Gaussian mixture model (GMM) and Dirichlet process mixture model (DPMM) are the primary techniques used to characterize uncertainties in power systems, which are commonly solved by expectation-maximization (EM) algorithm. However, for the massive data of uncertain variables, the algorithm encounters challenges in accurately obtaining GMM and DPMM with a lower time consumption. To address this issue, we propose a method for GMM uncertainty modeling in power systems considering the mutual assistance of latent variables. Specifically, the GMM of uncertain variables is first constructed, and the conditional probability is employed to characterize the mutual assistance of latent variables. Then, an improved EM algorithm is developed to obtain the optimal GMM parameters, in which the expectation step (E-step) and maximization step (M-step) of the algorithm are revised using the conditional probability. Importantly, the closed-form solutions for GMM parameters are rederived in the revised E-step and M-step. Finally, the proposed uncertainty modeling method is compared with the traditional GMM and DPMM on actual wind power and load data from Australia. The proposed method performs the efficiency and accuracy in characterizing uncertainty.

Aiming at problems of power allocation and economic scheduling for independent multi-microgrid systems, a bi-level optimization method based on optimal power flow and consensus algorithm is proposed. The novelty of the method is that an independent multi-microgrid system is divided into two layers: in the upper layer, with the predicted output range of the microgrids as the input data, each microgrid is considered as a virtual power supply or virtual load, and taking the minimum network loss as the goal, the energy mutual aid and power allocation among the microgrids are transformed into solving the optimal power flow; in the lower layer, taking the upper layer power distribution scheme as the constraint condition, considering load fluctuation and wind/solar generation uncertainty, the optimal dispatch model of the controllable distributed generator is established based on the distributed theory and the consensus algorithm of equal cost increment, and the "plug and play" of the distributed generator is also realized. An islanded multi-microgrid cluster is taken as an example to verify the economy, security, and reliability of the proposed scheme. The advantages of the scheme have been shown by the simulation example. Simulation results show that the upper-layer method not only realizes the optimal power allocation of microgrids, but also reduces the power loss of the energy mutual aid among the microgrids; through the optimal scheduling of controllable power supply in the microgrid, the lower-level scheme not only improves the economic benefit of the microgrid, but also well suppresses the negative effects of the uncertainties, prediction errors and power fault removal on the multi-microgrid system, which improves the robustness of the system.

The accurate characterization of net load uncertainty can ensure the economy and stability of renewable power systems operation. In existing studies, Gaussian mixture model (GMM) and Dirichlet process mixture model (DPMM) are powerful tools for characterizing the uncertainty of monthly net load. However, the correlation among the Gaussian components and the common distribution characteristics among the monthly net load are not considered in these studies, which may lead to lower characterization accuracy. To solve these issues, we propose a hierarchical uncertainty characterization method for monthly net load considering Gaussian component reduction and correlation. Firstly, the method obtains the Gaussian mixture components and the label information of the annual net load by the Expectation-Maximum (EM) algorithm. Then, the Gaussian components of the monthly net load are corrected by the label information, which helps extract the common distribution characteristics among the monthly net load. On this basis, the corrected Gaussian components are considered as the base distribution. A component-preserving Expectation-Maximum (CPEM) algorithm is developed for component reduction. This realizes the uncertainty characterization of the monthly net load with high accuracy and lower time consumption. Importantly, the temporal correlation of net load is converted into the correlation among the Gaussian components, which are explicitly characterized by the Spearman coefficient. Finally, the superiority of the proposed method is verified with actual data collected in Australia. Note to Practitioners —In this article, we address the issue of monthly net load uncertainty characterization in renewable power systems. A hierarchical uncertainty characterization method is proposed to enable the distribution characteristic extraction of monthly net load. Most existing works on uncertainty characterization use GMM and DPMM, where the internal information in the net load data (e.g., distribution and component correlation information) is poorly utilized. This poses a considerable challenge for accurately analyzing the net load uncertainty. To this end, the article proposes a hierarchical uncertainty characterization method for monthly net load considering Gaussian component reduction and correlation. In this method, the common and typical distribution characteristics of monthly net load are considered by the CPEM algorithm. Moreover, the temporal information of the net load is transformed as the correlation among the Gaussian components, which are integrated into the E-step in the proposed CPEM algorithm. Since more internal information is utilized, the proposed method is helpful in characterizing the net load uncertainty by means of high accuracy and lower time consumption. The results of the uncertainty characterization can be readily implemented in the planning operation and real-time dispatch of renewable power systems.

The accurate quantification of risk caused by uncertainty forms a crucial foundation for formulating the generation maintenance scheduling (GMS) of power systems. However, the probability distribution functions (PDFs) of uncertain variables such as wind power and load are challenging to model accurately or are unknown, which makes it difficult to measure their economic risk and formulate appropriate GMS for power systems. To address this issue, we consider hybrid uncertainty from wind power and load, and propose a novel interval-probabilistic worst conditional value-at-risk (IP-WCVaR)-based generation maintenance scheduling method. Firstly, a novel IP-WCVaR method is proposed to measure the risk of the interval and probabilistic hybrid uncertainty, and the analytical mathematical model of the IP-WCVaR is derived through typical scenarios of probability correction. On this basis, the positive and negative spinning reserve models are established using the IP-WCVaR and then integrated into the GMS model, which enhances the resilience of the power system. Finally, the new risk-averse GMS model is formulated as the lower and upper boundary optimal models, which are transformed into tractable mixed integer linear programming problems based on the interval extreme value theory. The effectiveness and superiority of the proposed IP-WCVaR method are verified on the modified IEEE 24-bus and IEEE 118-bus power systems.

Interconnecting multiple combined heat and power (CHP) microgrids with the distribution network to form a CHP multi-microgrid system can promote the complementarity of various energy forms effectively and improve energy utilization efficiency. Therefore, a dual-layer stochastic optimal scheduling strategy of a multi-microgrid system interconnected with multiple CHP microgrids and multi-node distribution networks is proposed in this paper. In this strategy, the whole system is divided into two layers: (1) The lower layer adopts a stochastic simulation method based on versatile distribution to randomly generate a large number of scenarios and then uses a backward scenario reduction method to reduce the generated scenarios to a target number. Furthermore, with the goal of minimizing the sum of the weighted costs of the microgrid in each scenario, an economically optimal mixed integer programming model for a CHP microgrid with multiple energy storage devices is established. The output power of the electro-thermal units is optimized in each microgrid, and then the power to buy or sell power for each microgrid is determined. (2) The upper layer takes the minimum loss of the distribution system connected to multiple CHP microgrids as the goal and considers the power of each microgrid optimized by the lower layer as constraints. The optimal power flow model of the multi-node power distribution system is formulated based on the second-order cone relaxation technology, and the global optimal solution of the power node is obtained. Finally, the daily dispatch in which the IEEE-30 node power system connected to multiple CHP microgrids is used as an example for analysis. The results show that considering the uncertain factors of wind and solar output can effectively improve the reliability and robustness of the system when multiple CHP microgrids are connected to distribution network. Meanwhile, the coordinated use of multiple flexible coupling devices can reduce economic costs and promote energy consumption.