities between different variables of the vector at a timepoint (spatial relationship) and the causalities between multiple historical vectors and future vector (temporal relationship). Existing FCM-based MTS forecasting models often fail to treat the vectors as a whole and do not distinctly reflect the temporal relationship and spatial relationship in MTS. To address these limitations, this paper introduces the concept of composite fuzzy cognitive maps (CFCMs). A CFCM comprises two layers of FCMs: the layer-1 FCM describes the temporal relationship in an MTS, while the layer-2 FCM describes the spatial relationship. By embedding the layer-2 FCMs into the nodes of the layer-1 FCM, the relationships within the MTS can be separately reflected while still treating each vector as a whole. In this structure, the nodes of the layer-1 FCM represent historical vectors used to forecast the future vector, and each node of the layer-1 FCM corresponds to a layer-2 FCM whose nodes represent the variables of the vector at a specific historical timepoint in the MTS. Based on the novel CFCM concept, this paper proposes a new MTS forecasting model that can distinctly reflect the temporal and spatial relationships in an MTS and utilize multiple historical vectors to forecast the future vector. Experimental results demonstrate the effectiveness of the proposed MTS forecasting model.

Along with the abundant appearance of the interval-valued time series (ITS), the study on ITS clustering, especially shape-based ITS clustering, is becoming increasingly important. As an effective approach to extracting trend information in time series, fuzzy trend granulation addresses the needs of shape-based ITS clustering. However, when extracting trend information in ITS, unequal-size granules are inevitably produced, which makes ITS clustering difficult and challenging. Facing this issue, this article aims to generalize the widely used fuzzy C-means (FCM) algorithm to a fuzzy trend-granulation-based FCM algorithm for ITS clustering. To this end, a suite of algorithms, including ITS segmenting, segment merging, and granule building algorithms, are first developed for fuzzy trend-granulation of ITS, with which the given ITS is transformed into granular ITS, which consists of double linear fuzzy information granules (DLFIGs) and may be of different lengths. With the defined distance between DLFIGs, the distance between granular ITS is further developed through the dynamic time warping (DTW) algorithm. In designing the fuzzy trend-granulation-based FCM algorithm, the key step is to design the method for updating cluster prototypes to cope with the unequal lengths of granular ITS. The weighted DTW barycenter averaging method is a previously adopted prototype updating approach with the drawback of hardly changing the lengths of prototypes, which often makes prototypes less representative. Thus, a granule splitting and merging algorithm is designed to resolve this issue. Additionally, a prototype initialization method is also proposed to improve the clustering performance. The proposed fuzzy trend-granulation-based FCM algorithm for clustering ITS, being a typical shape-based clustering algorithm, exhibits superior performance, which is validated by the ablation experiments as well as the comparative experiments.